Optimal Designs - Solid Mechanics

Optimal Designs
- Structures and Materials - Problems and Tools -
Pauli Pedersen
Department of Mechanical Engineering, Solid Mechanics
Technical University of Denmark
Nils Koppels Allè, Building 404, DK-2800 Kgs.Lyngby, Denmark
email: pauli@mek.dtu.dk
PRE-PRINT
May 28, 2003

2
Optimal Designs
- Structures and Materials - Problems and Tools -
Copyright c○2003 by Pauli Pedersen,
but feel free to copy for personal use.
Colour-pages available from homepage
ISBN 87-90416-06-6

Contents
Contents 3
1 Preface and Introduction 11
1.0.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1 Words of the title . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.1.1 Specific models . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.1.2 Specific problems . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 The chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Names and concepts 17
2.1 Major classification of names and concepts . . . . . . . . . . . . . . . . 17
2.2 Design variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.1 Alternative classifications . . . . . . . . . . . . . . . . . . . . . 18
2.3 Design objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Design constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Design space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6 Optimization formulations . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.7 Optimization methods and procedures . . . . . . . . . . . . . . . . . . 25
2.7.1 The sub-problems of optimal redesign . . . . . . . . . . . . . . 25
2.7.2 The overall strategy . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Design of trusses 31
3.1 Theoretical and numerical background . . . . . . . . . . . . . . . . . . 32
3.1.1 Common parameters for the examples . . . . . . . . . . . . . . 32
3.1.2 Truss members with circular cross-section . . . . . . . . . . . . 33
3.2 Size and topology for optimal plane trusses . . . . . . . . . . . . . . . 33
3.2.1 A cantilever truss with 9 joints . . . . . . . . . . . . . . . . . . 33
3.2.2 A cantilever truss with 17 joints . . . . . . . . . . . . . . . . . 38
3

4 CONTENTS
3.2.3 A bridge truss with 16 joints . . . . . . . . . . . . . . . . . . . 39
3.2.4 A beam/bridge truss with 24 joints . . . . . . . . . . . . . . . . 41
3.3 Size, supports and topology for
optimal single load case space trusses . . . . . . . . . . . . . . . . . . . 44
3.3.1 A tower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4 Size, shape and topology for optimal 2D-trusses . . . . . . . . . . . . . 44
3.4.1 The cantilever truss with 7 to 12 joints . . . . . . . . . . . . . . 46
3.4.2 The bridge truss with 16 joints . . . . . . . . . . . . . . . . . . 46
3.5 Size and shape for optimal multi-purpose plane trusses . . . . . . . . . 46
3.6 Size and shape for optimal multi-purpose space trusses . . . . . . . . . 50
3.7 A large space truss with constraints on
eigenfrequencies, displacements, stresses and buckling . . . . . . . . . 54
3.8 Approach to topology optimization for
multi-purpose space trusses . . . . . . . . . . . . . . . . . . . . . . . . 54
4 Continua of
uniform energy density 55
4.1 General comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1.1 Design variables and design constraints . . . . . . . . . . . . . 55
4.1.2 Presentation of results . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 Bars, beams and beam-bars in 2D-formulation . . . . . . . . . . . . . 57
4.3 Bars: long, medium and short . . . . . . . . . . . . . . . . . . . . . . . 59
4.4 Beam-bars: long, medium and short . . . . . . . . . . . . . . . . . . . 62
4.5 Beams: long, medium and short . . . . . . . . . . . . . . . . . . . . . . 65
4.6 ”Bridge”s with different load distributions and supports . . . . . . . . 65
4.7 A ”knee” domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.8 Biaxially loaded hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.9 A foundation problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.10 Uniformly loaded, non-rectangular cantilever . . . . . . . . . . . . . . 75
4.11 The numerical procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.12 Summing up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.12.1 Influence from the maximum thickness constraint . . . . . . . . 78
4.12.2 Relation to free material design . . . . . . . . . . . . . . . . . . 78
4.12.3 Relation to non-linear and non-isotropic material . . . . . . . . 79
4.12.4 The specific examples . . . . . . . . . . . . . . . . . . . . . . . 79
5 Design of beams and frames 81
5.1 Explicit analytical optimal designs . . . . . . . . . . . . . . . . . . . . 82
5.1.1 Statically determined elementary cases . . . . . . . . . . . . . . 83
5.2 Implicit ”analytical” optimal designs . . . . . . . . . . . . . . . . . . . 85

CONTENTS 5
5.2.1 Side constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2.2 Statically indetermined cases . . . . . . . . . . . . . . . . . . . 85
5.3 Beam design with eigenfrequency constraints . . . . . . . . . . . . . . 85
5.3.1 Sensitivity analysis for eigenfrequencies . . . . . . . . . . . . . 85
5.3.2 Optimal design with one eigenfrequency constraint . . . . . . . 87
5.3.3 Optimal design with several eigenfrequency constraints . . . . . 87
5.3.4 Optimality criterion with only a single constraint . . . . . . . . 87
5.3.5 Sensitivity analysis for Timoshenko beam theory . . . . . . . . 89
5.3.6 Optimization with different gradients . . . . . . . . . . . . . . . 90
5.4 2D-Frame design with many load cases . . . . . . . . . . . . . . . . . . 91
5.4.1 Design problem, analysis and sensitivity analysis . . . . . . . . 92
5.4.2 Only stress constraints and influence from stiffness of columns 93
5.4.3 Including displacement constraints . . . . . . . . . . . . . . . . 98
5.5 3D-Frame design with optimal joint positions . . . . . . . . . . . . . . 98
5.5.1 Design problem, analysis and sensitivity analysis . . . . . . . . 99
5.5.2 Optimal design of a dome frame . . . . . . . . . . . . . . . . . 100
5.5.3 Optimal design of a mobile crane frame . . . . . . . . . . . . . 103
6 Design of laminates and plates 107
6.1 Laminate analysis and rotational transformations . . . . . . . . . . . . 108
6.1.1 Rotational transformations of stress and strain vectors . . . . . 108
6.1.2 Rotational transformations of constitutive matrices . . . . . . . 109
6.1.3 Alternative description by practical parameters . . . . . . . . . 110
6.1.4 Laminate stiffnesses and lamination parameters . . . . . . . . . 113
6.2 Elastic energy sensitivity analysis . . . . . . . . . . . . . . . . . . . . . 115
6.3 Optimal orientation of material in
simply supported plates in bending . . . . . . . . . . . . . . . . . . . . 116
6.4 Laminate point-wise design . . . . . . . . . . . . . . . . . . . . . . . . 120
6.4.1 Thickness distribution only . . . . . . . . . . . . . . . . . . . . 122
6.4.2 Orientations also . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.5 Strength optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.5.1 Global design parameters . . . . . . . . . . . . . . . . . . . . . 125
7 Shapes of minimum stress concentration 129
7.1 Statement of the actual problems . . . . . . . . . . . . . . . . . . . . . 130
7.2 The 2D-fillet in tension . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.2.1 Circular connection fillet . . . . . . . . . . . . . . . . . . . . . . 132
7.2.2 Super-circular connection fillet . . . . . . . . . . . . . . . . . . 135
7.2.3 Extended length of the fillet . . . . . . . . . . . . . . . . . . . . 135
7.2.4 Multi-parameter optimal shape design . . . . . . . . . . . . . . 135

6 CONTENTS
7.3 The 3D-fillet in tension, bending and torsion . . . . . . . . . . . . . . 138
7.4 The 2D-hole subjected to biaxial stress . . . . . . . . . . . . . . . . . . 144
7.4.1 The classic solution for stress concentration around small elliptical
holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.4.2 Numerical solutions for finite domains . . . . . . . . . . . . . . 146
7.5 The 2D-hole in an orthotropic material . . . . . . . . . . . . . . . . . . 148
7.6 The 2D-hole in a non-linear elastic material . . . . . . . . . . . . . . . 148
7.7 The 3D-cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.8 Shape design in materials for maximum bulk modulus . . . . . . . . . 151
7.9 Stress release at a crack tip . . . . . . . . . . . . . . . . . . . . . . . . 151
7.9.1 Influence from the external load and size of the hole . . . . . . 154
7.9.2 Influence from material non-isotropy . . . . . . . . . . . . . . . 155
7.9.3 Influence from material power law non-linearity . . . . . . . . . 155
7.9.4 Influence from the allowable domain of the hole . . . . . . . . . 157
8 Material design 159
8.1 Free material design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
8.2 Design of density distribution . . . . . . . . . . . . . . . . . . . . . . . 161
8.2.1 ”Black and white” designs . . . . . . . . . . . . . . . . . . . . . 162
8.2.2 An illustrative example . . . . . . . . . . . . . . . . . . . . . . 163
8.3 Design of a two parameter cell model . . . . . . . . . . . . . . . . . . . 170
8.4 Design of material with a single hole . . . . . . . . . . . . . . . . . . . 172
8.4.1 Influence from the volume constraint . . . . . . . . . . . . . . . 173
8.4.2 Influence from boundary conditions . . . . . . . . . . . . . . . 174
8.4.3 Isotropic cases with influence from Poisson’s ratio . . . . . . . 178
8.4.4 Influence from non-linear elasticity . . . . . . . . . . . . . . . . 180
8.5 Design with multi-physics constraints . . . . . . . . . . . . . . . . . . . 184
9 Bone mechanics and damage evolution 185
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
9.1.1 Points of view . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
9.1.2 Contents of the chapter . . . . . . . . . . . . . . . . . . . . . . 187
9.2 Bone layouts from an optimization process . . . . . . . . . . . . . . . . 187
9.3 Finite element analysis and load modelling . . . . . . . . . . . . . . . . 188
9.4 Homogenization and inverse homogenization . . . . . . . . . . . . . . . 190
9.5 Bone layout based on
one parameter optimization . . . . . . . . . . . . . . . . . . . . . . . . 191
9.6 Multi-parameter evolution . . . . . . . . . . . . . . . . . . . . . . . . . 192
9.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

CONTENTS 7
10 Identification (estimation) and inverse problems 197
10.1 Estimation of orthotropic material parameters . . . . . . . . . . . . . 197
10.1.1 Identification formulation . . . . . . . . . . . . . . . . . . . . . 199
10.1.2 The experimental setup . . . . . . . . . . . . . . . . . . . . . . 200
10.1.3 Plate eigenfrequency analysis and sensitivity analysis . . . . . . 202
10.1.4 Example results . . . . . . . . . . . . . . . . . . . . . . . . . . 203
10.1.5 Uncertainties and optimal experiments . . . . . . . . . . . . . . 204
10.1.6 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
10.2 Laminate inverse problems . . . . . . . . . . . . . . . . . . . . . . . . . 207
10.2.1 Laminate with given bending stiffness . . . . . . . . . . . . . . 207
10.2.2 A redesign procedure . . . . . . . . . . . . . . . . . . . . . . . . 208
10.3 Material identification and inverse homogenization . . . . . . . . . . . 209
10.3.1 Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . 209
10.3.2 Inverse homogenization . . . . . . . . . . . . . . . . . . . . . . 209
11 Alternative Descriptions of Constitutive Parameter 211
11.1 Stress/strain relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
11.2 Two-dimensional formulations . . . . . . . . . . . . . . . . . . . . . . . 211
11.3 Three-dimensional formulations . . . . . . . . . . . . . . . . . . . . . . 214
11.3.1 Spectral decomposition . . . . . . . . . . . . . . . . . . . . . . 214
11.3.2 Constitutive parameters by energy densities . . . . . . . . . . . 214
11.4 Bulk modulus and alternative isotropic parameters . . . . . . . . . . . 216
11.4.1 Three-dimensional case . . . . . . . . . . . . . . . . . . . . . . 216
11.4.2 Two-dimensional case . . . . . . . . . . . . . . . . . . . . . . . 216
11.4.3 Relations between other alternative moduli . . . . . . . . . . . 217
11.5 Summing up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
12 Effective stress/strain and energy densities 219
12.1 Analysis by secant formulation . . . . . . . . . . . . . . . . . . . . . . 219
12.1.1 Non-dimensional compliance matrix . . . . . . . . . . . . . . . 220
12.1.2 The von Mises effective stress . . . . . . . . . . . . . . . . . . . 221
12.1.3 The Hill strength measure . . . . . . . . . . . . . . . . . . . . . 222
12.2 Strain energy density and
stress energy density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
12.3 Summing up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
13 Elastic potentials, relations and sensitivities 225
13.1 Elastic potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
13.2 Derivatives of elastic potentials . . . . . . . . . . . . . . . . . . . . . . 226
13.2.1 Constant constitutive matrix, a specific case . . . . . . . . . . . 227

8 CONTENTS
13.2.2 Constant volume, a specific case . . . . . . . . . . . . . . . . . 227
13.3 Summing up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
14 Some necessary conditions for optimality 229
14.1 Non-constrained problems . . . . . . . . . . . . . . . . . . . . . . . . . 229
14.2 Problems with a single constraint . . . . . . . . . . . . . . . . . . . . . 230
14.3 Size optimization for stiffness and strength . . . . . . . . . . . . . . . . 231
14.3.1 Size design with optimal stiffness . . . . . . . . . . . . . . . . . 231
14.3.2 Size design with optimal strength . . . . . . . . . . . . . . . . . 232
14.4 Shape optimization for stiffness and strength . . . . . . . . . . . . . . 232
14.4.1 Shape design with optimal stiffness . . . . . . . . . . . . . . . . 232
14.4.2 Shape design with optimal strength . . . . . . . . . . . . . . . 234
14.5 Conditions with a
simple shape parametrization . . . . . . . . . . . . . . . . . . . . . . . 235
14.5.1 Possible iterative procedure . . . . . . . . . . . . . . . . . . . . 239
14.5.2 Extended design space . . . . . . . . . . . . . . . . . . . . . . . 239
14.6 Summing up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
15 The ultimate optimal material 241
15.1 The individual constitutive parameters . . . . . . . . . . . . . . . . . . 241
15.2 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
15.3 Final optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
15.4 Numerical aspects and comparison with isotropic material . . . . . . . 243
15.5 Summing up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
16 Conditions for statically determinate trusses 245
16.1 Theorem for the single load case . . . . . . . . . . . . . . . . . . . . . 246
16.2 Proof with supports as further unknowns . . . . . . . . . . . . . . . . 246
16.2.1 Force equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 246
16.2.2 The total cost with all forces . . . . . . . . . . . . . . . . . . . 247
16.2.3 The change in total cost . . . . . . . . . . . . . . . . . . . . . . 248
16.2.4 Monotonous behaviour and new basis solution . . . . . . . . . 248
16.3 Optimality condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
16.4 Alternative proofs and solution procedures . . . . . . . . . . . . . . . . 249
16.5 A truss member model . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
16.5.1 Tensile bars, i.e. P>0 . . . . . . . . . . . . . . . . . . . . . . 250
16.5.2 Slender columns, i.e. PL

CONTENTS 9
16.5.6 Cost functions for the supports . . . . . . . . . . . . . . . . . . 252
16.6 Summing up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
17 Orientation of orthotropic material 253
17.1 Inherent practical problems . . . . . . . . . . . . . . . . . . . . . . . . 253
17.2 From global to local optimality criterion . . . . . . . . . . . . . . . . . 254
17.3 Multiplicity of extremum . . . . . . . . . . . . . . . . . . . . . . . . . 255
17.4 The match problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
17.5 Numerical iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
17.6 Summing up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
18 Sensitivity analysis for dynamic problems 261
18.1 Including non-conservative problems . . . . . . . . . . . . . . . . . . . 261
18.2 Classification of dynamic systems and their behaviour . . . . . . . . . 262
18.2.1 System classification . . . . . . . . . . . . . . . . . . . . . . . . 262
18.2.2 Classification of the behaviour . . . . . . . . . . . . . . . . . . 263
18.3 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
18.3.1 General variational analysis . . . . . . . . . . . . . . . . . . . . 264
18.3.2 Results with finite element formulation . . . . . . . . . . . . . 266
18.3.3 Results with Galerkin modelling (global expansion) . . . . . . . 267
18.3.4 Example of a cantilever beam . . . . . . . . . . . . . . . . . . . 268
18.3.5 Multiple eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . 270
18.3.6 Double eigenvalue with only a single eigenvector . . . . . . . . 271
18.4 Summing up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
19 Simplex and a modified simplex procedure 273
19.1 Standard linear programming form . . . . . . . . . . . . . . . . . . . . 274
19.2 Effect of changing the basis set . . . . . . . . . . . . . . . . . . . . . . 274
19.3 Non-linear objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
19.4 Positive and negative variables . . . . . . . . . . . . . . . . . . . . . . 276
19.5 Summing up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
20 Linear programming reformulations with SLP 279
20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
20.2 Actual optimization problem after sign reformulations . . . . . . . . . 280
20.3 From non-linear to sequential linear problem . . . . . . . . . . . . . . 280
20.4 From inequalities to equalities . . . . . . . . . . . . . . . . . . . . . . . 281
20.5 Obtaining non-negative variables . . . . . . . . . . . . . . . . . . . . . 281
20.6 Move-limits and non-negative variables . . . . . . . . . . . . . . . . . . 282
20.7 Problem constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
20.8 Total LP coefficient matrix for the constraints . . . . . . . . . . . . . . 284

10 CONTENTS
20.9 The vector of cost coefficients . . . . . . . . . . . . . . . . . . . . . . . 284
20.10 Sequential linear programming . . . . . . . . . . . . . . . . . . . . . . 285
20.11 Summing up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
Bibliography 287
Index 297

Chapter 1
Preface and Introduction
1.0.1 Preface
This book grew from a Ph.D.course given at Politecnico di Milano in July 1999. For
more than 30 years I have been involved with optimal designs and identifications, but
this was the first chance to put together such a course. From July 1999 until finishing
the book many new aspects have been examined and are included.
My personal line through these problems started out with beam design in 1967,
followed by a Ph.D. study concentrated on truss design. From 1970 the importance
of combinations with the finite element method became clear with focus on design for
minimum stress concentration, first 2D-problems and later 3D-problems. In fact, the
interest in the finite element method itself took over.
Later, i.e., from 1980, vibrational problems were my primary area of research,
with focus on non-conservative dynamic stability. Related sensitivity analysis was
(and still is) a fascinating subject. Since 1985 the design of composite materials
like laminates was given a lot of attention. The step to material design is a natural
extension with inverse homogenization and bone mechanics now in focus.
It is a challenge trying to put all this together, so that the main results can be
communicated to graduate students and engineers in practice. This is a first layout
that will be freely available on the Internet, thus making colour presentations and
even animations possible (later). Critics and suggestions for improvements are most
welcome, e.g. by email to pauli@mek.dtu.dk Email
pauli@mek.dtu.dk
11

12 Copyright c○Pauli Pedersen: Optimal designs ....
Thanks to students and colleagues without whom the job was not done. For
critical comments on early manuscripts I thank Alexander Seyranian from Moscow
State University, Russia; John E. Taylor from University of Michigan, Ann Arbor,
USA; Ciro A. Soto from Ford research laboratory in Dearborn, USA; Alija Picuga from
Mostar, Bosnia; and my young colleagues at Aalborg University Niels L. Pedersen and
Jan Stegmann.
Lyngby, January 2003
Pauli Pedersen

Preface and Introduction 13
1.1 Words of the title
Optimal
designs
This book is about optimal designs. Optimization and optimization methods are
necessary tools but are not put in focus. There are a number of good books that
cover methods to a large extent. From the optimal design point of view we mention:
(Banichuk 1983), (Save, Prager and Sacchi 1985), (Arora 1989), (Brandt 1989),
(Rozvany 1989), (Borkowski, Jendo and Reitman 1990), (Haftka, Gurdal and Kamat
1990), (Kirch 1993), (Bendsøe 1995), and (Bendsøe and Sigmund 2003). In the
present book we focus on the resulting designs and the possible general knowledge
connected to these. Identification
The subject of identification, or in more weak form estimation, is also of major
importance, but to make the title short it is not included in the title. Identification is Inverse
a subset of inverse problems, that from the author’s point of view are of even greater problems
practical importance than the optimal design problems. Traditionally, engineers analyse
a given structure (or material) to determine a response. In inverse problem we
seek a structure that gives a wanted response and in identification this response may
be obtained experimentally. Stated in other words, we seek a model that matches
measured values. Theory and tools
Theory and tools (procedures) are included as the last 10 chapters and may serve
as a kind of appendices. The hope is that the reader, motivated by the many examples
in the first chapters, is interested in the more general aspects of optimal design.
1.1.1 Specific models
In the second line of the title, structures and materials are mentioned in parallel.
Most research on optimal design is related to structures or we may be more specific
and say structural models such as bars, rods, trusses, beams, frames, grids, plates,
Structural
models
shells, axisymmetric models or in general 2D- and 3D-continuum models. In recent Material
years a major focus has been put on design of materials and again we should say on
models
material models. Points or
domains
There is not a clear distinction between structure and material. Is a laminate
a structure or a material? In reality even a traditional material is a structure if we
look close enough. The distinction could be if we model what is related to a point (a
material point) or we model what is related to a domain (a structural domain). Inverse
Methods of homogenization transfer modelling from domain to point. Modelling
a material point as a finite domain (a cell model) we have a structural/continuum
problem, and the material design is then related to structural/continuum design.
Thus we are directly lead to what is named inverse homogenization.
homogenization

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1.1.2 Specific problems
The further words of the title, - problems and tools -, again stress the fact that formulation
and solution of specific problems are put in focus. We want to learn by solving,
hopefully well-defined problems, and from this extrapolate our possibilities. This
General
means that the problems are not just examples to illustrate a theory or a procedure.
knowledge However, the specific problems often teach us about or indicate that general
knowledge can be obtained. Then the focus is naturally turned in that direction,
but the more theoretical aspects and proofs are placed in chapters that more serve as
appendices, say the theory behind statically determined optimal trusses, the theory
behind uniformly stressed continuum domains, and the theory related to shapes with
uniform energy density.
1.2 References
Not proper
references The book is intended to be self-containing, although this naturally depends on the
readers background. A list of references is given, but not with proper references to
the large international literature. The book is mainly based on ”in house” research,
often carried out in cooperation with students and colleagues, also outside Denmark.
A number of extensive review papers are available, for early reviews see (Wasiutynski
and Brandt 1963), (Sheu and Prager 1968), (Niordson and Pedersen 1973-74),
(Olhoff 1980), (Olhoff and Taylor 1983), and for a more recent review see (Eschenauer
and Olhoff 2001). Also the books mentioned in the beginning of this introduction have
valuable lists of references. A book with all proper references would be difficult to
write, and hopefully, the reader will agree and appreciate the present style.
Glossary
Trusses
Uniform
energy
density
1.3 The chapters
Like most areas of research, optimal design has its own names and concepts. Chapter
2 may serve as a glossary without being ordered. Then already in chapter 3 we go to
a specific problem, statically determined trusses.
Besides being the most simple structural model it is also the most effective structure,
at least when it is up to us to define the word effective. A number of rather
different aspects are covered including the account for local stability. The proof for
the statically determined, one load case is given in chapter 16.
In chapter 4 continua of uniform energy density are designed, restricted to 2Dproblems.
The theoretical knowledge of uniform elastic energy density is applied in a
simple recursive procedure that iteratively gives an optimal continuum, optimal with
respect to stiffness as well as to strength. In chapters 12, 13, 14, and 15 the theory

Preface and Introduction 15
behind these examples shows the extension to non-isotropic elasticity, to power law
non-linear elasticity, and to free material design. Energy
Energy principles of mechanics play a major role in the derivations, but even principles
without knowing the details of these derivations, the resulting conclusions can be
used, say in a standard finite element program which then directly extends from
analysis to synthesis. Beams
From continuum we go back to the one-dimensional models of static beam problems
in chapter 5, where the knowledge from energy principles is used to obtain some
analytical solutions for optimal design. More extended problems, like frames (made
of beams) must be solved numerically. On the other hand, the extension to dynamic
problem for the frames is not a big step, assuming that results from eigenvalue analysis
and eigenvalue sensitivity analysis are known. In agreement with the general Vibrations
idea of the book, this theory is described later in chapter 18, so chapter 5 can directly
discuss the solution to some specific problems.
and frames

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Chapter 2
Names and concepts
2.1 Major classification of names and concepts
Like most areas of research, optimal design has its own names and concepts. This
chapter may serve as a glossary without being ordered. From this chapter we go
directly to specific problems. The major classes of concepts are: Glossary
• Design VARIABLES
• Design OBJECTIVE
• Design CONSTRAINTS and Design SPACE
Optimal design problems may have one solution, several solutions or no solution.
Even when there is only one solution, the optimal design problem may be presented
in many different formulations. In addition a given formulation may be solved with
many different methods and procedures. Thus we add as major classes of concepts:
• Optimization FORMULATIONS
• Optimization METHODS and PROCEDURES
2.2 Design variables
The most important classification of design variables is into:
• SIZE design variables
• SHAPE design variables
• TOPOLOGY design variables
17

Size
Shape
Topology
Examples
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stated here in order of difficulty to solve but also in order of increasing importance
for the obtained objective value. It is therefore not surprising that recent research to
some extent concentrates on topology design variables.
The notion of size design variable, relates to the thickness of a beam, a plate or
a shell (although this is often termed the shape of a beam, a plate or a shell). The
area of a bar in a truss is also a size design variable, and the definition of size variable
is related to the fact that the modelling domain is not changed. So, the line of the
beam, rod or bar is unchanged, just like the reference surface of a plate or a shell is
assumed unchanged when the concept of size design variable is used. In 3D-problems
the mass density or a relative volume density is size. The orientations of non-isotropic
material we also treat as size design variables.
The notion of shape design variable, relates to the reference domain of the actual
model. For beams, rods and bars we may treat the length as a design variable,
which is then a shape design variable. Also the curvature of the reference line for
these one-dimensional models is a shape design variable. For 2D-models likewise the
boundary curve or the curvature of the reference surface are shape design variables.
For 3D-models the boundary surface (including internal boundaries like holes) is a
shape design variable. Stress concentration problems are often related to shapes of
boundaries.
Finally, the notion of topology design variable, relates to presence or absence of
a certain design aspect. Should two joints in a truss be connected with a bar, - yes or
no ?. Should a continuum like a plate have a hole, - yes or no ?. The complications
in treating topology design variables are due to the fact that a change in topology
results in a discontinuous change in the design response, while a continuous change
in size or shape design variables normally results in continuous change in the design
response.
Let us exemplify the difference between size, shape and topology design variables.
In a truss (2D as well as 3D), the bar areas (uniform or non-uniform) are the sizes,
the positions of the joints determine the shape, and the chosen bars (among many
possibilities) give the topology. In a shell the thickness and material density distributions
are the sizes, the boundaries of the reference surface and its curvatures are the
shapes, and the number of holes in the reference surface is the topology.
2.2.1 Alternative classifications
Many alternative names to classify design variables can be found in the literature, like
cross-sectional, geometrical, configuration, layout etc. We try to avoid these names
Other
in order to avoid unnecessary confusion.
classifications The design variables may also be classified from other points of view. Let us first
discuss the distinction between continuous and discrete design variables. If only a

Names and concepts 19
number of specific values for the design variable is acceptable, say when catalog values
must be used, then the notion of discrete design variables is used, and procedures Continuous
related to what is called Integer programming come into focus. This is not covered discrete
in the present book that concentrates on continuous descriptions, but we include
absolute limitations for the values of the design variables. Distributed
Another meaning of the ”continuous” and ”discrete” relates to the modelling parameter
of the design domain. A complete continuous description in space means design
variables related to a point (like a design function) and not to a domain. Often this
is termed distributed parameter description, in contrast to say a truss description
where each bar is described as a unit. In a finite element modelling of a continuum,
the element domains may be related to a number of design values, so in reality this
is a discrete description. However, with the extensive number of elements and the
fact that everything in a computer is discrete, the distinction between continuous and
discrete related to the modelling of the design domain is of no practical importance. Parametrization
For a successful optimization the choice of design parametrization is of vital importance,
perhaps the most important decision to take. In the experience of the
author it is wise to start with as few design variables as possible. A hierarchical description
is suggested, and also it is important to make sure that the design variables
serve different purposes. It is asking for practical problems, if the design variables are
chosen such that different combinations of design variables can give the same design.
The parametrization is also related to the chosen optimization procedure, so with
an optimality criterion method large quantities, say 50.000 design variables, can be
handled without problems.
2.3 Design objective
The design objective is a function or a functional that returns a single value from
which different designs can be compared. The optimal design is then the design with
a minimum (or maximum) value of the objective. In this book we often use the
notation Φ to denote the objective. We shall not treat multi-objective formulations,
which in most cases are reformulated into a single objective anyhow.
Alternative names for the objective include criterion, cost, merit, goal as well as
many others. The name ”criterion” is in this book used extensively in relation to
optimality criterion formulations (see chapter 14), so we try only to use the name
objective, although a name like cost may be more appealing. In fact, the objective
Single value
value is often a measure of the cost of the design. Minimum
A minimum and maximum formulation may be interchanged by simply changing
the sign of the objective. However, it is important to notice that many methods just
locate a stationary value of the objective, which means that the convergence of the
Maximum

Local
Global
Optimized
design
Existence
Convergence
tests
Constraints
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procedure must be followed and the final design justified.
A much more severe problem is related to the existence of local stationary solutions,
and in reality very few (and often non-practical) methods are able to find a
global optimal solution. Starting an optimal design procedure from different initial
designs and always ending up in the same optimal design may be the most practical
procedure for improving the probability of an obtained solution being the global optimal
solution. It should be noticed that a number of optimal design formulations for
idealized problems may include a proof of global optimal solution.
However, for problems where a large number of practical constraints need to be
taken into account, it is more safe to state that we have optimized the design as an
alternative to obtaining the optimal design. Furthermore, it is not always easy to see
from the formulation whether an optimal design exists. If an optimal design does not
exist we talk about a not well formulated problem. Even so a procedure may return
an optimized design, and the convergence often reveals the missing aspect(s) in the
formulation.
An important part of an optimization procedure is to decide when to stop. We
talk about convergence tests. Two different aspects of convergence must be clarified,
convergence of the design objective and convergence of the design variables. Often
the rates of these two convergences are very different. Also the formulation of the
specific stop condition can be mathematically formulated more or less complicated.
The favourite formulation of the present author is as follows: When the design changes
are somewhat smaller than the actual accuracy in the design production, then the
design procedure should be stopped. At that instant the design objective is often
converged at a much earlier step.
2.4 Design constraints
The design constraints serve many purposes and more or less reflect the goal of the
design. In order to get some overview let us discuss them according to the following
names:
• LOAD conditions and equilibrium
• BEHAVIOURAL constraints
• SIDE constraints, absolute limits and move-limits
• EQUALITY and INEQUALITY constraints
• FULLY STRESSED design
• SATISFIED and VIOLATED constraints

Names and concepts 21
Often the loads on a structure are given and are design independent. However,
loads like inertia forces and pressure are in most cases design dependent. Boundary
conditions may also be given or may be part of the design variables. In the most
simple cases as for the truss design in chapter 3, we in reality solve a problem of
Loads and
boundary
conditions
transfer of given loads to given possible supports. Load cases
The main condition of all the design problems is that equilibrium must be satisfied
for static as well as for dynamic problems. In most cases the analysis to obtain
equilibrium is solved by the finite element method, a method that we assume the
reader to be familiar with. Note that in most practical problems, we need to take a
number of different load cases into account. Implicit
A structure is subjected to a large number of behavioural constraints, that more or
less are all calculated based on the results from the equilibrium. All these constraints
depend in a very implicit manner on the design variables, and must be calculated for
behaviour
all load cases. Displacements
and stiffness
The stiffness or flexibility can be described locally by displacements or globally by
total elastic energy. The displacements are restricted in magnitude, but very specific
prescribed displacements may also be part of the design purpose. A large amount of Compliance
optimal designs are based on minimum compliance, which is the same as minimum
elastic energy (as defined more specifically in chapter 13). Stresses
strength
From displacements follow strains and then with a constitutive model the stresses.
Constraints on stresses are related to the strength, as expressed by the von Mises stress
or the strain/stress energy density. To account for these strength constraints we must
find the most critical points in a structure, i.e. these constraints are formulated as
point (element) constraints, where the dangerous domains must first be located (active
set strategy). Eigenfrequencies
For dynamic problems, the eigenvalue analysis needed to determine eigenfrequencies
can be a finite element analysis, that returns sets of frequencies and corresponding
vibration modes. The constraint can specify a minimum allowable frequency and/or
a minimum gap between eigenfrequencies. The sensitivity analysis in chapter 18 gives
the extended background for the iterative redesign of such design problems, and ex-
tended experience is available. Stability
Another eigenvalue problem is related to the analysis which determines the stabil- (buckling)
ity of the structure against buckling. To a large extent these aspects of optimal design
are similar to dealing with eigenfrequencies. More indirect behavioural constraints Fracture, creep
are related to fracture, fatigue, creep and in reality to aspects that determine the life
time of the structure. Absolute
The notion of side constraints covers two different aspects. The first one we term
absolute side limits and gives the designer the possibility of specifying a range of
acceptable values for each design variable. The second aspect is directly related to
side limits

Move-limits
Active
constraints
or inactive ?
Fully stressed
Satisfied
or violated ?
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the control of the iterative design process. The total optimal design is carried out
in a number of steps, or we may say redesigns, i.e. the optimization is an iterative
process.
In each redesign it is often a good idea to describe so called move-limits, that
control the change from one design to the next iterative design. By adjusting in
an actual redesign the move-limits we can indirectly take the absolute limits into
account, so that these are not involved in the formulation. When move-limits are
given in terms of physical quantities the designer treats these as probably the most
important parameters to control the design process.
Constraints specified as equality constraints are conditions that must be satisfied,
say the equilibrium constraints. Most behavioural constraints are specified as inequality
constraints, or we may say bounds on the behaviour. Likewise the absolute limits
and the move-limits, and these bound constraints may be active or inactive, and in
general it is not known in advance what will be the case. Only when the optimal
design is obtained, the final information about the active constraints is known.
A fully stressed design reflects in reality a natural consequence, more than a
described constraint. We will see that a number of idealized design problems lead to
fully stressed designs, and if this can be assumed in advance, the design problem is
much less complicated. Even with multiple load cases, the notion of fully stressed
is meaningful; it describes that each structural member (or structural region) is fully
stressed at least in relation to one of the load cases.
The design problem is a very non-linear problem, but dividing the problem into
an iterative process of optimal redesigns we linearize the description relative to the
actual design, like a Taylor expansion of the behaviour. This means that errors in
the analysis are involved, and we may have constraints that are violated. During the
design process we have both satisfied and violated constraints. Ending the iterative
process with small move-limits we can control these violations.

Names and concepts 23
2.5 Design space
The notion of design space gives a geometrical interpretation of our possibilities for
an actual design. Dealing with many design parameters this design space is multidimensional,
but anyhow, for most of us is an advantageously interpretation. By the
following concepts we describe this design space in more detail:
• FEASIBLE and NON-FEASIBLE design space
• INTERIOR and EXTERIOR design space
• CONSTRAINT BOUNDARIES
• CONVEX design space
• DIRECTION in design space
• GRADIENTS for a design direction
• STEP SIZE and LINE-SEARCH
• JUMPS in design space
Feasibility
A number of these names are more or less self-explaining. The feasible design
space is the collection of all design points (sets of design parameters) that satisfy all
the design constraints. In the non-feasible design space at least one design constraint
is violated. In the interior of the design space all neighbouring points are also feasible
design points, while in the exterior design space all neighbouring points are also nonfeasible
points. The separation between these two design spaces is the constraint
boundary, where at least one constraint is active. Convex
Changing the parametrization of the design description we change the design
space, and if it is possible to enforce a parametrization where the design space is
convex, then valuable conclusions can be drawn. The notion of convex means that
any linear combination between two feasible design points is a feasible design point. Moving in
design space
A design change involves a movement in the design space. This can be specified
by a direction and a step size. The change may be continuous or it may be a jump.
If several step sizes are possible, then a search along a proposed direction may be
actual.
An advantageous move in the design space, corresponding to a redesign, is determined
by the actual gradients corresponding to the different directions in the design
space. The gradients relate to the objective as well as to all the constraints, so a lot of
information is necessary and in most cases this information is only available in an im- Sensitivity
plicit manner. Important results from sensitivity analysis may give this information,
see chapters 14 and 18.
analysis

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2.6 Optimization formulations
Choosing a formulation for an optimal design problem, we more or less restrict ourselves
and we do so in order to obtain specific results. Modifications and extensions
of a chosen formulation are in most cases necessary in the process of learning more
about our specific problem. A number of names and concepts related to different
formulations are:
• DETERMINISTIC or PROBABILITY based optimal designs
• ANALYTICAL or NUMERICAL formulation
• LINEAR or NON-LINEAR behavioural constraints involved
• STATIC or DYNAMIC constraints involved
• NECESSARY and SUFFICIENT conditions for optimality
• AUTOMATIC design, INVERSE problems and IDENTIFICATION
• SIMULTANEOUSLY ANALYSIS and DESIGN
Only
deterministic A few comments to this list is given. In the present book we only deal with
optimal designs based on deterministic behaviour.
Mostly
Only few analytically determined optimal designs are available, but because of
numerical
the insight they give, we describe some of these. However, the major part of the book
focus on numerically obtained optimal designs. This does not imply that analytical
results are not involved, because the basic theory behind a given numerical procedure
Also
is often based on analytically obtained knowledge, i.e., theorems and others.
non-linear The importance of non-linearity in behavioural description along with our ability
to day to analyse with the finite element method mean that to optimal designs, based
on non-linear behaviour must be given proper attention. We strive to obtain this
Multiple
goal.
eigenvalues Static and dynamic constraints are treated, and eigenvalue problems are put in
focus. The specific problem with multiple eigenvalues forces us to go in more details
Stationarity with these problems, that physically relate to both vibration and stability problems.
extremum The energy theorems of mechanics are divided into stationarity principles and
extremum principles. Likewise are the optimality conditions, and furthermore we
need to distinguish between necessary and sufficient conditions for optimality. The
close relation between the principles of mechanics and the principles of optimality is
shown.
Inverse
In practice the restricted problem of finding just a feasible design may be of
problems
major importance. In the subject of optimal design this sub-problem is often termed
automatic design. Also the very important problems related to formulation of inverse

Names and concepts 25
problems and to problems of identification (estimation) are shown to involve the same
tools and procedures as necessary for optimal design.
2.7 Optimization methods and procedures
Most optimal designs can only be obtained by an iterative solution of a sequence of
sub-problems as shown in figure 2.1. Each of these sub-problems may be solved by
a number of different methods and, furthermore, the selection of design parameters
is by no means unique. Viewing a specific combination of methods of solution for
these sub-problems as a ”new” optimal design method, we get too many methods
and lose our overview of the subject. Therefore it is necessary to concentrate on the
sub-problems, separately.
From the author’s point of view, the splitting of our approaches into analytical
methods versus numerical methods, or our subject into distributed parameter problems
versus discrete parameter problems is also confusing. This splitting is more a
matter of taste than necessity since almost all optimal design problems are solved
numerically with a finite number of parameters.
2.7.1 The sub-problems of optimal redesign
Sub-problems
The box of optimal redesign in figure 2.1 includes three sub-problems: Optimal
redesign
• ANALYSIS by finite element method (FEM)
• SENSITIVITY ANALYSIS
• REDESIGN
The term response analysis is a very wide one, which may include displacement-
, stress-, vibration- and stability-analysis. It is a characteristic of optimal design
that we are confronted with non-uniform thickness distribution, shapes which cannot
be expressed by classical functions, and often with extreme behaviour like multiple
eigenvalues. Thus, it is essential that we use a very flexible method and one in which
we are experienced. It is advantageous if this method for response analysis can be
used with different levels of accuracy.
Many methods are available for the different response analyses, but none serves
our purpose as well as the finite element method (FEM). It is therefore important to
state: the unified approach to optimal design is based on the finite element method for
response analysis. Advanced
The process of optimal design should always terminate with a response analysis of
the resulting optimal design. Stated in other terms: optimal design is not a separate
subject, but an advanced extension of the response analysis.
FEM-analysis

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SELECTION OF
DESIGN PARAMETERS
TEMPORARY INITIAL
DESIGN
RESPONSE ANALYSIS
OF THE DESIGN
SENSITIVITY ANALYSIS
OF THE DESIGN
DECISION OF REDESIGN
DESIGN
YES CHANGED NO
OPTIMIZED
DESIGN
Problem of optimal design:
A highly non−linear problem
Problem of optimal redesign:
A linear programming problem
DESIGN
Figure 2.1: The approach to optimal design presented from the point of view of a
number of sub-problems.

Names and concepts 27
The advanced extension of response analysis has two aspects. Firstly, because
of the iterative process of redesign, we have to do repeated response analysis. An
efficient optimal design approach naturally takes this into account.
Secondly and more important, for the decision of redesign we need to know how
the response of a design changes with the design parameters. The analysis necessary
for determining this is often named as sensitivity analysis. This analysis may be
carried out numerically by differences, but often it is our goal to perform this analysis
more analytically.
As it is seen also from the present book, sensitivity analysis plays a leading role
and many results are available. Entire books (Kleiber, Antunez, Hien and Kowalczyk
1997) are written on the subject. Always
sensitivities
Some approaches to optimal design do not seem to involve the sensitivity analysis
but a closer look shows that it is almost always present. As an example, the terms
of an optimality criterion are gradients and must be obtained by a sensitivity analysis.
It is therefore the opinion of the present author that: sensitivity analysis is the
”cornerstone” of almost any approach to optimal design. Redesign
When we focus on the third sub-problem, i.e. the problem of deciding the redesign,
we find methods that are very much different in nature. There is no doubt that the
decision of redesign is an optimization problem. Nevertheless, it is often attempted
with ad hoc methods and without an objective, and it is difficult to assess the degree
to which the results approximate the optimal redesign. Move-limits
The problem of optimal design as whole is highly non-linear, and even the problem
of optimal redesign is, in fact, non-linear. However, by move-limits we restrict the
design changes so that linear expansions are acceptable. These move-limits are of
major importance, as they are the key for the designer, to control the design process.
In the primary redesigns we choose rather large move-limits, whereas in the final
redesigns, small move-limits may be necessary. Classes of
Still, if we choose to treat the decision of redesign as an optimization problem, we
have many possible methods to choose among. For the optimal designs in the present
book, the two major classes are:
• OPTIMALITY CRITERIA methods
• MATHEMATICAL PROGRAMMING methods
and in the later class we shall primarily use
• Linear Programming (LP), used sequentially (SLP)
Further methods in the class of mathematical programming are:
• Quadratic Programming (QP), used sequentially (SQP)
• Dynamic Programming
optimization
procedures

Local/global
parametrization
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• Geometric Programming
• Method of Moving Asymptotes (MMA)
• Interior or Exterior point methods
• Stochastic methods
• Generic Algorithms
• Simulated Annealing
• Neural Network
• Evolutionary methods
but the more detailed description of these procedures is not given in the present book.
2.7.2 The overall strategy
The selection of design parameters is by no means unique, and in relation to the
optimization approach the manner in which the design is described is very important.
This is especially true in relation to shape design.
The one extreme is the local design variables, like thickness at a given point,
position of a joint (node), or in relation to FEM, the element variables. The other
extreme is the global design variables, which influence the design everywhere. The
use of local design variables seems to dominate, although the use of global design
variables has many advantages.
One example of the use of global design variables is the description of a shape by a
linear combination of a few known functions, in a way very parallel to mode expansion.
The discrete design parameters are then the factors of the linear combination. Such
an approach has the advantage of dealing only with a few design variables, and it is
easy to ensure a smooth connection to regions of fixed design. The disadvantage is
that the sensitivity analysis is enlarged, which counteracts the small number of design
variables.
With regard to the use of local design variables, the main advantage is that the
gradients are often directly available, being, say specific energies that do not have to be
computed separately. Naturally, the use of local design variables gives the possibility
of a very detailed design but at the expenses of having to deal with many design
variables. Furthermore, the use of local design variables gives the risk of obtaining
degenerate designs and solutions that depend drastically on the discretization level.
A final comment on the important question of selection of design variables is
that many combinations of local and global design variables are possible, and that

Names and concepts 29
the accuracy in description of the design need not be the same at the initial design
iterations as at the final ones. This is easily obtained with local as well as with global
design variables. The approach of design variable linking should also be kept in mind.
The optimality condition for the optimal design in total (not for the individual
redesigns) may also be viewed as a stopping condition. When to stop the sequence
of redesigns ? As indicated in figure 2.1 there is a general practical answer to this
question, which is: When the solution to the linear programming problem of redesign
is to make no design changes, then the design is optimal. This is, of course, true
in the local sense only, but we have to deal with the question of local versus global
optimal solution anyhow, and base our beliefs on engineering judgment and optimal
design from different initial designs.
There is no contradiction between this stopping condition and other statements
like stationarity or Kuhn-Tucker condition. The nice aspects of the simple condition
are that the stopping condition is valid also for solutions with non-stationarity and
the condition does not have to be expressed explicitly.
Stopping
condition

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Chapter 3
Design of trusses
Trusses are structures that are simple to analyse, because each point of the structure
is only subjected to a unidirectional state of stress/strain. The basic element of a
truss is a bar and the bars are connected through joints that only transfer a normal
force (tensile or compression) to each bar. In this chapter we only deal with these
simple models, but local stability of each bar is also considered. Simple
In addition to being simple structures, trusses are also very efficient structures
with the possibility of designing fully stressed structures, where each point in the
structure is used to its limit. For a statically determined truss each bar can be designed
independently and therefore in an optimal design must be fully stressed. The notion
of fully stressed for multiple load cases means that each bar is fully stressed, at least
in one of the load cases.
With reference to chapters 16, 19, and 20 we shortly discuss the theoretical and
numerical background and choose bar members with circular cross-section as our basic
Bars and joints
unidirectional
element. The influence of hollow cross-sections in relation to design for local stability Compressive
is studied.
Through a number of specific examples (cantilever and bridge types) the importance
of the difference between tensile bars and compression bars is put in focus. All
the results for plane trusses as well as for space trusses are obtained by an interactive
stability
program that also includes the design of supports as an integrated part. Single
Early results for cantilevers and bridges (Pedersen 1969) show how size and topol- load case
ogy optimal design (bar areas and selection of bars) are combined with shape optimal
design (joint positions), but these examples are restricted to single load case models.
In the extensions to multiple load cases and/or including constraints on displacements
(stiffness in addition to strength), the design parameters will only be size and
shape, and we use a more traditional, iterative optimal design procedure with anal- Multi-purpose
trusses
31

32 Copyright c○Pauli Pedersen: Optimal designs ...
ysis, sensitivity analysis and linear programming. Some attempts to include also
topology is discussed, but in general these practical important design problems are
still open for new research initiatives.
3.1 Theoretical and numerical background
If we assume that allowable stresses and joint positions are given, then the problem
of finding a truss of minimum mass is a linear programming problem. Such optimal
design problems can be solved using the simplex optimization procedure, and this was
Statically
done already by (Dorn, Gomory and Greenberg 1964) and by (Fleron 1964).
determined With only a single load case, local stability is taken into account, and we can prove
that a statically determined truss is a solution. This is proven in chapter 16 in a more
general setting of minimum cost and starting also without a complete specification of
supports.
The modified simplex procedure, described in chapter 19, is used to obtain the
solutions to a number of illustrative problems, which we discuss in sections 3.2 and
Size, shape
and topology
3.3, but without going into detail about the optimization method itself. Also the size,
shape, and topology combined solutions in section 3.4 are obtained with a similar
procedure combined with a more traditional mathematical programming solution for
the joint positions (shape variables).
In the last sections 3.5 and 3.6, all practical constraints can be accounted for,
but the truss topology is then assumed to be given. Sensitivity analysis plays an
important role and the formulation as well as the solution procedure are described
in chapter 20. In the present chapter we concentrate on the discussion related to the
specific examples.
3.1.1 Common parameters for the examples
In addition to some specified joint positions and the actual loads to be transferred,
only few parameters are needed for the optimization. In most of the examples we use
the same values for these data and therefore list them here:
General data
for examples Modulus of elasticity E 2.0 · 10 11 Pa
Stress limit of proportionality σP 1.6 · 10 8 Pa
Allowable tensile stress σT 2.0 · 10 8 Pa
Maximum allowable compressive stress σC 1.6 · 10 8 Pa
Factor of safety for slender columns ξ 2.5
Mass density ρ 7500 kg/m 3
Cross-sectional parameter α := √ I/A see section 3.1.2

Design of trusses 33
3.1.2 Truss members with circular cross-section
We shall study the influence of local stability on the optimal topology by changing
a cross-sectional parameter α. Also for visualization of the results, we need crosssectional
outer dimensions. Therefore we specialize to truss members with circular
cross-section (pipes) with outer diameter do and inner diameter di, specified by the
ratio µ
di = µdo with 0 ≤ µ

34 Copyright c○Pauli Pedersen: Optimal designs ...
3.4 including optimal position of the free joints and with different number of free
joints.
3m
3m
80000 N 40000 N 20000 N 20000 N
3m
Figure 3.1: Load and geometry assumptions with the possible supports for the cantilever
test example. In addition to the loaded and supported joints this example has
three joints.
The initial chosen topology with only size optimization (fully stressed) is shown
in figure 3.2 together with the optimized topology. For this example the self-weight
of the truss is not taken into account. The used colour code in all the examples are
blue for tensile members, orange for compressive slender columns, red for compressive
short columns, and green for zero strings. For definition of these different types, see
chapter 16.
Fully
stressed As commented in general, both topology’s in figure 3.2 are fully stressed and statically
determined. We note the repeated sub-structure in the optimal design, which without
Self-weight
the string support also has a column of double length. This problem is not actual in
the optimal design shown in figure 3.3, where the self-weight is taken into account.
included Note, that with self-weight we totally add loads in the same direction as the
given external loads and still the optimal design has a smaller total mass (685 kg to
compare with 707 kg). The explanation for this is that the vertical columns when
self-weights are taken into account, then transfer smaller compressive forces, and in
3m
3m

Design of trusses 35
Figure 3.2: Cantilever with given loads in gravity directions, but without self-weights.
Initial size optimized design (mass = 902 kg) at the top and optimal topology design
(mass = 707 kg) at the bottom. Solid cross-sections (α =0.3).
this manner make a lighter structure possible. We notice that the topology is changed
when self-weight is taken into account.
Figure 3.3: Cantilever with given loads in gravity directions, and with self-weights.
Initial size optimized design (mass = 922 kg) as in figure 3.2 and the optimal topology
design here results in mass = 685 kg. Solid cross-sections (α =0.3).
In a formulation without local stability constraints and with equal allowable tensile
and compressive stress, the optimal design will be the same with opposite external
Changing
load direction

Changing
type of
cross-section
36 Copyright c○Pauli Pedersen: Optimal designs ...
loads. However, when compressive members are more expensive the optimal topology
will normally be different. This is illustrated in figure 3.4. The self-weights now
counteract the given external loads, and it is possible to obtain a lighter structure, for
the initial topology as well as for the optimal topology. Comparing with the optimal
topology in figure 3.3 we notice a completely different solution.
Figure 3.4: Cantilever with given loads opposite gravity directions (opposite direction
of forces in figure 3.1), and with self-weights. Initial size optimized design (mass =
683 kg) at the top and optimal topology design (mass = 605 kg) at the bottom. Solid
cross-sections (α =0.3).
The solutions presented in figures 3.2, 3.3 and 3.4 are based on an assumption of a
solid cross-section (α =0.3), and thus compressive members dominate the total mass.
With other cross-sectional assumptions we get different optimal designs, as illustrated
in figures 3.5, 3.6, 3.7 and 3.8. Comparing the designs in figure 3.5 corresponding to
α = 1 (a hollow circular cross-section with di =0.92d0) with the results in figure 3.2
and 3.3, we first of all notice the much less necessary mass, but also the change in the
optimal topology. Figure 3.5 shows that a number of the compressive members are
now classified as short columns (in colour: red not orange). See chapter 16 for the
classification.

Design of trusses 37
Figure 3.5: Cantilever with given loads in gravity directions, and with self-weights.
Initial size optimized design (mass = 336 kg) at the top and optimal topology design
(mass = 280 kg) at the bottom. Cross-sections corresponding to α =1.
Figure 3.6: Cantilever with given loads opposite gravity directions, and with selfweights.
Initial design size optimized (mass = 282 kg) at the top and optimal topology
design (mass = 264 kg) at the bottom. Cross-sections corresponding to α =1.

38 Copyright c○Pauli Pedersen: Optimal designs ...
The results corresponding to extreme thin-walled cross-sections are shown in figure
3.7 and 3.8. In these designs, all compressive members are short columns (red, not
orange) and taking local stability into account is then not so important. However,
with compressive allowable stress only 80% of allowable tensile stress, see section
3.1.1, we still find different design when the directions of the given external loads are
changed.
Figure 3.7: Cantilever with given loads in gravity directions, and with self-weights.
Initial size optimized design (mass = 223 kg) at the top and optimal topology design
(mass = 188 kg) at the bottom. Thin walled cross-sections (α = 3).
3.2.2 A cantilever truss with 17 joints
The optimal cantilever designs presented depend on the basic 9 joints and their positions.
Later we shall discuss the possibility of optimizing the positions of some of the
joints, but let us primarily just double the number of joints in the cantilever length
direction. The total given loads are not changed, with each load force just divided

Design of trusses 39
Figure 3.8: Cantilever with given loads opposite gravity directions, and with selfweights.
Initial size optimized design (mass = 216 kg) at the top and optimal topology
design (mass = 180 kg) at the bottom. Thin walled cross-sections (α = 3).
into two neighboring joints. Two cases are shown both with loads in the gravity direction,
both with self-weights included, differing only in cross-sections corresponding
to α =0.3 and α = 1, respectively.
Comparing the results in figure 3.9 with the results in figure 3.3 certain similarities
in topology are recognized and the resulting masses are lower with the many joints.
This is due to the possibility of shorter columns, but in practice the higher cost of
many connections may counteract this.
Comparing the results in figure 3.10 with the result in figure 3.5 certain similarities
in topology are again recognized and the resulting masses are again lower with the
many joints, only 3 kg for the initial topology but 36 kg for the optimal topology. Repeated
Notice again the repeated sub-structures.
sub-structures
3.2.3 A bridge truss with 16 joints
This example is also an optimization taken from (Pedersen 1969), in order to illustrate
a classical civil-engineering design problem in a simple model. Figure 3.11 shows the

40 Copyright c○Pauli Pedersen: Optimal designs ...
Figure 3.9: Cantilever with given loads in gravity directions, and with self-weights.
Initial size optimized design (mass = 836 kg) at the top and optimal topology design
(mass = 548 kg) at the bottom. Solid cross-sections corresponding to α =0.3.
Figure 3.10: Cantilever with given loads in gravity directions, and with self-weights.
Initial size optimized design (mass = 333 kg) at the top and optimal topology design
(mass = 244 kg) at the bottom. Cross-sections corresponding to α =1.

Design of trusses 41
load and geometry assumptions.
L/8
7 forces, each 100000 N
L =32m
Figure 3.11: Load and geometry assumptions with the possible supports for the
bridge test example. In addition to the loaded and supported joints this example has
seven joints.
Figure 3.12 shows the optimal topology based on joint positions with a parabolic
reference and length/height = 5.5. The resulting total mass of the optimal bridge
is 4157 kg. This bridge truss optimization is extended in section 3.4 to include the
positions of the seven upper joints as design parameters. Furthermore, the solutions
for different total length of the bridge are then presented. For all these problems the Same optimal
optimal topology in figure 3.12 is unchanged.
topology
Figure 3.12: Optimal topology for the bridge problem defined in figure 3.11, with
compressive bars along the parabola and tensile bars in the remaining part of the
truss.
3.2.4 A beam/bridge truss with 24 joints
We design a beam/bridge that span 44 meter. At the lower joints the beam/bridge
truss is loaded with a uniformly distributed load of 2500 N/m (totally 110000 N),
plus the weight of the structure itself, which adds about 10%. The design depends
on the possible height of the beam/bridge and especially on the possible supports.

42 Copyright c○Pauli Pedersen: Optimal designs ...
With symmetry assumed we only model (and only show in the figures) half of the Half a model
by symmetry
structure, using for this 24 joints.
In figure 3.13 the possible height is only 1 meter and fixed supports are assumed
possible at the two left boundary joints in addition to the supports that model the
symmetry line at the right boundary joints. The characteristic of the optimal design
in figure 3.13 is a combination of a cantilever part that supports a mid-part of mostly
tensile members.
Figure 3.13: Half of a low beam with uniform loads at the lower joints, two left
joints fixed and two right joints at symmetry line. Self-weights included. Initial size
optimized design (mass = 845 kg) at the top and optimal topology design (mass =
532 kg) at the bottom. Solid cross-sections corresponding to α =0.3.
Cantilever +
hanging part The relative length of the cantilever part and the tensile hanging part is depending
upon the relative mass (cost) of tensile to compressive members (here the α parameter
is equal to 0.3), and upon the possible height of the beam/bridge. Figure 3.14 shows
the optimal design for a height of 3 meter, and for this case only the hanging part
remains. Thus, we end up with a structure with only tensile members, a very mass
effective structure.
Importance
of supports
However, the total reaction at the left top support may be a limiting factor. We
therefore try to optimize the design without this support possibility, and then obtain
the result in figure 3.15. Now, also the optimal design is very much increased in mass.
Note, that again in this optimal design we find repeated sub-structures, and in the
mid-part of the upper structure short columns appear. The conclusion is that the
cost and possibility of supports must be carefully evaluated, before the truss design
is decided.
It is concluded that parameter studies are necessary and parameter studies based
on optimal designs are more conclusive than parameter studies based on non-optimal
designs.

Design of trusses 43
Figure 3.14: Half of a high beam with uniform loads at the lower joints, two left joints
as possible supports and two right joints at symmetry line. Self-weights included.
Initial size optimized design (mass = 897 kg) at the top and optimal topology design
(mass = 195 kg) at the bottom. Solid cross-sections corresponding to α =0.3.
Figure 3.15: Half of a bridge with uniform loads at the lower joints, only lower left
joint fixed and two right joints at symmetry line. Self-weights included. Initial size
optimized design (mass = 1320 kg) at the top and optimal topology design (mass =
933 kg) at the bottom. Solid cross-sections corresponding to α =0.3.

Combined
44 Copyright c○Pauli Pedersen: Optimal designs ...
3.3 Size, supports and topology for
optimal single load case space trusses
The theoretical formulation in chapter 16 and the computer program based on this
is for three dimensional trusses, and in section 3.2 just applied to plane trusses. The
main challenge for space trusses is to present in graphics the results. For this reason
only one example is presented here.
3.3.1 A tower
A tower space truss is shown in the perspective drawing in figure 3.16. The height of
the tower is 16 meter and the assumed single load case consists of horizontal forces of
magnitude 10000 N at each of the three top joints, all forces in the same direction. The
further data are as for the plane trusses given in section 3.1.1 and the cross-sectional
parameter α is set to α =1.
This space truss has 15 joints and 39 bars which must be chosen out of 105 possible
bars. The resulting optimal truss only gives 15 active bars and 24 ”zero strings”. This
is often the case when only a few joints are subjected to external forces.
As expected, this optimization results in two tensile bars connected directly to
the foundation. The mass of the initial, size optimized topology is 373 kg, while
the optimized topology returns a structure of 255 kg. The example is taken from
(Pedersen 1993).
3.4 Size, shape and topology for optimal 2D-trusses
The size and topology optimized designs in the previous sections of this chapter are
obtained by a fully stressed design, and topology optimization is obtainable due to
the basic knowledge of a statically determined truss, proven in chapter 16.
shape design This holds independent of the position of the actual joints and it is thus possible
to combine the size and topology optimization with optimization of positions for joints
that do not have a fixed position. This combination with shape optimization is applied
in (Pedersen 1969) for cantilever trusses as well as for bridge trusses, corresponding
to the cases in sections 3.2.1 and 3.2.3.
The extended formulation where a number of joint positions are also treated as
design variables thus includes shape optimization. The solutions are obtained by
Two level
iterative
optimization
iterative redesign where improved joint positions by analysis, sensitivity analysis and
linear programming, are followed by a new size and topology optimization. Optimal
designs for the two specific cases in sections 3.2.1 and 3.2.3 are presented.

Design of trusses 45
Figure 3.16: Perspective drawing of tower space truss. Left: initial topology resulting
in 373 kg. Right: optimal topology resulting in 255 kg. Cross-sectional pipes
corresponding to α =1.

Influence of
number of
joints
Better
46 Copyright c○Pauli Pedersen: Optimal designs ...
3.4.1 The cantilever truss with 7 to 12 joints
With reference to the optimal design in figure 3.3 that by size and topology optimization
resulted in a total mass of 685 kg, we optimize also the positions of the three
”free” joints and obtain now a total mass of 582 kg. This design is shown in figure
3.17 together with solutions for alternative number of free joints.
We note in figure 3.17 that the improvement from 5 to 6 free joints is larger
than the improvements both from 3 to 4 free joints and from 4 to 5 free joints. The
explanation is that with 6 free joints a third joint can be supported directly from
the fixed upper supports, which with ”expensive” compressive bars is of dominating
influence.
The decision of the number of joints must be taken on the basic of cost of joints,
and this is not included in the present formulation.
3.4.2 The bridge truss with 16 joints
Similar to the extension of the cantilever problem, the bridge problem described in
figures 3.11 and 3.12 is also optimized with respect to joint positions, and figure 3.18
shows the result. The general tendency in the shape optimization is that the free
joints move closer to the supports and that the height of the structure increases. This
increase in height is a balance between longer bars and the possibility to get the forces
more vertically into the supports.
force flow Figure 3.18 shows the free joint coordinates xi and the actual values can be found
in figure 3.19. Relative values are given for different total length of the bridge, and
the resulting masses as function of the total load and of the total length can be found
in (Pedersen 1969).
3.5 Size and shape for optimal multi-purpose plane
trusses
Also with shape optimization the optimal designs are fully stressed designs, and topology
optimization is possible. The severe limitation behind these solutions is the
assumption of only one load case, and the fact that only strength constraints are
Not statically considered although also local stability of each bar is accounted for.
determined With multiple load cases and/or with displacement constraints the knowledge of a
statically determined optimal truss is lost. We are therefore forced to use the general
procedure described in chapter 20, and the topology design problem can not be solved
as done previously in the present chapter. In the following sections of this chapter
we therefore restrict to size and shape optimization. On the other hand all kind of

Design of trusses 47
1136 kg
756 kg
582 kg
520 kg
473 kg
401 kg
Figure 3.17: Optimal topology and shape design with different number of free joints.
Same problem as in figure 3.3. Tensile members with thin lines, and compressive
members with thicker lines.

Not fully
stressed
Results with
different
assumptions
48 Copyright c○Pauli Pedersen: Optimal designs ...
(x3,x4)
(x7,x8)
(x11,x12)
(L/2,x16)
Figure 3.18: Optimal shape design compared to the parabolic initial shape design.
The optimal topology is unchanged, but the new sizes decrease the total mass from
4157 kg to 3852 kg.
practical constraints can then be involved, and in fact the more constraints, the more
stable is the optimization process normally.
As specific examples we again choose a bridge and a cantilever for the plane trusses
and for space trusses a dome problem.
Figure 3.20 shows a 12 joint bridge subjected to five equal forces, that act independently,
say to model a moving load. This problem is solved in different versions, with
and without local stability constraints, with and without displacement constraints.
The optimal shape solutions are collected in figure 3.21.
For comparison we list the results with only size optimization for the shape in
figure 3.20. The following total masses are obtained (in the parentheses relative to
the optimal design in figure 3.21):
• without stability constraints and without displacement constraints 1876 kg
(+13.3 %) for a fully stressed design and 1868 kg (+12.8 %) for an optimal
size design.
• without stability constraints but with maximum displacement constraint 3730
kg (+28.1 %) for a proportional change of the optimal design without displacement
constraint and 3526 kg (+21.1 %) for an optimal size design.
• with stability constraints but without displacement constraints 3141 kg (+8.1
%) for a fully stressed design equal to the optimal size design.
• with stability constraints and with maximum displacement constraint 4030 kg
(+21.6 %) for a proportional change of the optimal design without displacement

Design of trusses 49
0.35
0.30
0.25
0.20
0.15
0.10
0.07
xi/L (relative position)
i = 11
i = 16
i = 12
i = 7
i = 8
i = 4
i = 3
16 24 32 48 64
L meter
Figure 3.19: Optimal shape design for different total lengths of the bridge, given as
relative positions of the free joints with numbering according to figure 3.18
1st 2nd 3rd 4th 5th
Figure 3.20: Initial parabolic shape design for a multi-purpose bridge truss with
fixed topology. Five independent load cases.

50 Copyright c○Pauli Pedersen: Optimal designs ...
constraint and 3668 kg (+10.6 %) for an optimal size design.
3.6 Size and shape for optimal multi-purpose space
trusses
A dome space truss problem is solved in (Pedersen 1973) and we only comment the
different solutions, which are shown in figures 3.22 and 3.23.
Figure 3.22-left shows a paraboloid shape design for a dome with a diameter of 30
meter. This dome is assumed to be loaded by four vertical load cases (forces toward
the plane of the supports):
1) a force at the central joint # 1 with the value 300000 N.
2) at all the free joints, each force has the value 30000 N.
3) a force at the central point # 1 with the value 150000 N, and forces at the joints
# 4 and 5, each force has the value 100000 N.
4) a force at the central point # 1 with the value 150000 N, and forces at the joints
# 2, 3 and 4, each force has the value 70000 N.
For fixed paraboloid shape, the optimal design of the bar areas results in a fully
stressed design, i.e. at least one bar in each linked group is stressed to the allowable
limit for at least one of the loading conditions. Such a design is easily obtainable
by four to five fully stressed iterations, where the mutual connections are neglected.
For heights of the paraboloid shape equal to 6, 9, 12, and 15 meter, the mass of
the optimal size domes is 7247, 6679, 6929 and 7638 kg, respectively. The optimal
paraboloid shape design is the one shown in figure 3.22-left with height 9.25 m and
optimal mass 6675 kg.
Figure 3.22-mid shows the optimal topology not restricted to the paraboloid
shape. The total mass for this design is 5623 kg, i.e. the paraboloid shape implies
18.7 % more mass. For this comparison, it must be noted that the load systems
are assumed design independent, i.e. in the structure in figure 3.22-mid the forces
acting on the joints # 2 to 13 are closer to the supports, but the more important
forces acting on joint # 1 have fixed horizontal position.
An objection to the design in figure 3.22-mid may well be that it is too high (11.34
meter), and therefore the problem is optimized with a maximum height of the 9.25 m
as the result in figure 3.22-left. We then get the design in figure 3.22-right with a total
mass of 5733 kg, i.e. only 1.7 % more than the result without the height constraint.
We also optimize the dome with a constraint on displacements. The maximum
vertical displacement of joint # 1 for the designs in figure 3.22 are 0.026 m, 0.026 m,
and 0.032 m, respectively. If we only allow this displacement to be 0.01 m we find
the optimal design in figure 3.23-left. The total mass for this design is 5989 kg, and

Design of trusses 51
.
.
.
.
(0, 7.53)
1656 kg
No stability and displacement constraints
(0, 8.64)
(5.53, 7.80)
(6.33, 6.72)
(11.53, 4.37)
2911 kg
No stability constraints, maximum displacement 0.01 m
(10.65, 5.05)
(0, 5.00) (6.29, 4.85)
(11.38, 2.66)
2905 kg
Stability constraints, without displacement constraints
(0, 7.39)
(5.78, 6.53)
(10.81, 4.37)
3315 kg
Stability constraints, maximum displacement 0.01 m
Figure 3.21: Optimal size and shape design of a multi-purpose truss, subjected to
five load cases. The topology is fixed and not optimized.

52 Copyright c○Pauli Pedersen: Optimal designs ...
Figure 3.22: Optimal size and shape design of a multi-purpose dome space truss,
subjected to four load cases. The topology is fixed and not optimized. Top: Enlarged
picture to see the linking. Left: paraboloid shape resulting in 6675 kg. Mid: optimal
shape without height constraint resulting in 5623 kg. Right: optimal shape with
height constraint resulting in 5733 kg.

Design of trusses 53
Figure 3.23: Optimal size and shape design of a multi-purpose dome space truss,
subjected to four load cases. The topology is fixed and not optimized. Left: optimal
shape with α =1.0 5989 kg. Mid: optimal shape with α =0.5 9832 kg. Right:
optimal shape with α =1.5 4242 kg.

54 Copyright c○Pauli Pedersen: Optimal designs ...
compared with the design in figure 3.22-mid we have reduced the actual displacement
by a factor of 2.6 by an increase of only 6.5 % in the total mass.
The optimal designs fulfill a balance between a favourable transmission of external
forces to the supports (which normally give rise to rather high structures), and the
influence of local stability which tends to shorten the bars. Therefore, the optimal
design depends on the characteristics of the bar cross-section, here described by the
parameter α. The dome results shown so far correspond to α =1.0. In figure 3.23-mid
the optimal design is given for α =0.5 with increased total mass to 9832 kg, and in
figure 3.23-right the optimal design is given for α =1.5 with decreased total mass to
4242 kg.
The optimal design for α =1.0 in figure 3.22-mid is in no way a design ”between”
the optimal design for α =0.5 in figure 3.23-mid and the optimal design for α =1.5
in figure 3.23-right. As we have no guaranty of global optima, a further ”test of
the designs” was performed by optimizing based on α =1.0, starting from both the
optimal designs corresponding to α =0.5 and α =1.5. In both cases we again ended
up with the result of figure 3.22-mid.
The same structural dome problem is in chapter 5 modelled as a frame structure
with account for the much more complicated stress distributions.
3.7 A large space truss with constraints on
eigenfrequencies, displacements, stresses and buckling
Section to be included, based on (Pedersen and Nielsen 2002)
3.8 Approach to topology optimization for
multi-purpose space trusses
Section to be included, based on (da Silva Smith 1997)

Chapter 4
Continua of
uniform energy density
4.1 General comments
In this chapter we show a number of continua designed to obtain uniform (or almost
uniform) energy density, restricted to only one load case. From chapter 14 we will
know that these designs are optimal with respect to stiffness as well as with respect
to strength, when the total volume is not changed.
Stiffness is a global measure for the whole continuum and is here set equal to the
total elastic energy, which in itself is equal to the work of the external loads. The
objective in stiffness optimization is to minimize the stored elastic energy, equivalent
Only single
load case
to minimization of the work of the external forces, which is often named compliance. Stiffness
Strength is a local measure and thus focus is put on the most critical point(s) in a
continuum. As a general measure for a critical quantity we choose the elastic energy
density, and thus strength optimization is to minimize the maximum elastic energy
density, located in the continuum. If uniform energy density is obtained, then the
compliance is stationary, see chapter 12.
4.1.1 Design variables and design constraints
A continuum example is shown in figure 4.1, and with reference to this figure we
define some quantities that are general for the different examples. The continuum is
modelled by finite elements, and absolute quantities depend on the chosen model, thus
we mainly present results by relative quantities and keep the finite element modelling
55
and strength

56 Copyright c○Pauli Pedersen: Optimal designs ...
Figure 4.1: A cantilever shown as a continuum example, modelled by finite elements.
Total volume ¯ V and total area Ā are kept constant during optimization. (Isolines for
energy densities shown for a uniform thickness field with the indicated uniform line
load).
constant during optimization. The boundary conditions are indicated by green (gray)
arrow heads, thus the cantilever in figure 4.1 is completely fixed at the left boundary.
The actual load condition is indicated by black arrows, in figure 4.1 a uniform
line-load on the top boundary. Blue (dark) contour line shows the non-deformed, and
red (gray) contour line shows the deformed contour (magnified). Total volume ¯ V and
total area Ā are kept constant, so, the mean thickness Given
¯t is
volume ¯t = ¯ V/ Ā and ¯ V =
(4.1)
Thicknesses
or densities
Side
constraints
where te is the thickness in element (domain) e and area ae is the corresponding area.
Optimization is started from a uniform thickness field te = ¯t.
The actual design variables are then te, giving the thickness field. We may as an
alternative take the relative volume densities ρe as design variables, and this would be
the appropriate quantities for 3D-problems. However, in this chapter we only show
2D-problems. In addition to the global constraint on given volume, we have local side
constraints
αmin¯t = tmin ≤ te ≤ tmax = αmax¯t (4.2)
with typical relative values being αmin =0.001 and αmax = 10. The lower limit has
very little influence on the results and is seen as ”holes”, while the upper relative limit
may be responsible for not reaching a complete uniform energy density. However, this
limit is necessary because large gradients in thickness design put severe limitations
to the accuracy of the finite element modelling, and often such a constraint is also
important from a more practical point of view.
e
teae

Continua of uniform energy density 57
4.1.2 Presentation of results
In addition to graphical illustrations of thickness fields and fields of strain energy
density, a few relative numbers are needed to clarify the obtained results. The starting
analysis with a uniform thickness design gives an initial total elastic energy U0, from
which an initial mean energy density umean0 = U0/ ¯ V is given. Also this starting
analysis gives a maximum energy density umax0, and an energy concentration factor,
defined as umax0/umean0.
The total elastic energy of the optimized design is U and the goal of optimization
is to obtain uniform energy density umax = umean = U/ ¯ V , from which follows that
a gain factor for improvement in stiffness can be defined as umean0/umean = U0/U,
and a gain factor for improvement in strength can be defined as umax0/umax. Reference
Especially the upper bound tmax should also be seen in relation to the reliability
of the finite element analysis. When the gradients of the thickness field get extended,
the accuracy in modelling may be violated.
The effect of the upper bound is that in most cases umax/umean > 1 results even
for the optimized continuum that gives almost uniform energy density. A good measure
for the influence from the upper bound tmax is given by this ratio umax/umean.
The procedure for obtaining the optimized designs is rather simple and is described
in section 4.11.
values
4.2 Bars, beams and beam-bars in 2D-formulation
Axial load
or bending
Bars, beams and beam-bars are effectively analysed by one dimensional models when a
number of conditions are satisfied. These conditions relate to slenderness, to simplicity
in design, and to loads giving rise to a not too complicated displacement field.
With two dimensional modelling we can analyse more complex bars, beams and
beam-bars, but naturally also with limitations, before three dimensional modelling is
necessary. The name bar merely reflects that the axial stress flow is of major concern.
With 2D-modelling we can also analyse and design very short beams, where also the
name of beam is questionable. Slender
Figure 4.2 illustrates nine problems, for which we show thickness fields, that return
almost uniform energy density. The pure beam problems and the pure bar problems
might be solved with only half the models due to symmetry and skew-symmetry,
but due to the bar-beam problems full models are used. The actual finite element
models have 11042 degrees of freedom and 10800 design variables (thicknesses of 10800
or short
triangular elements). Finite element
models
Immediately, a suggested design for the bar problem would be a uniform bar
only of the width corresponding to the area of uniform external load. However, such
a design is questionable at the support where stress concentration can result. The

58 Copyright c○Pauli Pedersen: Optimal designs ...
isolines in figure 4.2 for this case illustrate the complexity. Also the Saint-Venant
principle is nicely illustrated in figure 4.2, and in the solutions to follow.
Figure 4.2: The basic models for optimization of bars, beams, and beam-bars in
three different lengths, i.e. in combination 9 cases. Isolines for energy densities are
shown for the uniform thickness design with the loads shown at the free ends. The
colour scale is red (gray) for low values, green for medium values and blue/violet
(dark) for large values.

Continua of uniform energy density 59
4.3 Bars: long, medium and short
Figure 4.3 shows the results for a rectangular domain, which is fixed at the left end
and loaded at the free right end, only at the mid third part and in the length direction,
i.e. orthogonal to the supports.
The optimized thickness design at the top of figure 4.3 is illustrated by isolines
together with five cross-sectional ”cuts” that more clearly show the thickness distribution.
The maximum thickness is set to ten times the average thickness (initial
design), but is not reached.
The mid figure illustrates directions of principal stresses for the optimized design,
where isolines of energy density disappear due to the uniformity in design domain.
The bottom figure shows the isolines for energy density distribution in the uniform
Only axial
forces
initial design, added the principal stress directions. Theoretical
For a design of uniform thickness (only in the part directly from the uniform
external stress σ) a finite element model results in total elastic energy Uref = ¯ Vσ 2 /E
with also uniform energy density uref = σ 2 /E, where E is the modulus of elasticity.
Such a design can result in stress concentration at a support, which is wider than the
width of a uniform thickness, rectangular domain, see the discussion in section 4.9.
The obtained optimal designs give values which, within a few tenth of a percent, are
equal to these reference values.
In table 4.1 values for the initial design with uniform thickness over the whole
rectangular domain, relative to the values for the optimal design are given. Compliance
U0/U =1.18 is thus 18% more for the initial design and the stress concentration
umax0/umean0 =11.8 is more than a factor of ten, with the actual value depending
on the specific finite element model.
For the optimized design with thickness design shown at the top in figure 4.3,
the energy density field (shown in the mid of figure 4.3) is almost uniform and the
relevant ratios are umax0/umax =13.8 and umax/umean =1.006. With the same
finite element mesh the maximum energy density is thus decreased with a factor of
14.9, corresponding to a stress decrease with a factor of 3.9 (= √ 14.9). A uniform
stress distribution is obtained.
Figure 4.4 shows the similar results for a shorter rectangular domain and for a
squared domain, respectively. In table 4.1 we have collected the important ratios
for energy (stiffness) and for energy density (strength). In general we note that the
optimization has a stronger influence for the shorter domains.
With uniform thickness over the full design domain, the compliances are 18%,
27%, and 54% too high, and the energy concentrations are factors of 14.9, 14.0, and
13.8 too high, respectively for the different lengths of the design domain. With optimal
thickness design we obtain in all three cases uniform energy density within less than
1% (umax/umean =1.006).
reference

60 Copyright c○Pauli Pedersen: Optimal designs ...
Figure 4.3: A bar with a 3 × 9 rectangular design domain. Top: resulting optimized
thickness design. Mid: corresponding resulting field of energy densities (almost uniform).
Bottom: the field of energy densities for a uniform thickness design. The black
”hatch” is an attempt to show the principal stress directions.

Continua of uniform energy density 61
Figure 4.4: Bars with a 3 × 3anda3× 6 rectangular design domains. Top: resulting
optimized thickness designs. Mid: corresponding resulting fields of energy densities
(almost uniform). Bottom: the fields of energy densities for uniform thickness design.
The black ”hatch” is an attempt to show the principal stress directions.

Combined
bending and
axial forces
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Designs Uniform thickness Optimized design
Resulting ratios U0/U umax0/umean0 umax0/umax umax/umean
Design domain 9 × 3 1.18 11.8 14.9 1.006
Design domain 6 × 3 1.27 11.1 14.0 1.006
Design domain 3 × 3 1.54 9.8 13.8 1.006
Table 4.1: Resulting ratios for energy and energy density, corresponding to three
different lengths of rectangular domains, subjected to tension external load. For each
case the results for two designs, which are uniform thickness design and optimized
thickness design.
4.4 Beam-bars: long, medium and short
Figure 4.5 shows the results for a rectangular domain, which is fixed at the left end
and loaded at the free short end, only at the mid third part. The load is in the length
direction and in the transverse direction, i.e. relative to the previous case we have
added bending, with equal magnitudes in the two directions.
Bending has the dominating influence and for this case it is advantageous to
use the full width of the design domain. We compare again the uniform thickness
solution with the optimized thickness design. For shorter design domains the results
are presented in figure 4.6, and all the numerical obtained ratios are contained in
table 4.2. The illustrations in figures 4.5 and 4.6 are arranged in the same manner as
used for the bars.
Designs Uniform thickness Optimized design
Resulting ratios U0/U umax0/umean0 umax0/umax umax/umean
Design domain 9 × 3 2.15 10.8 7.1 3.3
Design domain 6 × 3 1.87 11.2 5.9 5.9
Design domain 3 × 3 1.60 10.4 3.5 4.8
Table 4.2: Resulting ratios for energy and energy density, corresponding to three
different lengths of rectangular domains, subjected to combined tension and bending
external load, i.e. a non-symmetric case. For each case the results for two designs,
which are uniform thickness design and optimized thickness design.
We notice that the limitation of tmax =10tmean implies that even for the optimized
designs we get an energy concentration (ratios 3.3, 5.9 and 4.8), and in general
depending upon the actual finite element modelling, because of the high gradients of
thicknesses.

Continua of uniform energy density 63
Figure 4.5: A beam-bar with 3 × 9 rectangular design domain. Top: resulting
optimized thickness design. Mid: corresponding resulting field of energy densities
(almost uniform). Bottom: the field of energy densities for a uniform thickness design.
The black ”hatch” is an attempt to show the principal stress directions.

64 Copyright c○Pauli Pedersen: Optimal designs ...
Figure 4.6: Beam-bars with 3×3 and 3×6 rectangular design domain. Top: resulting
optimized thickness designs. Mid: corresponding resulting fields of energy densities
(almost uniform). Bottom: the fields of energy densities for uniform thickness design.
The black ”hatch” is an attempt to show the principal stress directions.

Continua of uniform energy density 65
4.5 Beams: long, medium and short
Figure 4.7 shows the results for a rectangular domain, which is fixed at the left end
and loaded at the free short end, only at the mid third part. The load is in the Only bending
forces
transverse direction as a bending load. As for the two previous cases the results
for shorter design domains are also shown, here in figure 4.8. The illustrations in
figures 4.7 and 4.8 are arranged in the same manner as used for the bars and for the
beam-bars.
Table 4.3 gives the ratios for the corresponding numerical results. The similarities
in these numbers with the combined tension bending case are striking, although the
optimized designs are quite different, being symmetric for the pure bending case.
Designs Uniform thickness Optimized design
Resulting ratios U0/U umax0/umean0 umax0/umax umax/umean
Design domain 9 × 3 2.12 10.1 3.8 3.1
Design domain 6 × 3 1.80 10.5 5.5 3.4
Design domain 3 × 3 1.40 9.9 6.9 3.6
Table 4.3: Resulting ratios for energy and energy density, corresponding to three
different lengths of rectangular domains, subjected to bending external load. For each
case the results for two designs, which are uniform thickness design and optimized
thickness design.
4.6 ”Bridge”s with different load distributions and
supports
Nine
problems
Figure 4.9 shows the results for a model of half a continuum bridge, i.e. a rectangular
domain with one node supported at the left boundary and all the nodes at the right
boundary supported to model the symmetry. The design depends on the position of
the left supported node, in figure 4.9 at the top. The design also depends on the
distribution of the load.
We analyse three load cases, corresponding to: a) uniform loads at the full bottom
boundary, b) uniform loads at only the mid half of the bottom boundary, and c)
uniform loads at only the mid eights of the bottom boundary. We also analyse three
different support cases at the left boundary, which are a single point support at the
lower part, at the mid, and at the upper part, respectively. Totally nine cases, for
which resulting numerical ratios are presented in table 4.4. Three
figures

66 Copyright c○Pauli Pedersen: Optimal designs ...
Figure 4.7: A beam with 3 × 9 rectangular design domain. Top: resulting optimized
thickness design. Mid: corresponding resulting field of energy densities (almost uniform).
Bottom: the field of energy densities for a uniform thickness design. The black
”hatch” is an attempt to show the principal stress directions.

Continua of uniform energy density 67
Figure 4.8: Beams with 3 × 3 and 3 × 6 rectangular design domain. Top: resulting
optimized thickness designs. Mid: corresponding resulting fields of energy densities
(almost uniform). Bottom: the fields of energy densities for uniform thickness design.
The black ”hatch” is an attempt to show the principal stress directions.

68 Copyright c○Pauli Pedersen: Optimal designs ...
The results of three designs are shown in the three figures 4.9, 4.10, and 4.11. In
figure 4.9 corresponding to upper support and full load, in figure 4.10 mid support
and load at only the half center of the bridge, and in figure 4.11 lower support and
load at only an eights of the bottom boundary, concentrated on the center of the
bridge.
Designs Uniform thickness Optimized design
Resulting ratios U0/U umax0/umean0 umax0/umax umax/umean
Lower full load 2.12 10.1 6.9 3.1
support half load 1.80 15.7 5.5 5.1
eights load 1.40 29.8 3.8 10.9
Mid full load 2.24 223.2 76.1 6.6
support half load 2.19 120.0 62.1 4.2
eights load 2.15 100.2 56.0 3.9
Upper full load 2.23 224.0 75.5 6.6
support half load 2.19 120.0 62.1 4.2
eights load 2.05 23.0 12.0 3.9
Table 4.4: Resulting ratios for energy and energy density, corresponding to nine
different support and load cases of a bridge. For each case the results for two designs,
which are uniform thickness design and optimized thickness design.
Especially the energy concentration values in table 4.4 depend on the chosen finite
element model, in particular due to the single point support, and also depend on the
chosen tmax =10tmean. The modelling is the same for all nine cases, and thus the
relative ratios give information.
We notice that the improvement in compliance is almost a factor 2 (first number
column), with less gained for the more concentrated load and lower support. Energy
concentration is for the optimized designs brought down to values from 3 to 11 (last
number column). The energy concentration factors for the initial uniform thickness
design is described in second and third column, but as mentioned these values are
related to the very detailed modelling (25000 elements) and the actual point supports.

Continua of uniform energy density 69
Figure 4.9: A bridge model with upper left support and uniform loads. Top: resulting
optimized thickness designs with red (dark) inserts to clarify the thickness design.
Mid: corresponding resulting field of energy densities (almost uniform) with rough
illustration of principle stress directions. Bottom: the field of energy densities for
uniform thickness design.

70 Copyright c○Pauli Pedersen: Optimal designs ...
Figure 4.10: A bridge model with mid left support and concentrated loads on mid
half of the span. Top: resulting optimized thickness designs with red (dark) inserts
to clarify the thickness design. Mid: corresponding resulting field of energy densities
(almost uniform) with rough illustration of principle stress directions. Bottom: the
field of energy densities for uniform thickness design.

Continua of uniform energy density 71
Figure 4.11: A bridge model with lower left support and rather concentrated loads
on mid eights of the span. Top: resulting optimized thickness designs with red (dark)
inserts to clarify the thickness design. Mid: corresponding resulting field of energy
densities (almost uniform) with rough illustration of principle stress directions. Bottom:
the field of energy densities for uniform thickness design.

Approximately
optimal hole
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4.7 A ”knee” domain
Figure 4.12 shows as for the other examples at the top the optimal thickness design,
at the mid the obtained almost uniform energy density field, and at the bottom the
energy density field for the design with uniform thickness. In addition to the isolines
we here and in the remaining examples add a full colour (gray scale) display.
By the thickness optimization, we have improved the stiffness of the continuum
as given by the ratio U0/U =2.52. Furthermore, the strength is much improved with
specific ratios being umax0/umax =20.0 and umax/umean =2.15 as alternative to
umax0/umean0 =9.33
Completely uniform energy density is not obtained with an active design constraint
of tmax/tmean = 10. This maximum thickness is reached along the inner
curved boundary and except for the non-effective corner also at the outer straight
sides.
4.8 Biaxially loaded hole
Figure 4.13 shows the results of optimizing the thickness field close to a hole that
is biaxially stressed. A few initial comments are necessary before these results are
discussed.
The shape of the hole is described by an ellipse with ratio for the axes equal to the
ratio of the given external stresses, in this case 2:1. As will be discussed in chapter 7,
specifically devoted to shape design, this is the optimal shape for a very small hole (a
hole in an infinite domain), but for a finite domain, as the present, this shape is only
approximately optimal. Anyhow, the stress field, even for the uniform thickness, is
expected to be rather uniform along the boundary of the hole. This is illustrated in
bottom of figure 4.13.
However, by thickness design we can obtain almost uniform energy density in the
total domain, as illustrated in the mid figure. By the thickness optimization, we have
improved the stiffness of the continuum only little (because of the almost optimal
hole shape). The improvement is specified by the ratio U0/U =1.02. However, the
strength is improved with specific ratios being umax0/umax =2.47 and umax/umean =
1.06 as alternative to umax0/umean0 =2.26.
Completely uniform energy density is not obtained with an active design constraint
of tmax/tmean = 10. The optimal design in the figure top shows a clear
reinforcement around the elliptic boundary shape with maximum thickness at the
lower part, where the larger stresses must be released.

Continua of uniform energy density 73
Figure 4.12: A knee model with bending external stress. Top: the optimal thickness
design. Mid: resulting uniform energy density field. Bottom: Starting energy density
field corresponding to uniform thickness design.

74 Copyright c○Pauli Pedersen: Optimal designs ...
Figure 4.13: A hole model with biaxially external stress. Top: the optimal thickness
design. Mid: resulting uniform energy density field. Bottom: Starting energy density
field corresponding to uniform thickness design.

Continua of uniform energy density 75
4.9 A foundation problem
In order to relate to the bar problems studied initially, we here see how a uniform
stress load can be distributed in a kind of foundation, modelled by only half due to
symmetry. Figure 4.14 shows at the bottom the very non-uniform energy density
field for a homogeneous foundation (uniform thickness design). The characteristic
ratio is umax0/umean0 =26.4, which for the optimal thickness design is brought down
to umax/umean =1.3
The improvement of strength for the foundation can then be measured by umax0/umax =
11.78
Homogeneous
nonhomogeneous
4.10 Uniformly loaded, non-rectangular cantilever
Non-rectangular
cantilever
The cantilever used as a generic problem in figure 4.1 is finally optimized, and within
the design constraint of tmax/tmean = 10 we can obtain umax/umean =1.10, i.e. in
reality uniform energy density.
The results are presented in figure 4.15, where the top figure shows a kind of I
beam design that gets smaller toward the free tip. The green inserts illustrate the
thickness design. The mid figure shows the uniform energy density field, while the
bottom figure shows the energy density field corresponding to the uniform thickness
design.
The improvement in stiffness is described by the obtained ratio U0/U =1.86, and
the improvement in strength is characterized by umax0/umax =38.5.
4.11 The numerical procedure
As in general, optimal designs are obtained through a number of iterations. In each
iteration a finite element analysis is performed and based on the information from this
analysis (element energy densities) a better design is suggested. The simple recursive
formula in agreement with optimality criteria methods is an update of element
thicknesses by
0.8
(ue)actual
¯V
(te)new =(te)actual
(4.3)
ūactual V
with proper adjustment to the limits tmin and tmax and iteratively updating the
actual volume V , so that the volume constraint V = ¯ V is satisfied in each redesign.
Redesign
iterations

76 Copyright c○Pauli Pedersen: Optimal designs ...
Figure 4.14: Half of a foundation model with short uniform external line load (left
lower corner). Top: the optimal thickness design. Mid: resulting uniform energy density
field. Bottom: Starting energy density field corresponding to uniform thickness
design.

Continua of uniform energy density 77
Figure 4.15: A cantilever model with uniform external line load. Top: the optimal
thickness design with green (gray) inserts to illustrate the thickness design. Mid:
resulting uniform energy density field. Bottom: Starting energy density field corresponding
to uniform thickness design

Modelling
accuracy
78 Copyright c○Pauli Pedersen: Optimal designs ...
The relaxation power of 0.8 is used by most researchers but naturally it is not a
physical constant. However without this relaxation, design cycling is often observed
while with the relaxation power monotonic convergence is often obtained.
To find the optimized designs shown we have only used about 10 design iteration
for each, but this number is strongly related to the actual convergence test.
4.12 Summing up
We end this chapters with the major conclusions about the influence from maximum
thickness constraint, the direct relation to free material design and design with alternative
materials, and finally about the specific examples dealt with in this chapter.
4.12.1 Influence from the maximum thickness constraint
As mentioned before we might have used mass density or relative volume density ρe
for each element as design variables, and then naturally ρmax = 1 would be the upper
side constraint. With thicknesses te as design variables the upper side constraint must
be set from practical point of view (say manufacturing possibilities) and/or estimated
from the reliability of the finite element analysis of a continuum with large slopes in
the thickness field.
In some of the examples treated we end up with designs that call for a kind
of stiffener enforcement. Then the results for the optimized designs depend on the
chosen upper side constraint tmax, and a completely uniform energy density field is
not obtained. The measure of the maximum energy density relative to the mean
energy density umax/umean gives information about this situation.
4.12.2 Relation to free material design
All the examples are optimized for an isotropic material with zero Poisson’s ratio.
This means that the stress field and the strain field are equal, except for the constant
E, the modulus of elasticity (see chapter 11). The energy density ue in element e is
therefore given simply by
ue =(σ 2 1 + σ 2 2)e/E =(ɛ 2 1 + ɛ 2 2)eE (4.4)
with the two principal stresses (σ1,σ2)e, and the two principal strains (ɛ1,ɛ2)e.
Behind this choice of material description is the fact, that the ultimately optimal
continua are then obtained by using for each element (in reality each point) the
constitutive parameters
C1111 = Eɛ 2 1,C2222 = Eɛ 2 2,C1122 = Eɛ1ɛ2, C1212 = 0 (4.5)

Continua of uniform energy density 79
as described in detail in chapter 15.
4.12.3 Relation to non-linear and non-isotropic material
Although we have here focused on these extreme materials, the optimization procedure
would be the same for non-zero Poisson’s ratio, isotropic material and the resulting
optimized designs are not too different.
For non-isotropic materials we may combine with optimal material orientation as
described in chapter 17. Even for non-linear elastic materials, described with a power
law, the procedure is valid as described in chapter 12. So, the examples do by no
means indicate actual restrictions.
4.12.4 The specific examples
The examples in sections 4.3, 4.4, and 4.5 show the influence from rather concentrated
external loads. A single force on a continuum results in a singularity and the finite
element analysis of this depends on the specific modelling. We have chosen to apply
the external load over a finite domain, but still only part of the actual external
boundary. Away from loads a more homogeneous state is obtained and we notice this
for the longer rectangular domains.
This is reflected in the optimized designs for tension, for bending and for combined
tension bending. For the squared domain the whole continuum is in a kind of
transition zone. For the tension case we have compared with a narrowed domain that
is externally loaded over the full boundary. This idealized situation is in reality questionable
at the support, and the optimized designs based on the full domain agree
in stiffness and in strength within a less than a percent. The cases which include
bending result in designs comparable to I-beam and the maximum thickness then has
a strong influence on the obtained results.
In section 4.6 we optimized continua that transfer loads to two end supports,
and in reality these also act like beams. We find again I-beam solutions with end
modifications near the supports The examples include different load distributions
and different positions of the end support. Again the maximum thickness constraint
influences the obtained results.
Bending is also active in the ”knee” problem where stress concentration also is
active around a curved inner boundary. Large domains of minimum thickness result,
indicating that shape design at inner holes as well as at outer boundaries would be
valuable. This leads to the hole problem optimized in section 4.8, which for a biaxially
loaded rectangular domain is based on an assumed almost optimal shape of the hole
boundary. As the hole has finite size in the domain, the shape of the hole is not

80 Copyright c○Pauli Pedersen: Optimal designs ...
completely optimal, and we find that the optimized thickness design gives different
enforcement along the hole boundary.
To illustrate the complexity in transferring loads to a foundation, we optimize
a simple foundation problem in section 4.9. It is interesting to see the resulting
combination of areas of enlarged stiffness (large thickness) with areas of enlarged
flexibility (small thickness) which makes it possible to obtain almost uniform energy
density.

Chapter 5
Design of beams and frames
Bars are one dimensional structural elements from which we can construct two- and
three-dimensional trusses; beams are one-dimensional structural elements from which Structures
we can construct two- and three-dimensional frames, including grids, curved beams
and non-uniform beams.
In bars the cross-sectional points are in a uniform state of stress, strain, and energy
density, but this is not the case for beams. In beams and frames the aspect of bending
is dominating, and point-wise uniform energy density can not be obtained. However,
optimality criteria expressed in terms of finite design regions are still applicable,
of 1D-elements
and a number of important optimal design problems can be solved, if not explicit Finite design
regions
analytically, then implicit analytically.
The early literature related to structural optimization contains a large number
of specific results, which also include two-dimensional plate problems. For general
reference, see the course notes by (Prager 1974) together with the books by (Banichuk
1983) and by (Rozvany 1989). Extended lists of references are available in these Important
books.
In this chapter we, based on the theoretical results presented in chapter 14, show
a very direct way to obtain explicit analytical optimal designs and then indicate the
iterative solution to obtain implicit analytical optimal designs, but without repeating
references
the results found in the two books, just mentioned. Main contents
of chapter
We then concentrate on problems that are solved with application of methods
from mathematical programming. At first a specific example of beam design with
several eigenfrequency constraints is described in detail with focus on the difference
between using simple beam theory and taking shear deformations into account.
An example with a portal frame illustrates the inherent problem with unknown
active constraints when multiple load cases are prescribed. This problem also illus-
81

Focus on
stress energy
Optimality
criterion b
82 Copyright c○Pauli Pedersen: Optimal designs ...
trates the influence on the resulting optimal designs from given boundary conditions.
Shape optimization of frames involves positions of beam connections as design
variables. For 3D-frames this important structural design problem is presented in the
final section of this chapter. Analytical sensitivity analysis is still possible, but the
”bookkeeping” is extensive. Two specific optimal designs are shown.
5.1 Explicit analytical optimal designs
For a number of problems explicit analytical optimal designs can be obtained. The
derivation of these results is based on the optimality criterion for problems of minimum
compliance with a given mass (volume), which is stated in chapter 14, section 14.3.1
as: the ratio between sub-region energy and sub-region mass should be the same in all
the design sub-regions.
With a given ratio between strain energy, stress energy, and thus total elastic
energy we may choose what is most convenient for the actual formulation. Here we
choose stress energy. The optimality criterion also holds for non-linear elasticity,
modelled by power law non-linearity.
Taking a part of a beam, say from a to b with length (b − a), where the crosssectional
area A is to be designed (uniform over the length, which may be infinitesimal).
The sub-region stress energy Uσ for pure bending with a simple beam model
and linear elasticity is known to be
Uσ =
b
a
M 2 b
dx =
2EI a
M 2
2Eα2 dx (5.1)
A2 where M is the bending moment, E is the modulus of elasticity, and I is the crosssectional
moment of inertia, for which we assume the simple relation I = α 2 A 2 to be
valid with the same cross-sectional constant α as used in chapter 3.
With ρ as a given mass density, the mass m for the design sub-region is m =
b
a ρAdx and the ratio to be constant according to the optimality criterion (dUσ/dA) =
˜C(dm/dA), see chapter 14, is thus
a
A = C
2 M
d
2Eα 2 A 2
b
a M 2 dx
b − a
/dA dx = ˜ C
1/3
b
a
(d(ρA)/dA) dx ⇒
⇒ A = CM 2/3 for (b − a) → 0 (5.2)
where ˜ C and C are constants when 2Eα 2 ρ is assumed constant too. The constant C

Design of beams and frames 83
is determined by the given total volume ¯ V , i.e.
¯V =
L
0
Adx =
L
0
CM 2/3 ¯V
dx ⇒ C = L
0 M 2/3dx (5.3)
where L is the total length of the beam (or alternatively with the summation over
design sub-regions).
5.1.1 Statically determined elementary cases
A number of elementary cases such as cantilever beams (or frames) and simply supported
beams are statically determined, which means that the moment distribution
is determined independently of the design, and then (5.3) directly gives explicit Known
analytical optimal designs.
26% energy
39% energy
65% energy
100% energy
Figure 5.1: Left: Optimal designs with obtained compliance. Right: the corresponding
cantilever elementary load cases.
Figure 5.1 shows the optimal designs for cantilever cases where the moment distribution
is cubic dependent on x, quadratic dependent on x, linear dependent on x,
and independent on x (constant), respectively. Characterizing these four cases by the
power n, the optimal designs are
x1
A(x) =Cx 2n/3 with C determined by C
L
0
x1
x 2n/3 dx = ¯ V (5.4)
where L is the length of the actual beam. As expected, the best improvement can be
obtained for the triangular load distribution (almost a factor 4), and the pure moment
moment
distribution
load for which the uniform beam is the optimal design (no improvements possible). Alternative
The assumption of I = α 2 A 2 may not always be applicable, and alternatively we
look at the case of a rectangular cross-section with cross-sectional moment of inertia
design
variables

Explicit
optimal
beam design
Optimal
energy ratio
84 Copyright c○Pauli Pedersen: Optimal designs ...
I = bh 3 /12 and area A = bh, where the width b or the height h might be the actual
design parameters. In terms of width b and height h, the sub-region energy and
sub-region mass are
Uσ =
2 M
dx =
2EI
2 M
2Eh3 dx and m = hbdx (5.5)
b/12
with resulting optimality conditions for the two alternative optimality criteria
M 2 /b 2 = constant for given h
M 2 /h 4 = constant for given b (5.6)
Thus, the only difference relative to the solution (5.4) is that the power 2n/3 must be
changed either to 2n/2 =n or to 2n/4 =n/2, as seen by comparing the optimality
criterion (5.3) with (5.6).
In general for all the cases mentioned in this section, the optimal solution is of
the type d(x) =Cx q , where d(x) is width, height, radius, or area at position x from
the free tip of the cantilever. The power q depends on the load case as well as on the
actual design parameter.
The percentages in figure 5.1 are determined as follows directly from (5.1). For
an uniform beam with moment distribution M(x) = ˜ Mx n and area Ā = ¯ V/L the
total stress energy Ūσ is
Ūσ =
L
0
˜M 2 x 2n
2Eα 2 ¯ V 2 /L 2 dx = ˜ M 2 L 2
2Eα 2 ¯ V 2
L
0
x 2n dx (5.7)
For an optimal beam with the same moment distribution and area distribution A =
¯V
L
0 x2n/3 dx x2n/3 the total stress energy Uσ is
Uσ = ˜ M 2 ( L
0 x2n/3dx) 2
2Eα2 ¯ V 2
L
0
which together with (5.7) gives the ratio
Uσ
Ūσ
¯V
=
3
C3L2 L
0 x2ndx = (2n +1)¯ V 3
C3L2n+3 x2n dx (5.8)
x4n/3 (5.9)

Design of beams and frames 85
5.2 Implicit ”analytical” optimal designs
Two aspects prevent explicit analytically optimal designs, but implicit solutions are
then obtained by a few iterations. The first aspect is related to size constraints, due to
the missing knowledge about the regions where these size constraints are active. The
second aspect concerns statically indetermined cases, where the moment distribution
is depending on the actual design. Often both aspects must be taken into account, Simple
which can be done simultaneously, but in the description here we separate them. iterations
5.2.1 Side constraints
We notice in the optimal solutions of figure 5.1, that when the moment in a beam crosssection
is zero, the optimal cross-sectional area is also zero. To obtain more practical
solutions we impose a minimum area constraint, say Amin =0.1Amean =0.1 ¯ V/L.
The location of design regions where this constant is active is found by iteration
and the optimal design in the remaining regions is described by (5.4), which we use
as an example. In (5.4) the given volume ¯ V is substituted by the ”active” volume Active
V = volume
¯ V − Vconstraint of free design.
5.2.2 Statically indetermined cases
With statically indetermined boundary conditions, we can also use iterations to find
the optimal design, very much similar to the procedure for obtaining the designs
of uniform energy density in chapter 4, section ??. For multiple load cases and with
different constraints we can not expect the optimal design to be statically determined.
For a number of specific examples see (Rozvany 1989).
5.3 Beam design with eigenfrequency constraints
5.3.1 Sensitivity analysis for eigenfrequencies
In chapter 18 the sensitivity analysis, for dynamic problems in general, is described,
so here we concentrate on the actual beam problem, treating this as a finite element
model with a limited number of beam elements. The finite element formulation for
the structural eigenfrequencies is Problem of
analysis
[S]{D} = ω 2 [M]{D} (5.10)
where ω 2 and {D} are, respectively, the squared eigenfrequency and the corresponding
eigenvector. The total stiffness matrix [S] and the total mass matrix [M] are both

Sensitivity
result
Sensitivity for
simple beams
86 Copyright c○Pauli Pedersen: Optimal designs ...
obtained by summation of the individual beam element matrices
[S] =
[Se] and [M] =
[Me] (5.11)
e
This origin of the system matrices is essential.
Inverse iteration as described in (Bathe 1982) (and later editions), including shift,
sub-space iteration, and Sturm sequence check, provides a powerful analysis tool
which allows the user to focus on an actual frequency domain and control the rate of
convergence. Close or even equal eigenfrequencies are also taken care of.
Assuming that the eigenpair ω 2 , {D} is known, the important point to note, as
proven in chapter 18, is that this also contains the information necessary to determine
the eigenfrequency sensitivities, dω 2 /dae, i.e. the change in eigenfrequency due to a
change in the area of beam element e. The result, derived in chapter 18 and given in
(18.42) is (with the normalization condition {D} T [M]{D} =1)
dω2 = {D}
dae
T ( d[S] 2 d[M]
− ω ){D} (5.12)
dae dae
With ae being a local design parameter, this result with the assumed normalization
can be calculated on the element level with {De} contained in {D}
dω2 = {De}
dae
T ( d[Se] 2 ∂[Me]
− ω ){De} (5.13)
∂ae ∂ae
For simple beam theory we have d[Se]/dae =2[Se]/ae and d[Me]/dae =[Me]/ae, and
therefore get a result which can be described in element elastic and element kinetic
energies Ue,Te.
dω2 dae
= {De} T (2[Se] − ω 2 [Me]){De}/ae =(2Ue − ω 2 Te)/ae
e
(5.14)
With other cross-sectional design parameters than area, the only difference, relative
to (5.14), is another factor than 2 in front of Ue and a factor also in front of Te,
see section 18.3.2.
More complicated modifications of the sensitivity results are necessary when
higher order beam theories are used. They are discussed in detail in section 5.3.5.

Design of beams and frames 87
5.3.2 Optimal design with one eigenfrequency constraint
Let us first consider a simply supported beam whose volume we want to minimize
subject to the constraints that some of the eigenfrequencies must be greater than or
equal to those of the corresponding uniform beam. We choose a slender beam, and
thus the gradients (5.14) are valid. Putting constraint only on the first eigenfrequency
we obtain the design in figure 5.2. This result agrees well with the early result of
(Niordson 1965), although we do not reach zero area at the support when shear Classical
deformations are taken into account.
problem
ω1 optimal = ω1 uniform
ω2 optimal =0.928 ω2 uniform
ω3 optimal =0.910 ω3 uniform
ω4 optimal =0.902 ω4 uniform
Voptimal =0.883 Vuniform
Figure 5.2: Optimal design of slender, simply supported beam for minimum volume
with the constraint put only on the first eigenfrequency.
The volume is through the optimization minimized to 88.3% of that for the uniform
beam, and as expected the second, third and fourth eigenfrequencies are lowered.
5.3.3 Optimal design with several eigenfrequency constraints
Extended
Controlling also the second eigenfrequency, then the first three eigenfrequencies and problems
finally all four eigenfrequencies, we obtain the optimal designs shown in figure 5.3.
Solutions for other boundary conditions than simply supported, like clamped
- simply supported or flexible clamped - simply supported, and also for different
medium slender beams can be found in (Pedersen 1981).
5.3.4 Optimality criterion with only a single constraint
With only a single constraint the general optimality criterion of proportional gradients
(chapter 14 (14.4)) gives for the actual problem
dω2 dae
= C dV
= Cle
dae
(5.15)

88 Copyright c○Pauli Pedersen: Optimal designs ...
ω1 optimal = ω1 uniform
ω2 optimal = ω2 uniform
ω3 optimal =0.959 ω3 uniform
ω4 optimal =0.952 ω4 uniform
ω1 optimal = ω1 uniform
ω2 optimal = ω2 uniform
ω3 optimal = ω3 uniform
ω4 optimal =0.972 ω4 uniform
ω1 optimal = ω1 uniform
ω2 optimal = ω2 uniform
ω3 optimal = ω3 uniform
ω4 optimal = ω4 uniform
Voptimal =0.934 Vuniform
Voptimal =0.955 Vuniform
Voptimal =0.967 Vuniform
Figure 5.3: Optimal design of slender, simply supported beams for minimum volume
with constraint on several eigenfrequencies. Top: constraints on the first two eigenfrequencies.
Mid: constraints on the first three eigenfrequencies. Bottom: constraints
on the first four eigenfrequencies.

Design of beams and frames 89
where the constant C (Lagrange multiplier) is the same for all beam elements e. The
element lengths le need not be equal. Substituting the simple gradient (5.14), we get
2Ue − ω 2 Te = Caele ⇒ 2ue − ω 2 te = C (5.16)
where ue,ω2te are the elastic energy density and the kinetic energy density, respectively.
By summation (integration) we can then obtain the value of the constant C
2Ue −
ω 2 Te = C
aele = CV (5.17)
e
e
e
and with normalization
e Te =1→
e 2Ue =2ω 2 we have for the optimal design Value of
multiplier
C = ω 2 /V (5.18)
For the results of this section we have used mathematical programming as the iteration
tool, but with only a single constraint, optimality criterion iterations based on (5.16)
with estimates of C from (5.18) would also be possible. Test of
solutions
Even though the optimality criterion is not used for finding the optimal design,
it is valuable to test whether an optimal design fulfills this criterion, i.e. if it is valid.
We do this in the examples to follow.
5.3.5 Sensitivity analysis for Timoshenko beam theory
The numerical analysis of beams and frames is often based on Timoshenko beam
theory and although it is possible still to use the simplified results for sensitivity
analysis (5.14), we present the extended result, as derived in (Pedersen 1982–83). Exact
The result is
gradients
βe
1+βe
dω 2
dae
= 1
ae ({De} T (2[Se] − ω 2 [Me]){De} +
{De} T (2[Se] − ω 2 [Me]){De}− EIe
le (θ1 − θ2) 2 e
−
ω 2 {De} T [ ˜ Me]){De}) (5.19)
where θ1,θ2 are the two end rotations of the beam element. The element coefficient
of shear βe is
βe = 12µEIe
Gael 2 e
(5.20)
where E,G are the material moduli and µ is the cross-sectional constant that follows
from shear distribution. For slender beams we have βe = 0 and thus the same formula
as (5.14). The matrix [ ˜ Me] is a modified mass matrix, that is listed in (Pedersen 1982–
83), together with all the involved matrices.

Influence of
gradients
90 Copyright c○Pauli Pedersen: Optimal designs ...
5.3.6 Optimization with different gradients
The beam eigenfrequency analyses are in all the present cases based on Timoshenko
beam theory, but it is possible to perform the optimization with the simple gradients
(5.14), here termed approximate gradients or with the more complicated gradients
(5.19), here termed exact gradients.
To study the influence of slenderness and the difference between using exact gradients
(5.19) and approximate gradients (5.14) in the case of a simply supported beam,
we maximize the second eigenfrequency for a given volume, but without constraints on
the first eigenfrequency. For Bernoulli-Euler beams this problem is treated in (Olhoff
1976), and the optimality criterion (5.16) is known to be valid. Choosing a uniform
beam as our initial design, figure 5.4 presents the results for a total slenderness ratio
of 25. Based on these results we conclude:
1. The design obtained by exact gradients is better with the eigenfrequency improved
7.5% as compared to 6.9%, but for larger slenderness ratio this difference
disappears as expected.
2. All the designs agree with the optimality criterion, where (dω2 /dae)exact has
the same value for all beam elements e. Thus, we have designs which differ
only slightly, but which both are stationary.
Figure 5.4: Two optimal designs of a simply supported beam for maximum second
eigenfrequency with given volume. Thin line boundary: initial uniform design and
result with approximate gradients. Heavy line boundary: Result with exact gradients.
Taking the initial design to be uniform (symmetric design), we end up with symmetric
optimal designs, which is to be expected since every re-design problem itself is

Design of beams and frames 91
symmetric. However, taking an initial design with linear tapered thickness, we find
the optimal design in figure 5.5, which are better than the designs in figure 5.4. Again
the results fulfill optimality criterion (5.16). The improvement in eigenfrequency is
now 22.8%, but the first eigenfrequency is then close to zero. Symmetry or
non-symmetry
Figure 5.5: Optimal beam design for maximizing the second eigenfrequency with
given volume, and without constraint on symmetry. The first eigenfrequency is not
constrained and exact gradients are used for redesigns.
It is a general feature of optimal design that many local optima exist, and optimization
should therefore be attempted from different initial designs. However, it
should be noted that if we start with a good engineering design, and use the approach
of mathematical programming, then we can not get a less favourable design.
5.4 2D-Frame design with many load cases
Most structures in practice must be designed for several load cases. In relation to the
design problem, this means that a multiplicity of constraints have to be accounted
for, although most of them turn out to be non-critical. Therefore, we cannot choose
a design approach where the critical constraints have to be identified in advance.
The combination of finite element analysis and sequential linear programming
(SLP) constitutes an effective and reliable approach for these practical optimal designs.
We show and discuss the resulting design of a portal frame for a crane, where
the moving load is modelled by a number of single forces, acting independently, i.e.
multiple load cases (actually 15 cases).
Again sensitivity analysis plays a major role and analytical gradients are obtained.
Figure 5.6 shows a model of the portal frame and the constraint of a straight upper
Multiple
load cases

Eccentricity
Reference
stress
92 Copyright c○Pauli Pedersen: Optimal designs ...
surface means that eccentricities need to be taken into account. (In frame analysis
for ships this is also often the case.) Detail of the involved matrices can be found in
(Pedersen and Jørgensen 1984).
Figure 5.6: Model of a portal frame with one of the many load cases.
5.4.1 Design problem, analysis and sensitivity analysis
The objective of design is to minimize the total mass Φ
Minimize Φ=
cehe with given ce := ρelebe
e
(5.21)
where the heights he are the design variables while lengths le, widths be and mass
densities ρe are given.
The constraints of the problem are specified as a limit on the squared von Mises
stress (Fe)l in all elements e for all load cases l, i.e.
(Fe)l = 1
2 {σe} T l [Z]{σe}l ≤ Fmax
(5.22)
where {σe}l is the stress vector in element e for load case l and [Z] is a symmetric
matrix of integer constants. The specified limit is Fmax. The stress vector we evaluate
from
{σe}l =[Qe]{De}l
(5.23)

Design of beams and frames 93
where [Qe] is the element stress-displacement matrix and with element displacements
{De}l from the system equilibrium of load case l
[S]{D}l = {A}l
(5.24)
where {A}l constitutes the load case l.
The finite element analysis locates the critical (say within 5% from Fmax) stress
elements and corresponding load cases. For these we perform sensitivity analysis by
the chain rule of differentiation Chain rule of
differentiation
d(Fe)l/dhi =[Z]{dσe/dhi}l
{dσe/dhi}l =[dQe/dhi]{De}l +[Qe]{dDe/dhi}l
[S]{dD/dhi}l = {dA/dhi}l − [dS/dhi]{D}l
(5.25)
where [dQe/dhi] =0fore = i. Treating {dA/dhi}l − [dS/dhi]{D}l as a pseudo-load
we get {dD/dhi}l by what is named the semi-analytical approach. Normally this does
not give rise to numerical problems, but for possible errors see (Pedersen, Cheng and Semi-analytical
Rasmussen 1989).
The solution by SLP follows from the general description in chapter 19, and we
can go directly to the results.
5.4.2 Only stress constraints and influence from stiffness of
columns
The portal frame is modelled as a beam with boundary conditions given by the actual
columns, that influence the solution. Three different columns are applied as examples.
sensitivities
Timoshenko beam theory with Poisson’s ratio ν =0.3 and the coefficient of shear Timoshenko
distribution µ = (12 + 11ν)/(10 + 10ν) is the basis. A discretization in 30 uniform,
eccentrically connected elements of equal length is used and symmetry is enforced to
be able only to encounter 15 heights as design variables and 15 load cases, as figure
5.7 shows.
When the bending stiffnesses of the portal columns are large in relation to that of
beam theory
the portal beam, then the portal model corresponds to fixed-fixed beam, and when the Columns
portal columns are very flexible, the portal model corresponds to a simply supported
beam, i.e. a statically determinate structure. The optimal designs for three models
with increasing stiffness of the columns (relative stiffness of 1, 8 and 64) are shown
in figures 5.9, 5.10, and 5.11. In all cases, a uniform beam is taken as initial design,
and the characteristics of optimal design (mass M and squared reference stress F )
are given relative to those of the uniform beam. Only stress constraints are involved
in these solutions, and the allowable Fmax is taken as the maximum value for the
uniform beam.
at beam
boundaries

Initial
designs
94 Copyright c○Pauli Pedersen: Optimal designs ...
1 2 3 4 5 6 7 8 9101112131415
Figure 5.7: Design model of a portal frame with numbers on the 15 different load
cases.
Note that smooth stress results along the beam length are not to be expected,
because the maximum stresses may refer to different loading cases. By the numbers
(1 - 15) in the figures we have indicated the actual load case, also numbered in figure
5.7.
The optimal designs are not fully stressed, by which is meant that every beam
element is stressed to the allowable level in at least one load case. However, the case
of flexible columns is close to this condition and is discussed specifically.
Figure 5.8: Initial design models used for optimization.
The optimal designs are performed with the three different initial designs, shown
in figure 5.8 and in all cases the same optimal designs are obtained. Other initial
designs have also been tried, and everything points in the direction of the designs
being global optimal solutions, but no proof is available.

Design of beams and frames 95
Figure 5.9 shows the optimal design and the resulting maximum reference stresses
with the flexible columns. A 20% decrease in mass is obtained and the portal beam
is almost fully stressed. A fully stressed design can be found by simple fully stressed
iterations and for this design we find only slightly more mass, with small design
modifications only at the beam ends. Slender
columns
(Fe)l/Fmax
Optimal design
Moptimal/Muniform =0.805
with slender columns
Figure 5.9: Top: optimal design of portal beam on slender columns (hcolumns =
huniform/2). Bottom: Relative maximum squared von Mises stress and with numbers
of the corresponding load cases (see figure 5.7). Dotted line for the uniform beam.

Medium
slender
columns
96 Copyright c○Pauli Pedersen: Optimal designs ...
Figure 5.10 shows the optimal design and the resulting maximum reference stresses
with the stiffness of the columns increased with a factor 8. A 14% decrease in mass
is obtained and the portal beam is again almost fully stressed. If we iterate to find a
fully stressed design, then as for the flexible columns it has only slightly more mass,
with small design modifications only at the beam ends.
(Fe)l/Fmax
Optimal design
Moptimal/Muniform =0.862
with medium columns
Figure 5.10: Top: optimal design of portal beam on medium columns (hcolumns =
huniform). Bottom: Relative maximum squared von Mises stress and with numbers
of corresponding load cases (see figure 5.7). Dotted line for the uniform beam.

Design of beams and frames 97
Figure 5.11 shows the optimal design and the resulting maximum reference stresses
with large stiffness of the columns. A 12% decrease in mass is obtained and the portal
beam is again almost fully stressed. Non-slender
columns
(Fe)l/Fmax
Optimal design
Moptimal/Muniform =0.881
with heavy columns
Figure 5.11: Top: optimal design of portal beam on heavy columns (hcolumns =
2huniform). Bottom: Relative maximum squared von Mises stress and with numbers
of corresponding load cases (see figure 5.7). Dotted line for the uniform beam.

Further
designs
Size and shape
with linking
98 Copyright c○Pauli Pedersen: Optimal designs ...
5.4.3 Including displacement constraints
The presented optimal designs give a maximum displacement that is close to 20%
larger than for the uniform beam, and severe design modifications are necessary if
we add a constraint on the maximum displacement, equal to the displacement before
optimization. Still a save in mass of 16% is possible also for the case in figure 5.9.
For the case in figure 5.10 the save in total mass is 11% as compared to 14% without
displacement constraints, and for the case in figure 5.11 the corresponding percentages
are 9% instead of 12%. The changed optimal designs can be found in (Pedersen and
Jørgensen 1984).
A common conclusion from this portal frame problem is that displacement constraints
change the optimal design, but the structure generally has the ability to
adjust to the additional constraint with only a slight increase in mass.
The optimization of the portal frame is based on Timoshenko beam theory and
analytical sensitivity analysis is obtained. For slenderness ratio lower than about
20 we see a clear difference between optimal designs based on Bernoulli-Euler beam
theory and optimal designs based on Timoshenko beam theory.
5.5 3D-Frame design with optimal joint positions
This section mainly refers to the paper (Sergeyev and Pedersen 1996). The experience
of optimal design shows that the shape of a construction is a fundamental factor for
its mechanical characteristics. We thus here see the optimization of joint positions
as our primary interest, but naturally the beam optimization plays the role as an
important sub-problem, which with non-trivial beam cross-sections may be rather
complicated.
Linking design variables into groups we can take constructive requirements into
consideration. Furthermore, we may then decrease the number of load cases and
naturally we also decrease the total number of design variables. Constraints on stresses
have to account for stresses resulting from tension, bending, shear and torsion with a
definition of an effective stress measure.
At first the sensitivity of this stress measure to joint positions may seem very
complicated, but in fact the analytical sensitivity analysis is rather straight forward.
We prefer this analytical sensitivity analysis not only because of its effectiveness with
respect to computer time, but also for its robustness.
Firstly, a problem which was earlier solved in a truss formulation (Pedersen 1973)
is treated. This problem includes 52 beams and several load cases. Solutions with
different assumptions are compared and the influence of including self-weights is illustrated.
Then, a portal frame as defined in (Apostol, Santos and Goia 1995) is
optimized, again studying the influence from the basic assumptions.

Design of beams and frames 99
All solutions are found using a sequential quadratic programming algorithm, with
a possibility for including move-limits.
5.5.1 Design problem, analysis and sensitivity analysis
From a formulation point of view the present problem is almost like the 2D-frame
problem in section 5.4. The objective of design is to minimize the total mass Φ
Minimize Φ=
ρeaele =
e
k
ρkak
e
le
k
(5.26)
where the design variables are cross-sectional areas ae, and lengths le indirectly
through the joint positions xj, while densities ρk are assumed given. By index k
we denote a group of beams in order to get a formulation suited for linking.
All our design variables are scalar quantities contained in a design vector {h} that
includes cross-sectional parameters and joint coordinates. The influences of these two
different groups of design parameters are very different, but for the formulation it
is an advantage to treat them together. We may have side constraints on the size
parameters (say thickness of beam cross-sections) as well as on the shape parameters
(say a joint constrained to be in a certain domain). These side constraints are specified
by Side
{h}min ≤{h} ≤{h}max
(5.27)
For the iteration strategy it is often advantageous to specify move-limits
{∆hp}min ≤{∆hp} ≤{∆hp}max
(5.28)
constraints
indicating by index p that these may change from iteration step p to another iteration
step. Note that the absolute side constraints (5.27) can be included in the move-limit
constraints (5.28).
The more physically based constraints relate to stresses and displacements, i.e. to
strength and stiffness. As the formulation is based on multiple load cases we have to
add an index l indicating the load case and for the state of strength write constraints
like (5.22) Strength
(Fe)l ≤ Fmax for all elements e and all load cases l (5.29) constraints
The number of constraints is given by the ”nearby” critical cases. Thus, for every
beam there may be several critical points and each point may in fact be critical in
several load cases. Furthermore, during design iteration the critical cases may change
and in most problems the number of constraints increases with the design evolution.

Stiffness
100 Copyright c○Pauli Pedersen: Optimal designs ...
Another group of design constraints are related to the stiffness of the frame, as
specified by the resulting displacements {D} of the joints
{D}min ≤{D}l ≤{D}max for all load cases l (5.30)
with interpretation again in terms of active set strategy.
constraints The detail of analysis and of sensitivity analysis are given in (Sergeyev and Pedersen
1996), and especially the sensitivities to change in joint positions are rather
lengthy expressions, that should be tested against finite difference calculations.
In the following two illustrative examples are presented. A number of data are
common for all examples: Young’s modulus of elasticity E = 200 GPa, shear modulus
of elasticity G =80GPa, allowable effective stress σeff = √ Fmax = 147 M Pa,
density of mass ρ = 7799kg/m3 , and gravity acceleration in negative z-direction,
which is orthogonal to the plane of supports (the xy plane).
Specified
linking
Single
load case
5.5.2 Optimal design of a dome frame
Figure 5.12 shows a dome with linking of joint positions as well as beam areas, which
gives only 13 independent design variables. Furthermore, by this symmetry requirement,
we can deal with fewer load cases, in the following illustrated by load cases
restricted to a quarter part and a half part, respectively.
000000000
111111111
000 11100
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000 11100
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000 111 0000 1111 00 11
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0000 1111 0000 1111
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000 111
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000 111
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0000 1111
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000 1110000 1111 00 11
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0000 1111 00 11 000 111 000000
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0000 1111 00 11 0000 1111
00000
11111 000 111 01
0000 1111 5
00 11
01
000 1110000
1111 0000 1111 0000 1111
00 1101
000 1110000
1111 0000 1111
01
00 110000000
1111111
01
000 1110000
1111
01
0000 1111
0000 1111
01
Figure 5.12: Left: Illustrations of the dome topology. Right: Symmetry linking of
joint positions and their numbers together with symmetry linking of beam members
and letters for these.

Design of beams and frames 101
The first problem for the dome is a single load case, with only a force at the
central point # 1. The force is in the negative z-direction and has the value 632745
N. The resulting optimal joint positions are illustrated in figure 5.13 with numerical
values for beam thicknesses and joint positions in table 5.1.
00 11
00 11 00 11
00 11
Figure 5.13: Left: Horizontal view of the optimal dome with joint vertical positions.
Right: Projection on the xy plane of the optimal dome.
01
01
00 11
The second problem formulation has four load cases:
00 11
00 11
00 11
00 11
1. a force at the central joint # 1 in the negative z-direction with the value 300000
N.
2. at all the free joints, forces in the negative z-direction where each force has the
value 30000 N.
3. a force at the central point # 1 in the negative z-direction with the value 150000
N, and forces at the joints # 4 and 5 in the negative z-direction where each
force has the value 100000 N.
4. a force at the central point # 1 in the negative z-direction with the value 150000
N, and forces at the joints # 2, 3 and 4 in the negative z direction where each
force has the value 70000 N.
This problem is solved both without and with self-weight. The resulting optimal
joint positions are illustrated in figure 5.14 with numerical values for beam thicknesses
and joint positions again in table 5.1.
Multiple
load case
With and
without
self-weights

102 Copyright c○Pauli Pedersen: Optimal designs ...
Design Minimum Maximum Optimal values in mm
variables 1 load case 4 load cases 4 load cases
in mm no self-weight self-weight
ta 1.5 15.0 6.199 4.363 4.346
tb 1.5 15.0 1.5 1.5 1.5
tc 1.5 15.0 3.667 2.959 2.935
td 1.5 15.0 1.677 2.401 2.476
te 1.5 15.0 1.5 1.5 1.5
tf 1.5 15.0 2.968 1.607 2.375
tg 1.5 15.0 1.5 1.5 1.5
th 1.5 15.0 1.5 1.5 1.5
z1 7000. 11000. 11000. 11000. 11000.
x2 18000. 22500. 18000. 19390. 20281.
z2 6000. 10000. 8430. 8636. 8178.
x6 23000. 27500. 23000. 23000. 24350
z6 3000. 7000. 4305. 5583. 4613.
Mass kg 1887. 1703. 1875.
Table 5.1: Values for side constraints and resulting optimal dome designs for three
problems, differing only in applied loads.

Design of beams and frames 103
Including self-weight does not change for this case the optimal design much, but
in figure 5.14 we see a clear tendency for the joints to move closer toward the supports
(down and out).
00 11
00 11
00 11
00 11
00 11
00 11 0 0 1 1
00 11
Figure 5.14: Top-left: Horizontal view of the optimal dome with joint vertical positions.
Top-right: Projection on the xy plane of the optimal dome, without selfweights.
Bottom: the same but with self-weight of the optimal frame.
01
01
01
01
01
01
00 11
5.5.3 Optimal design of a mobile crane frame
00 11
00 11
A model of a mobile crane structure is suggested by (Apostol et al. 1995), and figure
5.15 shows the model with numbers and letters for the design variables. We optimize
01
01
01
01
01
01
01
01

Linking and
no-linking
104 Copyright c○Pauli Pedersen: Optimal designs ...
this structure in two versions, first with linking given only 9 independent design
variables and then without linking resulting in 22 independent design variables. Only
the ”roof” is subjected to optimization.
a
c
e
Figure 5.15: Mobile crane frame, with initial joint positions.
The loads are all acting in the negative z-direction at the six joints at the top.
The actual value at these joints are 123607 N , i.e. total load of 741642 N. Figure
5.16 shows the resulting optimal shapes of the mobile crane frame with actual design
values given in table 5.2.
Figure 5.16: Optimal shapes of mobile crane frame. Left: with linking to enforce a
straight roof line. Right: without linking.
From these 3D-frame design problems it is clear that by the shape design of space
frames in addition to the beam cross-sectional design we have gained a lot in the
possibilities for mass minimization.
For frames the shape parameters are the positions of the joints where the involved
beams meet. We have seen that the sensitivity analysis with respect to these joint
positions are much more involved than the sensitivity analysis with respect to the
b
1
d
f
2

Design of beams and frames 105
Design Minimum Initial Maximum Optimal values in mm
variables Without With
in mm linking linking
ta 0.8 8.0 15.0 5.54 5.73
tb 0.8 8.0 15.0 4.41 3.62
tc 0.8 8.0 15.0 1.32 1.36
td 0.8 8.0 15.0 1.08 1.18
te 0.8 8.0 15.0 1.70 1.16
tf 0.8 8.0 15.0 0.94 0.98
z1 4500. 5000. 6000. 5099. 5521.
y1 300. 1000. 2200. 545. 548.
x2 500. 1000. 4700. 500. 500.
z2 4500. 5000. 6000. 5531. 5521.
Mass kg 2457. 1679. 1684.
Table 5.2: Values for side constraints, initial design and resulting optimal crane
designs for two problems, differing only in linking.
cross-sectional beam parameters. This sensitivity analysis is done analytically, and
we obtain confidence and computational advantages at the same time. Design dependent
loads, like temperature loads and inertia loads, are included in a more general
formulation.

106 Copyright c○Pauli Pedersen: Optimal designs ...

Chapter 6
Design of laminates and
plates
In the science of composites the problems of major importance are manufacturing
and strength estimation. The natural wish to do things better, e.g. to optimize these
materials and their use, has also created interest among researchers with a primary
interest in optimal design, see (Gürdal, Haftka and Hajela 1999). Structure
Detailed design of laminates, like a point-wise design of orientation and thickness,
might be named material design. Therefore, there is no clear boundary between this
chapter and chapter 8 on material optimal design, and overlapping also occurs in
relation to chapter 10 on identification and inverse problems.
Effective analysis and analytical sensitivity analysis give a good background for
the performed optimization. We therefore first report some of these essential tools,
and primarily concentrate on laminates. Laminate analysis is an important subject
within the mechanics of composite materials. Among the many good textbooks on
laminates we mention (Jones 1975), (Tsai and Hahn 1980), (Vinson and Sierakowski
1989), (Whitney 1987).
From the book (Tsai and Hahn 1980) we quote from page 210:
”It is unfortunate that the use of composite materials is limited or penalized by the
non-availability of analytical tools. It is important to understand how anisotropy
and non-homogeneity arise in composite laminates and to what degree they can be
manipulated to perform functions not possible with conventional materials”.
or material
There is still some truth in this statement. Plies - layers
A laminate consists of two or more plies and acts as an integral plate. Plies are
also named layers or laminas, and often they are available as prepregs. The plies
may be made of different materials, although the same ply material is often used
107
- laminas
- prepregs

Stress/strain
vectors
108 Copyright c○Pauli Pedersen: Optimal designs ...
throughout the laminate. The main difference between the individual plies is thus
their orientation in the laminate, and these orientations are very important because
the plies are often strongly non-isotropic, with ratios of modulus of the order 5-40as
illustrated in table 6.1 in section 6.1.3. The increasing possibilities in manufacturing
with individual placement of fibers make optimization even more important.
We first concentrate on analysis, and we use the √ 2-contracted notation that
simplifies the rotational transformations, as argued in (Pedersen 1995).
6.1 Laminate analysis and rotational transformations
To a large extent the laminate analysis is based on ”book-keeping”, where the rotational
transformations of stresses, strains, and constitutive relations play a major role.
We list the necessary formulas by rather simple expressions, but without derivations,
and we restrict the description to 2D.
6.1.1 Rotational transformations of stress and strain vectors
Orthogonal transformations of stresses and strains are, as argued in chapter 11, obtained
simply with the vector definitions
√
2ɛ12} (6.1)
y2
{σ} T √
:= {σ11 σ22 2σ12}, {ɛ} T := {ɛ11 ɛ22
x2
Figure 6.1: The two Cartesian coordinate systems x1,x2 and y1,y2 with definition
of the relative angle θ.
θ
y1
x1

Design of laminates and plates 109
The orthogonal matrix [T ] that transforms second order tensors between the
Cartesian coordinate systems x to y, as shown in figure 6.1, is Orthogonal
[T ] matrix
[T ]= 1
⎡
√
1+c2 1 − c2 2s2
⎣ 1 − c2 1+c2 −
2
√ 2s2
− √ ⎤
⎦
√
with [T ]
2s2 2s2 2c2
−1 =[T ] T
(6.2)
with the short notation for the trigonometric functions being
c2 := cos(2θ), s2 := sin(2θ), c4 := cos(4θ), s4 := sin(4θ) (6.3)
In general for stresses as well as for strains (and other similar vectors) we have
(6.4)
{σ}y =[T ]{σ}x, {σ}x =[T ] T {σ}y, {ɛ}y =[T ]{ɛ}x, {ɛ}x =[T ] T {ɛ} T y
and conclude that only one matrix, the [T ] matrix, is necessary for these transformations.
Note, that if the stress and strain vectors are defined differently (traditionally
σ12 not √ 2σ12, and 2ɛ12 not √ 2ɛ12), then we need more than one matrix, which are
not orthogonal.
6.1.2 Rotational transformations of constitutive matrices
A general non-isotropic constitutive matrix [C] for two dimensional problems is defined
in relation to the vector definitions (6.1), and using the four index notation from
tensor notation we have
⎧ ⎫ ⎡
⎨ σ11 ⎬
= ⎣
⎩
σ22
√ 2σ12
⎭ x
C1111
C1122 √
2C1112
√
C1122 2C1112 √
C2222 2C2212
√
2C2212 2C1212
⎤
⎦
x
⎧
⎨
⎩
ɛ11
ɛ22
√ 2ɛ12
⎫
⎬
⎭ x
(6.5)
Constitutive
matrix
Constitutive
vector
In optimal material design it is very convenient to contract further the constitutive
matrix [C] into a vector {C} with six components (see chapter 11 for further
discussion),
√
{C} = {C1111 C2222 2C1212 2C1122 2C1112 2C2212} (6.6)
and the orthogonal matrix [R] that transforms fourth order tensors between the Cartesian
coordinate systems x to y is Orthogonal
[R] = 1
8 ·
⎡
√ √
3+4c2 + c4 3 − 4c2 + c4 2 − 2c4 2 − 2c4 4s2 +2s4 4s2 − 2s4
√ √
⎢
3 − 4c2 + c4 3+4c2 + c4 2 − 2c4 2 − 2c4 −4s2 +2s4 −4s2 − 2s4
⎢ 2 − 2c4 2 − 2c4 4+4c4 −2
⎢
⎣
√ 2+2 √ 2c4 −4s4 4s4
√ √ √ √
2 − 2c4 2 − 2c4 −2 √ 2+2 √ 2c4 6+2c4 −2 √ 2s4 2 √ 2s4
−4s2 − 2s4 4s2 − 2s4 4s4 2 √ ⎤
⎥
2s4 4c2 +4c4 4c2 − 4c4
⎦
−4s2 +2s4 4s2 +2s4 −4s4 −2 √ 2s4 4c2 − 4c4 4c2 +4c4
[R] matrix

Practical
definitions
110 Copyright c○Pauli Pedersen: Optimal designs ...
with [R] −1 =[R] T (6.7)
We can then by a simple matrix multiplication transform the constitutive parameters
from one Cartesian coordinate system to another by
{C}y =[R]{C}x, {C}x =[R] T {C}y
(6.8)
and similar transformations are valid for contracted compliance matrices [C] −1 as well
as for the stress strength matrices [F ] and the strain strength matrices [G], that is
used in relation to strength constraints.
6.1.3 Alternative description by practical parameters
With reference to the paper (Hammer, Bendsøe, Lipton and Pedersen 1997), we
now show an alternative description of the constitutive matrix [C] in a coordinate
system, rotated an angle θ counter-clockwise relative to a coordinate system where
the following practical parameters are defined
C1 := C1111 − C2 − C3
C2 := (C1111 − C2222)/2
C3 := (C1111 + C2222 − 2C1122 − 4C1212)/8
C4 := C1122 + C3
C5 := (C1 − C4)/2
C6 := (C1112 + C2212)/2
C7 := (C1112 − C2212)/2 (6.9)
These parameters are used in the book by (Tsai and Hahn 1980), and are in the
early work (Tsai and Pagano 1968) named invariants, which may be misleading. In a
non-dimensional form αn = Cn/C1111 some actual parameters are listed in table 6.1
for some orthotropic materials (C6 = C7 = 0).
With definitions (6.9) the description in terms of the following five symmetric
matrices, defined by
⎡
[Γ0] := ⎣
C1 C4 0
C4 C1 0
0 0 2C5
⎡
C2 0
[Γ1] := ⎣ 0 −C2
√ √
2C6 2C6
⎤
⎦ (6.10)
√
2C6 √
2C6
0
⎤
⎦ (6.11)

Design of laminates and plates 111
Materials GPa (Giga Pascal) Non-dimensional parameters
EL ET GLT νLT α0 α1 α2 α3 α4 α5
181.0 10.30 7.17 0.28 0.9955 0.4200 0.4716 0.1084 0.1244 0.1478
Graphite/Epoxy 138.0 8.96 7.10 0.30 0.9942 0.4298 0.4675 0.1027 0.1222 0.1538
207.0 5.17 2.59 0.25 0.9964 0.3922 0.4875 0.1203 0.1266 0.1328
204.0 18.50 5.59 0.23 0.9952 0.4278 0.4547 0.1175 0.1384 0.1447
Boron/Epoxy 207.0 20.70 6.90 0.30 0.9910 0.4365 0.4500 0.1135 0.1435 0.1465
213.7 23.44 5.17 0.28 0.9914 0.4358 0.4452 0.1190 0.1498 0.1430
Aramid/Epoxy 76.0 5.50 2.30 0.34 0.9916 0.4233 0.4638 0.1129 0.1375 0.1429
Glass/Epoxy 38.6 8.27 4.14 0.26 0.9855 0.5221 0.3929 0.0850 0.1407 0.1907
53.8 17.90 8.96 0.25 0.9792 0.6021 0.3336 0.0643 0.1474 0.2273
Isotropic E E E
2(1+ν) ν 1 − ν2 1 0 0 ν 1−ν
2
Table 6.1: Actual parameters for orthogonal materials, calculated from (Tsai and
Hahn 1980) using the definitions by αn := Cn/C1111. Note that α1 + α2 + α3 =1.
is simply written
⎡
[Γ2] := ⎣
⎡
[Γ3] := ⎣
[Γ4] :=
C3
−C3
√ 2C7
−C3 C3 − √ 2C7
√ 2C7 − √ 2C7 −2C3
⎤
⎦ (6.12)
2C6 0 −C2/ √ 0 −2C6
2
−C2/ √ −C2/
2
√ 2 −C2/ √ 2 0
⎦ (6.13)
⎡
⎣
C7 −C7 − √ −C7
−
C7
2C3 √
2C3
√ 2C3
√
2C3 −2C7
⎤
⎦ (6.14)
[C]x =[Γ0]+[Γ1] cos 2θ +[Γ2] cos 4θ +[Γ3] sin 2θ +[Γ4] sin 4θ (6.15)
The parameters C1 − C7 can be interpreted as follows: If the material is orthotropic,
we can choose a reference coordinate system where
C6 = C7 =0 with reference to orthotropic directions (6.16)
and such directions exist if the following condition can be satisfied in a coordinate Orthotropic
system, where (6.16) is not satisfied. The condition is
materials
C7C 2 2 − 4C7C 2 6 − 4C6C3C2 = 0 (6.17)
⎤

Orthotropic
engineering
4 parameters
112 Copyright c○Pauli Pedersen: Optimal designs ...
The parameters C6,C7 are therefore only necessary for non-orthotropic laminates
(materials). For an orthotropic material we can express the parameters in what is
normally named the engineering parameters EL,ET ,GLT ,νLT and get
C1 = EL/α0 − C2 − C3
C2 =(EL − ET )/(2α0)
C3 =(EL +(1− 2νLT )ET )/(8α0) − GLT /2
C4 = νLT ET /α0 + C3
C5 =(C1 − C4)/2
with the definition α0 := 1 − ν 2 LT ET /EL
(6.18)
The most important parameter C2 describes the level of non-isotropy, and the parameter
C3 describes the relative shear stiffness, as discussed in detail in chapter 17.
Figure 6.2 shows an overview of material classification, restricted to the two dimensional
case.
General
Non-isotropic
C7C 2 2 − 4C7C 2 6 − 4C6C3C2
Orthotropic
C2,C3
zero
Isotropic
both zero
not zero
not both zero
Non-orthotropic
C3
negative
High shear modulus
Low shear modulus
positive
Plane stress, isotropic
Plane strain, isotropic
Others
Figure 6.2: Overview of material classification in 2D.

Design of laminates and plates 113
If the laminate (the material) is isotropic, then in any coordinate system we have Isotropic
2 parameters
C2 = C3 = C6 = C7 =0 ⇒
C1 = C1111, C4 = C1122, C5 =(C1111 − C1122)/2 (6.19)
i.e. the only non-zero parameters C1, C4 and C5 are described by two parameters.
6.1.4 Laminate stiffnesses and lamination parameters
In the classical plate theory the global relation between the forces and moments per
unit length {N}, {M} and the mid-plane strains {ɛ0 } and curvatures {κ} is
0 {N} [A] [B] {ɛ }
=
(6.20)
{M} [B] [D] {κ}
In-plane and
out-of-plane
When the √ 2-notation is also used for {N}, {M}, {ɛ0 }, {κ},
i.e. {N} T √
:= {N11 N22 2N12} etc., then the laminate stiffness matrices [A], [B], [D]
are in terms of the material parameters C1 − C7 defined in (6.9) and the twelve Stiffnesses:
lamination parameters ξ membrane
coupling
bending
A,B,D
1,2,3,4 given as
[A] = h [Γ0]+[Γ1]ξ A 1 +[Γ2]ξ A 2 +[Γ3]ξ A 3 +[Γ4]ξ A 4
[B] = h 2 [Γ1]ξ B 1 +[Γ2]ξ B 2 +[Γ3]ξ B 3 +[Γ4]ξ B 4
[D] = h 3
1
12 [Γ0]+[Γ1]ξ D 1 +[Γ2]ξ D 2 +[Γ3]ξ D 3 +[Γ4]ξ D
4 (6.21)
where the lamination parameters in a global coordinate system x are defined as the
weighted trigonometric integrals over the thickness
ξ A,B,D
1,2,3,4 :=
1/2
z
−1/2
0,1,2 (cos(2θ(z)), cos(4θ(z)), sin(2θ(z)), sin(4θ(z))) dz (6.22)
This compact notation implies, as an example, that ξD 3 is given by Lamination
ξ D 1/2
3 :=
−1/2
z 2 sin(2θ(z))dz (6.23)
We use the lamination parameters as the design variables, mainly due to the fact
that the laminate stiffnesses are linear functions of these variables, as seen from (6.21).
The lamination parameters are not independent, as there exist trigonometric relations
between them. Governing the pure membrane stiffnesses or the pure bending
stiffnesses the constraints are
parameters
Linear
functions

114 Copyright c○Pauli Pedersen: Optimal designs ...
Figure 6.3: The design domains for the four lamination parameters with one parameter
equal to zero in turn. From Hammer et al. (1997).

Design of laminates and plates 115
2ξ 2 1(1 − ξ2)+2ξ 2 3(1 + ξ2)+ξ 2 2 + ξ 2 4 − 4ξ1ξ3ξ4 ≤ 1
ξ 2 1 + ξ 2 3 ≤ 1
−1 ≤ ξ2 ≤ 1 (6.24)
as discussed further in (Hammer et al. 1997). By rotating the reference coordinate
system for the lamination parameters, we can force one of these to be zero, and in this
way obtain a graphical illustration of the convex feasible space for these parameters,
as shown in figure 6.3. Mutual relations for the coupled problems with both A, B, D
lamination parameters are not yet available.
6.2 Elastic energy sensitivity analysis
In this section our general design parameter is taken to be the material orientation θe
in the region e, which could be the laminate as a whole, a specific ply in a laminate, or
more detailed an element of a ply. With power law elasticity and design independent
loads we have directly from (13.13) that
dUɛ
dθe
= − Ve
p
∂(ūɛ)e
∂θe
fixed strains
(6.25)
where (ūɛ)e is the mean strain energy density in region e with volume Ve, and for
linear elasticity we have p =1.
For coupled plate/disc problems, the strain energy per plate area ūɛt, where ūɛ
is the mean strain energy density through the thickness t of the plate, is for linear
elasticity given by
ūɛt = 1
2 {ɛ0 } T {N} + 1
2 {κ}T {M} (6.26)
which with relations (6.20) is written Strain energy
ūɛt = 1
2 {ɛ0 } T [A]{ɛ 0 } + {ɛ 0 } T [B]{κ} + 1
2 {κ}T [D]{κ} (6.27)
Inserting this in (6.25) we get (linear elasticity)
dUɛ 1
= −ae
dθe 2 {ɛ0 T ∂[Ae]
e} {ɛ
∂θe
0 e} + {ɛ 0 T ∂[Be]
e} {κe} +
∂θe
1
T ∂[De]
{κe} {κe}
2 ∂θe
(6.28)
where ae is the area of the actual laminate region, and index e on the involved
quantities relates to this actual region.
per area

Only four
stationary
solutions
Orthotropic
analytical
solutions
116 Copyright c○Pauli Pedersen: Optimal designs ...
Even for the fully coupled problems this result can be written
dUɛ
= U1 sin(2θe)+U2 sin(4θe)+U3 cos(2θe)+U4 cos(4θe) (6.29)
dθe
which follows from (6.21) or directly from the rotational matrix [R] in (6.7). The
energy coefficients U1, U2, U3, U4depend on the material as well as on the specific
strain state.
Before treating the specific and simplified problems, it should be appreciated that
according to (6.29) we can, in the general case, find at most four different solutions
to the stationarity condition dUɛ/dθe = 0. This follows from rewriting (6.29) as a
fourth order polynomial. However, analytical expressions for this general case are too
complicated to be shown here.
For orthotropic materials and models where only the cosine terms are involved in
(6.21), analytical solutions to dUɛ/dθe = 0 are not complicated. After differentiation
we keep in (6.29) only the sine terms, and have for these (often named specially
orthotropic or balanced) models
dUɛ
U1
=2U2sin(2θe) + cos(2θe)
(6.30)
dθe
2U2
Stationarity is then obtained for
θe =0, θe = π/2, θe = ± 1
2 arccos
− U1
(6.31)
2U2
and, furthermore, supplementary angles return the same energy density uɛ(π − θ) =
uɛ(θ) when only cos(2θ) and cos(4θ) appear in the stiffness expressions. Thus, for this
case the orientational dependence is described completely by the interval 0 ≤ θ ≤ π/2.
6.3 Optimal orientation of material in
simply supported plates in bending
Figure 6.4 illustrates a simply supported plate placed in a Cartesian coordinate system
x1,x2. The displacement field for the transverse displacement w = w(x1,x2) of this
plate in bending is given by
w(x1,x2) =wmn sin(mπx1/a) sin(nπx2/b) (6.32)
with number of half-waves m along the length a in the x1-direction and number of
half-waves n along the length b in the x2-direction, in agreement with the actual load.

Design of laminates and plates 117
x1
b
x3
w(x1,x2)
Figure 6.4: A rectangular plate with dimensions a, b, assumed simply supported
along all four boundaries.
a
For rectangular, simply supported plates in pure bending we introduce a mode
parameter η that describes the displacement pattern (for a square pattern we have
η = 1). The mode parameter is then defined as the ratio between the two actual
half-wave lengths of the deformation by Deformation
ηmn := b/n mb
=
a/m na
x2
(6.33)
A linear combination of the bending stiffnesses Φmn is for a specific deformation
mode ηmn defined by
mode
parameter
Φmn := 1 4
ηmnD1111 + D2222 +2η
8
2 mn(D1122 +2D1212)
(6.34)
where D1111,D2222,D1122,D1212 are the four components of the orthotropic laminate Objective
bending stiffness matrix, defined in (6.20). Φmn is our objective to extremize, because Φmn
eigenfrequencies, buckling loads, and displacement amplitudes for the corresponding
rectangular plate is then also extremized. The relations to these physical quantities
can be found in the book (Jones 1975) and in other classical books, and they are for
eigenfrequencies:
ωmn = n2π2 b2
Φmn
ρt
(6.35)

Eigen
frequencies
Buckling load
Maximum
displacement
Non-trivial
optimal
orientation
118 Copyright c○Pauli Pedersen: Optimal designs ...
where ρ is the mass density of the plate, t is the thickness of the plate, and m, n give
the actual mode.
For buckling load (per length) in the x1 − direction with most dangerous mode
corresponding to m, n =1, 1 we have
(Nx1 )buckling = π2
η2 Φ11
(6.36)
b2 and finally corresponding to a distributed load p = p(x1,x2) we get the maximum
displacement |w|max by
p = pmn sin(mπx1/a) sin(nπx2/b) ⇒|w|max = pmn
b 4
1
n4π4 Φmn
(6.37)
For laminates with orthotropic bending stiffness matrix , i.e. D1112 = D2212 =0
with reference to the rectangular side directions, the orientations that give stationary
objective Φ, (as defined in (6.34)), can be found analytically as by (6.31). The energies
U1 and U2 in (6.31) depend on the specific material as characterized by C2,C3 defined
in (6.9) and on the mode parameter ηmn. The derivation in (Pedersen 1987b) focus
on the result
(θe)optimal = 1
2 arccos
1 − η4
mn C2
(6.38)
4C3
1 − 6η 2 mn + η 4 mn
and figure 6.5 shows the results of optimizing the eigenfrequency, as also found in
(Bert 1977).
From an engineering point of view the main conclusions to be drawn from figure
Practical
6.5 are:
conclusions • The optimal orientation depends mainly on the mode parameter ηmn. Thus
if the deformation pattern is known, the optimal direction can be estimated
directly.
• Cases of inverse mode parameters ηa = η −1
b have complementary solutions
θb = π/2 − θa.
• For ”extreme” values of η the optimal orientation is perpendicular to the long
wavelength.
• The change of optimal direction to a skew direction is very sensitive to the
mode parameter.
• The optimal orientation is rather insensitive to the material parameters.
• The optimal orientation is independent of the position of the ply in the laminate,
and thus the same for all plies.

Design of laminates and plates 119
• It is not seen in the figure but local optima exist.
By simple numerical analysis, problems with combined modes can also be optimized,
(Pedersen 1986b) shows such solutions. Often a specific mode is dominating,
and then solutions from figure 6.5 are a good starting design for more detailed iterations.
Figure 6.5: Optimal orientation as a function of the mode parameter ηmn for plate
bending. Note that α2/α3 = C2/C3, defined in (6.9).

Simple
example
Large
improvements
No energy
concentration
120 Copyright c○Pauli Pedersen: Optimal designs ...
6.4 Laminate point-wise design
The importance of laminates has put increasing demands on the possibility to manufacture
graduated thickness as well as orientation, and even individual fiber laying
is possible. Thus the point-wise design of orientation and thickness is a realistic
optimization problem.
Figure 6.6: A uniformly loaded cantilever example, for which the thickness distribution
as well as the orientational field is optimized.
To exemplify the different effects of thickness and orientational optimization the
cantilever problem shown in figure 6.6 is solved. The objective is to minimize compliance
(minimize stored elastic energy) and at the same time minimize concentration
of energy density, but without involving this later aspect in the formulation. Simple
heuristic iterations based on optimality criteria have been applied. For the thickness
iterations this criterion is based on uniform energy density, and for the orientational
iterations alignment with principle stress/strain directions is applied, see chapter 17.
Figure 6.7 shows the results. In this simple figure a black and white visualization
of the results is used. The design is characterized by thickness and orientation, which
are shown by hatching the triangular finite elements in the direction of the larger
modulus direction and with the hatch density proportional to the thickness. Dark
areas are therefore areas with large thicknesses. The cantilever results shown in figure
6.7 are based on a 720 triangular constant stress element model, assuming constant
thickness and orientation in each element. For the uniform cantilever, the mean
(normalized to 1.0) and the maximum relative values of energy density are 1.0, 41.8
(relative measures of stiffness and stress concentration). Only thickness optimization
gives 0.53, 0.57, which means stiffness improved by a factor 1.9 and almost no energy
concentration. With simultaneous thickness and orientational optimization we obtain
0.23, 0.25, i.e. the stiffness improved by a factor of 4.3, and again almost no energy
concentration.

Design of laminates and plates 121
Figure 6.7: Above: the optimal thickness distribution (dark equal to large thickness)
for a uniformly loaded cantilever. Below: result of the combined thickness and
orientational optimization.

Non-linear
elasticity
Design
independence
122 Copyright c○Pauli Pedersen: Optimal designs ...
6.4.1 Thickness distribution only
Design of laminate thickness distribution for maximum stiffness is characterized by
resulting uniform energy density, and is with many examples illustrated in chapter
4. With three examples taken from (Pedersen and Taylor 1993) we give results that
show the solutions to be independent of a material non-linear elasticity, modelled by
a power law, see chapter 12.
Figure 6.8 gives the three specific examples and table 6.2 gives the relative resulting
values. The main result is that the optimal thickness distribution is independent
Figure 6.8: The three example cases: a) uniformly loaded cantilever of isotropic
material, b) circular hole loaded biaxially 2:1, isotropic material, c) optimal designed
shape of the hole, loaded biaxially 3:2, orthotropic material.
of the power p. Thus the results for the problems of figure 6.8 mainly illustrate the
influence from the power p on two given designs, i.e. the uniform thickness design
and the optimal thickness design as obtained from linear elasticity.
In terms of relative values table 6.2 gives the strain energy densities. Minimum,
mean and maximum values are related to the elements of the finite element models.
The table clearly shows the different results from uniform and optimal thickness distribution.
It is well known from optimization based on linear elasticity that certain
areas in a model cannot be fully stressed (too little energy density). Thus the minimum
values are of minor interest, and the agreement of the maximum values with the
mean values better shows the fulfillment of the optimality criterion of uniform energy
density.
The relative values of the objective function are given by the relative mean values
in table 6.2. The factor between energy in uniform design and energy in optimal
design is almost constant for the three cases, with a weak tendency to be more important
with increasing non-linearity (decreasing p). The stronger effect for the cantilever
problem reflects the initial less uniform/strain distribution. Also the actual
stress/strain level (higher for the cantilever problem) has an influence, and thus the

Design of laminates and plates 123
Uniform or Strain energy densities in % of reference energy density
optimal cantilever, isotropic circular hole, isotropic opt. hole, orthotropic
with p = min. mean max. min. mean max. min. mean max.
Uni. 1.0 0.5 100 577 4 100 677 47 100 387
Opt. 1.0 14 50 50 81 87 89 59 90 92
Uni. 0.9 0.7 163 935 4 140 951 60 138 528
Opt. 0.9 23 75 76 112 120 122 63 123 138
Uni. 0.8 0.9 280 1599 3 202 1391 79 196 737
Opt. 0.8 39 120 121 159 170 173 69 173 221
Uni. 0.7 1 512 2916 2 304 2127 108 288 1060
Opt. 0.7 68 201 202 236 251 256 80 253 369
Uni. 0.6 2 1005 5730 0.3 482 3430 153 445 1581
Opt. 0.6 126 357 359 365 387 394 90 385 641
Uni. 0.5 2 2151 12310 0.1 808 5891 227 722 2464
Opt. 0.5 251 683 686 595 630 640 107 618 1169
Table 6.2: Table of relative results with uniform thickness (Uni.) and with optimal
thickness distribution (Opt.) for linear elasticity (p = 1) and for five models of nonlinear
elasticity modelled by the power p

Multiple
124 Copyright c○Pauli Pedersen: Optimal designs ...
For computational examples based on first locating the optimal lamination parameters
and then identifying a laminate resulting in these parameters, see (Hammer
et al. 1997). The laminate identification problem can be solved as described in chapter
10, and a laminate with at most three plies can be constructed analytically, see
(Lipton 1994).
6.5 Strength optimization
On a more or less heuristic basis, designs for optimal strength have been taken to be
designs where principal stress direction is aligned with material orthotropy direction.
With a non-symmetric strength criterion like the Tsai-Wu criterion, such solutions
can not be the optimal ones, because optimal stiffness design (energy independent of
sign) is different from optimal strength design.
The strength problem is a local problem and therefore simple results from sensitivity
analysis can not be expected. To solve the optimal design problem in a proper way,
we must use mathematical programming. In this section we formulate the problem
and show results from (Hammer 1994).
load cases The optimization problem formulated in words is: maximize a common load factor,
subject to given strength constraints. Multiple plies, multiple loads, and multiple
strength constraints are included in the formulation. Let {Al} be the load distribution
vector corresponding to the load case l, then from finite element analysis the resulting
nodal displacement vector {Dl} is found by
[S]{Dl} = λ{Al} for l =1, 2,...,L (6.39)
where [S] is the stiffness matrix of the actual design (finally the optimal design) and
Multiple
λ is a load factor, common to all load cases L.
plies From {Dl}, directly strains and stresses in every ply k of every element e follow
Multiple
strength
constraints
{Dl} ⇒ {ɛekl}, {σekl}
for e =1, 2,...,E; k =1, 2,...,K; and l =1, 2,...,L (6.40)
Now the load strength Fn corresponding to a given strength criterion n can be determined
for each of these ekl and compared to the strength limit (F0)n. Formulating
our problem in relation to first ply failure (FPF) we thus have the constraints
(F/F0)ekln ≤ 1
for e =1, 2,...,E; for k =1, 2,...,K;
for l =1, 2,...,L; and for n =1, 2,...,N (6.41)

ective
Design of laminates and plates 125
i.e. strength limits should be satisfied in all K plies of all E elements, for all L load
cases, and this should be the fulfilled for all N specified strength constraints.
The objective of our optimization is by means of orientational design (and/or
thickness distribution design) to
Maximize λ subject to the strength constraints (6.42)
i.e. constraints (6.41) must be fulfilled. Analytical sensitivity analysis, as derived
in (Hammer 1994), can be carried through, and then the main practical problem is
related to the existence of a large number of local optima. In a way this problem
was expected. Different solutions to this problem include strategies for choosing the
initial design before mathematical programming is applied. A valuable alternative
is to describe the design with global design parameter. This approach is described
before an example is shown.
6.5.1 Global design parameters
The idea of a global design description used frequently in shape optimization, see
chapter 7, is extended from curve to surface parametrization. The design is given
as a linear combination of orthogonal functions. Application is here to orientational
design of laminates for strength optimization, but the technique is directly applicable
to design also of thickness distribution. The advantages by this design description are
many, including control on smoothness, control on slopes, and control on connections
at the design boundaries. This simplified parametrization of the design makes it
possible to work with only few design variables, say 25 for a whole design domain. 1D-design
description
Figure 6.9: Illustration of the mapping from reference line to the base curve.
Let us first illustrate the design description from (Pedersen, Tobiesen and Jensen
1992) in a form that makes our extension from curve to surface more clear. In figure

2D-design
description
126 Copyright c○Pauli Pedersen: Optimal designs ...
6.9 we show the reference domain −1 ≤ ξ ≤ 1, and the base curve C0 with the
mapping illustrated. The design curve C is then described by
C = C0 +
N
znfn(ξ) (6.43)
where fn(ξ) are given functions, and the linear combination factors zn are the design
variables. Mutual orthogonal functions may be chosen as the first vibration modes
to a clamped/clamped beam, if the positions and slopes at both ends of the curve C
should be identical to those of C0. Naturally, this selection of functions has no relation
to vibration at all - it is just a convenient way of obtaining well known orthogonal
functions, and surely other possibilities exist. Note that other conditions at the ends
of C0 can be obtained from other boundary conditions of the related beam problems.
Now for our 2D-problem we show in figure 6.10 the reference domain −1 ≤ ξ1 ≤
1, −1 ≤ ξ2 ≤ 1 and a finite element mesh put onto it. If a constant design (thickness
and/or orientation) is wanted in each element, this may be controlled by the values
in the center of the corresponding element in the reference domain. Mapping is
from element in reference model to corresponding element in actual model, and from
triangle center to corresponding triangle center.
Figure 6.10: Illustration of the mapping from reference domain to the actual finite
element model.
n
The material orientation θ in the reference domain is given by
θ(ξ1,ξ2) =
zmnfm(ξ1)fn(ξ2) form, n =0, 1, 2,...N (6.44)
m
n
where the f functions are given mutual orthogonal functions, and thus the zmn in
(6.44) are the design parameters. In this specific case we choose the f functions to be

Design of laminates and plates 127
f0(ξ) =1; f1(ξ) =ξ;
fm(ξ) = cosh(kmξ)+µm cos(km)ξ) for m =2, 4,...
fm(ξ) = sinh(kmξ)+µm sin(km)ξ) for m =3, 5,... (6.45)
where the constants km,µm are determined by
tan(km)+(−1) mtanh(km) =0 for m =2, 3, 4 ...
µm = cosh(km)/ cos(km) for m =2, 4,...
µm = sinh(km)/ sin(km) for m =3, 5,... (6.46)
Figure 6.11: Problem with an elliptic hole where symmetry is assumed, and the two
uniform loads act independently.
With this design description, i.e. using the zmn in (6.44) as design variables, an
example from (Hammer 1994) illustrates the possibilities. The example is a two ply,
two load cases problem with Tsai-Wu strength constraint as well as maximum strain
constraint. The problem and the quarter model are shown in figure 6.11, and the
optimal orientational designs with the optimized load factors for increasing number
Design
functions
of design variables are presented in figure 6.12. Manufacturing
The results in figure 6.12 show how the complexity of the optimized orientational
field is controlled by the global design description. Thus, this parametrization is a
tool that can take the possibilities for practical manufacturing into account. Note
also that the gain in going from 9 design variables to 16 design variables is rather
small (6.76 compared to 6.54).
constraints

128 Copyright c○Pauli Pedersen: Optimal designs ...
Figure 6.12: Optimal orientation with increasing number of design variables. Upper
ply orientations to the left and lower ply orientations to the right. Density of hatch
illustrates the resulting stress levels. From (Hammer 1994).

Chapter 7
Shapes of minimum stress
concentration
In the design of structural connections as well as in the micro-mechanics design of
materials, boundary shapes play an important role. The objective may be the stiffest
design, the strongest design or just a design of uniform energy density along the
shape. In an energy formulation in chapter 14 it is proven that these three objectives
have the same solution, at least within the limits of geometrical constraints, including
the parametrization. Without involving stress/strain fields, the proof holds for 3Dproblems,
for power-law non-linear elasticity, and for non-isotropic elasticity. The
proof is based on an assumption of a constant volume of material.
Many results are available within optimal shape design, but still a number of important
issues need to be addressed. The literature gives a picture of three different
Structures
or materials
directions of research. In mechanical and civil engineering focus has for more than Different
25 years been on design for minimum stress concentration, see (Ding 1986). In the
material and more mathematical oriented research the focus has been on design for
minimum compliance, see (Vigdergauz 1997). Some heuristic approaches have focused
on design for uniform stress, see (Xie and Steven 1997). Not too many mutual references
are given among these three research directions. A goal of the present chapter,
together with chapter 14, is to show that a unification is possible, because the three
different objectives give the same solution under certain conditions.
Studying the results of a variety of rather different shape design problems, it is
objectives
seen that very general knowledge can be obtained. The striking generality is the General
property of uniform energy density along the designed boundaries for all the different
formulations mentioned.
Mainly from formulations with minimum stress concentration we list here a num-
129
knowledge

Neighbouring
redesign
Modelling
issues
Minimum
of maximum
Only
minimum
130 Copyright c○Pauli Pedersen: Optimal designs ...
ber of analytically and/or numerically obtained results and the theoretical background
for this is presented in chapter 14.
7.1 Statement of the actual problems
Like most optimal design problems, as discussed in section 2.7, the problem of shape
design for minimum stress concentration is highly non-linear, and must be solved
iteratively. Thus, the problem is converted to a sequence of problems of optimal
redesigns; where a given shape is changed into a better neighbouring shape. The
solution to this problem involves three steps: finite element analysis for a given shape,
sensitivity analysis with respect to the parameters describing the design, and optimal
decision for redesign.
Furthermore, it should be kept in mind that the quantitative description of stress
concentration is very much depending on the finite element modelling, and very small
changes in shape can drastically influence the actual values. Modelling is therefore
related to analytically smooth shape modelling, and the element sizes in the critical
domains are to a large extent kept very small and with only small changes from one
iteration to the next iteration.
In mathematical terms, the objective of this chapter of shape design is to
Minimize Maximum σ 2 eff
(over feasible shapes) (over the structural space x and load cases) (7.1)
where the squared effective stress σ 2 eff
could also be the strain energy density uɛ.
Converting (7.1) to a pure minimization problem on redesign by increments of design
parameters ∆hi, and including for problems of hole design a volume constraint V = ¯ V ,
the actual redesign formulation is
Minimize σ2 max
(within move−limits on ∆hi)
subject to the constraints :
σ 2 eff (x)+
i
∂σ 2
eff (x)
∂hi
∆hi − σ 2 max ≤ 0
V +
i ∂V
∂hi ∆hi − ¯ V = 0 (7.2)
where σ 2 max is an unknown in addition to the unknown increments ∆hi. This general
technique of changing from a minimum-maximum formulation to a pure minimum
formulation is often used in optimal design, and there named the bound formulation.
The statement (7.2) is a problem within linear programming as described in general
in chapter 20.

Shapes of minimum stress concentration 131
Specific problems, with and without the volume constraint, are in the following
sections presented with focus on the obtained results.
7.2 The 2D-fillet in tension
The 2D-fillet-problem, defined with the goal of minimizing stress concentration was
solved by (Tvergaard 1973) using a finite difference method for stress analysis. In
(Kristensen and Madsen 1974) and in (Francavilla, Ramakrishnan and Zienkiewicz
1975) the same problem was solved using the finite element method for stress analysis.
Within the limits of the imposed geometrical constraints (length of the fillets and the
Early
references
parametrization) the results of optimization give constant tangential stress along the Constant
designed boundary. At this boundary we have an unidirectional state of stress and
therefore constant energy density as well as constant von Mises stress. Thus these
early papers point toward the importance of constant energy density.
In relation to the fillet model in figure 7.1 we illustrate the importance of design
to obtain remarkably better stress fields. Primarily we quantify the very negative
effect of small circular connection zones.
l1
l2
lf
l3
l4
Figure 7.1: Fillet model for the 2D-fillet in tension. All calculations and illustrations
to follow are related only to the lower left quarter part, due to symmetries.
energy
density

Overall
parameters
FE model
Severe
concentration
Influence of
stiffness
132 Copyright c○Pauli Pedersen: Optimal designs ...
In the examples to follow we use the following dimensions and parameters:
• length l1 =30mm, length l2 =60mm, length l3 =90mm
• length l4 =40→ 60 mm , length of fillet lf =10→ 60 mm
• isotropic material and Poisson’s ratio ν =0.3
• resulting maximum stress levels relative to the applied uniform stresses
7.2.1 Circular connection fillet
Figure 7.2 shows the results for 6 different models, all with circular shapes, but with
increasing radius and thus increasing fillet length and corresponding increasing larger
dimension l4, as defined in figure 7.1. Figure 7.2 shows by isolines the resulting fields
of energy densities. In this figure it is, by the green (gray) areas added to the boundary
of the hole, illustrated how the energy density varies along the boundary of the hole
(the same technique is applied in other figures of the chapter).
We conclude that a larger circular radius is advantageous, but even the largest
one shows a significant stress concentration of more than 50 %. In relation to specific
numbers it should be noted that the finite element model used, is in the range of
20.000 constant stress elements and 20.000 degrees of freedom, and the element sizes
close to the stress concentration areas have length of about 0.1 mm. We have listed
the results with four digits to show the relative tendencies. Note however, this is not
an indication of the actual accuracy for the model.
In the examples in figure 7.2 we note that the largest possible circular fillets are
used (radius = l4 − l1). The stress concentration is even larger, if smaller circular
fillets are applied to larger change in dimensions. To illustrate this we show in
figure 7.3 the circular fillet of 10 mm applied also to the case of l4 = 60 mm. The
stress concentration factor is then increased from 1.868 to 2.111. The explanation
for this is that larger dimension of l4 increases the stiffness close to the area of stress
concentration and thus has a negative influence.

Shapes of minimum stress concentration 133
Figure 7.2: Stress distributions for circular shapes with increasing fillet length, equal
to 10, 14, 18, 22, 26, 30 mm, respectively. The length of the dimension l4 in the
model figure is thus 40, 44, 48, 52, 56, 60 mm, respectively. The corresponding stress
concentration factors are 1.868, 1.774, 1.695, 1.627, 1.568, 1.516, respectively.

134 Copyright c○Pauli Pedersen: Optimal designs ...
Figure 7.3: Stress distributions for a circular shape with circular radius equal to 10
mm, although the change in dimensions is 30 mm. The corresponding stress concentration
factor is 2.111, to be compared to the value of 1.868 for the corresponding
case with change in dimensions equal to the fillet radius, i.e. 10 mm.

Shapes of minimum stress concentration 135
7.2.2 Super-circular connection fillet
A family of shapes is defined by
x η
1 + xη2
= r (7.3)
including the circle for η = 2 and in the limit the square for η →∞. For η>2we
name this super-circles, as described in section 14.5. Discontinuous
The circular connection to a straight domain is not the best solution, because a curvature
discontinuity in curvature takes place, with large jumps especially for small circular
radii. We have seen that large stress concentration factors result for the defined model.
The super-circular shapes have continuous change in curvature from the straight line,
and is therefore expected to perform better. Figure 7.4 shows the results with different
powers η for the case of a fillet length equal to the change in dimensions, here 30
mm. For this case we conclude that η =2.275 is to be preferred, and the stress
concentration is now minimized from 1.516 to 1.496, which is however, only a minor
improvement.
In relation to smaller fillets relatively more can be obtained by the super-circular
design, but in general the very important design change is to change from supercircular
design to super-elliptic design, which we illustrate in the next subsection.
7.2.3 Extended length of the fillet
Of major importance for minimization of stress concentration, is the possible length
of the fillet. With unlimited length we can totally avoid stress concentration, so an
optimization must be based on a given length of the fillet. Figure 7.5 illustrates this
major influence, here related to super-elliptic designs with η =2.25. We notice that
almost uniform stress (energy density) along the shape is obtained for the long elliptic
shapes, and thus according to chapter 14 confirms the versatility of the simple design
Major
importance
parametrization. Down to 15 %
concentration
Still, the best stress concentration is 15 % and especially for fillets in tension,
it is very difficult to avoid stress concentration. By adding more design variables
we are able to minimize the stress concentration further, as it is verified in the next
subsection.
7.2.4 Multi-parameter optimal shape design
With more or less single parameter design studies we have seen that the fillet design
is very sensitive to many aspects of design possibility. Based on this we now conclude
that the length of the fillet should be as long as possible, thus we assume this
parameter to be given. Discontinuity in change of curvature of the shape should be

136 Copyright c○Pauli Pedersen: Optimal designs ...
Figure 7.4: Stress distributions for super-circular shapes with increasing power of
the super-circle (η =2.0 for circle), equal to 2.0, 2.1, 2.2, 2.275, 2.3, 2.4, respectively.
The corresponding stress concentration factors are 1.516, 1.503, 1.497, 1.496, 1.496,
1.498, respectively.

Shapes of minimum stress concentration 137
Figure 7.5: Stress distributions for super-elliptic shapes with increasing relative
length of the super-ellipse (η =2.25). The lengths relative to the change in dimensions
are equal to 2/3, 2.5/3, 1, 4/3, 5/3, 6/3, respectively. The corresponding stress
concentration factors are 3.124, 1.625, 1.496, 1.327, 1.222, 1.153, respectively.

Down to 8 %
concentration
Axisymmetric
modelling
138 Copyright c○Pauli Pedersen: Optimal designs ...
avoided, thus we use super-elliptic power greater than two. The major optimization
parameter is the super-elliptic axis orthogonal to the length direction of the fillet,
taken relative to the change in dimensions. In the chosen specific case for which we
minimize the stress concentration we take, with reference to the model in figure 7.1,
l4 =2l1 =60mm and length of the fillet of the same size. Figure 7.6 shows the stress
field with the optimized shape, and a stress concentration of only 8 % is obtained.
Figure 7.6: Stress distributions for a three-parameter optimized shape with given
length of the fillet. Stress concentration factor for this design is 1.085
7.3 The 3D-fillet in tension, bending and torsion
The 3D-fillet of an axisymmetric solid is optimized in (Pedersen and Laursen 1982–83).
Tension, bending and torsion are all treated as separated load cases and the length of
the fillet is imposed as a geometrical constraint. The detailed stress distributions in
the paper (Pedersen and Laursen 1982–83) all show constant von Mises stress, until a
point of the designed boundary where the geometrical constraint imposes a decrease
in this stress. Note, that von Mises stress and energy density are only proportional
for isotropic, non-compressive materials, see (Pedersen 1998).
For the axisymmetric model shown in figure 7.7, the results are presented in figure
7.8, 7.9, 7.10, 7.11, 7.12, and 7.13. All these results are from (Pedersen and Laursen
1982–83).

Shapes of minimum stress concentration 139
Figure 7.7: Shaft with a shoulder fillet loaded in bending, tension, or torsion.
Stress concentration factor : 1.025 1.014 1.007 1.000
1.146 for circular design
Length of the fillet : 0.9 1.0 1.1 1.3 ×∆R
∆R = R2 − R1
R1
BENDING load
R2/R1 =11/6
Figure 7.8: Optimal shapes with minimum stress concentration for bending load.
Solutions for different lengths of the transition zone.
R2

140 Copyright c○Pauli Pedersen: Optimal designs ...
Figure 7.9: Stress levels corresponding to a bending load, for a circular design at the
top and for an optimal design at the bottom.

Shapes of minimum stress concentration 141
For the case of bending load, the optimal shapes as a function of fillet length
are shown in figure 7.8, indicating that with the actual finite element solution (1980)
no stress concentration is found for fillet length greater than 1.3 times the change in
radial dimension. We especially note that optimal fillets are obtained by removing
material relative to the circular design.
Stress levels for a specific fillet length are presented in figure 7.9, comparing the
stress field for a circular design with the field for the optimal design.
Stress concentration factor :1.120 1.099 1.080 1.056 1.012
1.288 for circular design
Length of the fillet : 0.9 1.0 1.1 1.3 2.0 ×∆R
∆R = R2 − R1
R1
T ENSION load
R2/R1 =11/6
Figure 7.10: Optimal shapes with minimum stress concentration for tension load.
Solutions for different lengths of the transition zone.
For the case of tension load, the optimal shapes as a function of fillet length are
shown in figure 7.10, indicating that stress concentration is still found for fillet length
greater than 2 times the change in radial dimension. Thus as found in section 7.2
with a two dimensional model, stress concentration with tension load is very difficult
to minimize. Stress levels for a specific fillet length are presented in figure 7.11,
comparing the stress field for a circular design with the field for the optimal design.
For the case of torsion load, the optimal shapes as a function of fillet length
are shown in figure 7.12, indicating that stress concentration is still found for fillet
length greater than 1.3 times the change in radial dimension, but not so severe as
for the tension loading case. The optimal shapes for the three different loading cases
are compared in figure 7.13. Further detail can be found in (Pedersen and Laursen
1982–83).
R2

142 Copyright c○Pauli Pedersen: Optimal designs ...
Figure 7.11: Stress levels corresponding to a tension load, for a circular design at
the top and for an optimal design at the bottom.

Shapes of minimum stress concentration 143
Stress concentration factor : 1.030 1.021 1.016 1.009
1.106 for circular design
Length of the fillet : 0.9 1.0 1.1 1.3 ×∆R
∆R = R2 − R1
R1
T ORSION load
R2/R1 =11/6
Figure 7.12: Optimal shapes with minimum stress concentration for torsion load.
Solutions for different lengths of the transition zone.
R1
Length of the fillet : 1.3 × ∆R
∆R = R2 − R1
TENSION
BENDING
TORSION
R2/R1 =11/6
Figure 7.13: Comparison of optimal shapes for different loads.
R2
R2

Load
direction
144 Copyright c○Pauli Pedersen: Optimal designs ...
7.4 The 2D-hole subjected to biaxial stress
2b
x2
2a
x1
σ2
σ1
Numerical model
Figure 7.14: Left: plane problem for the study of stress/strain concentration around
an elliptical hole in a finite domain, with ellipse axes a, b and external stresses σ1,σ2
in directions x1,x2, respectively. Right: the actual quarter model based on symmetry.
The 2D-hole-problem is illustrated in figure 7.14 and is shown for a biaxial load
case. Numerical results for this case are contained in (Kristensen and Madsen 1976),
and these are compared to what is mentioned as the ”analytical” results. Analytical
studies of the problem are reported by (Banichuk 1977) with reference to even
earlier results by (Cherepanov 1974). These analytical studies prove that a constant
tangential stress is obtained with an elliptical shape design where the ratio of the
two half axis a, b equals the ratio of the stresses, assuming equal signs for σ1 and σ2.
More recent studies by (Cherkaev, Grabovsky, Movchan and Serkov 1998) deal with
non-equal signs for σ1 and σ2. In all these cases constant energy density is obtained
but it should be remembered that the hole is assumed infinitesimally small.
The influence of a limited domain, i.e. holes of finite size, is of major importance,
and we discuss this aspect in more detail. First we present the analytical results.
7.4.1 The classic solution for stress concentration around small
elliptical holes
Figure 7.15 shows an elliptical hole with half-axes a, b in the x1,x2 directions, respectively.
Two angles are involved in the problem, the angle φ from the x1-direction to
the direction of the load. For the load a uniaxial stress state is assumed and the
stress level far away from the hole is σ∞. The other angle θ specifies the point of the
elliptical boundary at which we determine the actual tangential stress σθ, which is a
principal stress because the stress normal to and at the elliptical boundary is zero.

Shapes of minimum stress concentration 145
x2
σθ(φ)
θ
a
x1
b
Figure 7.15: Elliptical hole in an infinite plane domain, subjected to stress far away
from the hole.
The coordinates for the point at which the tangential stress is given are
x1 = a cos t, x2 = b sin t (7.4)
with the parameter 0 ≤ t

Concentration
factor of 2
Minimum
stress
concentration
Essential
assumptions
Influence from
finite size
Elliptical are
not optimal
146 Copyright c○Pauli Pedersen: Optimal designs ...
and we may talk about a stress concentration factor of 2 for this very simple case. In
the more general case of (7.6) the maximum tangential stress at the circular boundary
is
3σ2 − σ1 ≤ σθ ≤ 3σ1 − σ2 for σ1 >σ2
3σ1 − σ2 ≤ σθ ≤ 3σ2 − σ1 for σ2 >σ1
(7.9)
Let us go back to the elliptical hole and for the result (7.6) find the condition for
a constant tangential stress, i.e. dσθ/dθ = 0 for all θ. We find after some algebra
a/b = σ1/σ2 ⇒ σθ constant (7.10)
the result frequently used as a test example in optimal shape design for minimum
stress concentration. For this case the constant stress is
σθ(a/b = σ1/σ2) =σ1 + σ2
(7.11)
Now all these classical results are based on assumptions of plane, linear elasticity,
infinite domains, and isotropic material. We use some numerical examples to study
the influence of these assumptions.
7.4.2 Numerical solutions for finite domains
The essential influence from a finite width of a strip with a circular hole can be found
in (Savin 1961) based on early results using series expansions, i.e. before the finite
element method was developed. Today such results can be obtained with almost any
finite element program, and graphical illustrations of stress state, strain state, energy
density state give direct access to the result. With the reference model shown in
figure 7.14 we discuss such results for finite domains. The sizes of the ellipse axes are
specified as a percent of the corresponding lengths.
The results in 7.16 illustrate the influence of the ratio of the hole size to the finite
domain size. With isotropy and linear elasticity and relative stresses σ1 =3,σ2 =2
together with a/b =3/2, for infinite domain from (7.11) we have σθ = 5 and constant
along the hole boundary.
Figure 7.16 shows four solutions corresponding to relative hole sizes of 10%, 25%,
33%,and 50%, respectively. For the smallest hole model we get (σθ)max =5.18, and
for the largest hole model we get (σθ)max =9.4.
The general conclusion from the results in figure 7.16 is in good agreement with
the results in (Savin 1961). It tells us that the analytical results for infinite domains
should be used with great caution, because the ratio of hole size to domain size has
an influence which should not be neglected. For the largest hole the tangential stress
along the elliptical boundary varies from 4.1 to 9.4 , while for the smallest hole the
variation is only from 5.01 to 5.18.

Shapes of minimum stress concentration 147
50% size :4.1

Strong
influence from
orthotropy
148 Copyright c○Pauli Pedersen: Optimal designs ...
7.5 The 2D-hole in an orthotropic material
With the increasing use of non-isotropic material, like laminated composites, there is
a need for shape design also for non-isotropic material. The orthotropic case is studied
in (Pedersen et al. 1992). As expected the optimal shape design is very dependent on
the level of non-isotropy.
Results for the case illustrated in figure 7.14 (biaxial relative external stresses
σ1/σ2 =3/2), and with material data as listed in table 7.1 are determined.
Material EL/10 11 Pa ET /10 11 Pa νLT GLT /10 11 Pa EL/ET
Isotropic 1.0 1.0 0.3 0.3846 1.0
5% fiber 3.450 1.052 0.3 0.4044 3.281
10% fiber 5.900 1.109 0.3 0.4264 5.322
20% fiber 10.80 1.244 0.3 0.4784 8.683
30% fiber 15.70 1.416 0.3 0.5448 11.08
Strong orthotropic 60.91 1.450 0.3 0.1378 42.00
Table 7.1: Applied material data in shape optimal design.
The optimal boundary shapes for the six materials are shown in figure 7.17, and
the concentration of energy density relative to the original elliptical boundary are for
the four fiber materials
(uɛ)max elliptical shape
(7.12)
(uɛ)max optimal shape =1.8 5%fiber, 1.6 10%fiber, 1.5 20%fiber, 1.2 30%fiber
We conclude that as expected the level of orthotropy has a strong influence on
design. The important conclusion from this study is that we obtain constant strain
energy density along the designed boundary.
Another important conclusion from this study, discussed in (Pedersen et al. 1992),
is that the optimal design is not very influenced by the finite element modelling. This
holds for a change in element type as well as for a change in the number of elements
(modelling refinement). Naturally, the stress/strain/energy values are sensitive to the
accuracy of the finite element modelling, but the designed shape is not.
7.6 The 2D-hole in a non-linear elastic material
Here we discuss the 2D-hole with non-linear material. In a thesis by (Stokholm 1998)
we can find optimal shape designs based on a constitutive behaviour that follows from

Shapes of minimum stress concentration 149
Figure 7.17: Optimal boundary shapes for different ”degrees” of orthotropy.
the stress energy density uɛ given by uɛ = σ n+1
eff /((n +1)En ) where n ≥ 1 and σeff is
the effective stress, as defined in chapter 12. The result with this material constitutive
model is again uniform strain energy density along the designed boundary shape for
the problem in figure 7.14. In reality the optimal shapes differ so little from the shapes
obtained with linear elasticity that a more detailed numerical study is necessary to
conclude that there is a difference.
For the problem with the smallest hole (10 % size in figure 7.16), we study the
influence from possible material non-linearity, using a power law model as described Stress
in chapter 12. Concentrating on the field close to the hole boundary figure 7.18 shows
the stress results for four different values of the power p =1.0, 0.75, 0.50 and 0.25
(p =1/n).
The main effect of the non-linearity on the stress field is a releasing effect. The
maximum principal stress for the four cases is σmax =5.14, 4.69, 4.32, 3.94, respectively,
and the fields also clearly show more uniformity for increasing non-linearity
(power p decreasing).
release
The strain fields and the energy density fields show the opposite effect, which Strain
is increasing localization with increasing non-linearity. Figure 7.19 shows principal
strain fields with hatching proportional to strain intensity. The maximum principal
strain for the four cases is ɛmax =0.0026, 0.0028, 0.0033, 0.0051, respectively.
The variation of stresses and strains along the hole boundary is very limited for
concentration

150 Copyright c○Pauli Pedersen: Optimal designs ...
Figure 7.18: Stress field for solutions with non-linear elasticity.
Figure 7.19: Strain fields for solutions with non-linear elasticity.

Shapes of minimum stress concentration 151
all the cases, but this is naturally due to the initial choice of a/b = σ1/σ2. However,
it tells that the ”optimal” hole shape depends only a little on the power of the nonlinearity.
A more detailed discussions of this is contained in chapter 8, section 8.4.4.
7.7 The 3D-cavity
For the cavity problem in 3-D the shape of an inclusion with minimum stress concentration
has also been studied analytically as well as numerically. As expected
in an infinite model, the optimal shape is an ellipsoid, which follows from the work
of (Eshelby 1957). The von Mises reference stress is constant on the surface of the
cavity. In (Dybbro and Holm 1986) the optimal ellipsoidal half axes a, b and c are
given implicitly in terms of the three principal stresses, and the ellipsoidal axes are
aligned with the principal stress directions. A numerical study is also performed in
(Dybbro and Holm 1986) and within the accuracy of the applied finite element model
the uniform energy density is found on the boundary of the shape.
Also for composite materials we find three dimensional studies, as by (Vigdergauz
1994). However, the number of studies are limited, and it is not expected that the
analytical results for 2D-models (based on analytically determined stress fields) can
be extended to 3D-models.
7.8 Shape design in materials for maximum bulk
modulus
Chapter 8 is specifically devoted to material design and therefore is not contained
in the present chapter. For more extended explanation than given in chapter 8, see
(Pedersen 2001a).
7.9 Stress release at a crack tip
The method of hole drilling near the crack tip is often used in fatigue damage repair.
In the survey by (Shin, Wang and Song 1996) comparisons with alternative methods
are presented, and it is concluded that hole drilling is the most effective method. A
number of experimental results are presented.
From shape optimization we know that the circular shape is by no means optimal,
as seen in sections 7.2 and 7.3. It is therefore important to find the shape of a hole,
which in a very effective way reduces the stress concentration. This shape optimization
problem is in this section restricted to holes directly at the crack tip. Optimal
stress release

152 Copyright c○Pauli Pedersen: Optimal designs ...
2h
000000
111111
000000
111111
000000
111111
000000
111111
000000
111111
000000
111111
00 11
2a
2w
x2
θ
x1
crack tip
Figure 7.20: Left: the analyzed elementary case (uniform external stresses or forced
displacements) with holes at the crack tips indicating the area of analysis (hatched)
and the area of the graphs (cross-hatched) with hole size 1 mm. Right: the
Cartesian coordinate system with origo at the crack-tip.
r

Shapes of minimum stress concentration 153
In detail we analyze the elementary case shown in figure 7.20. The simple
parametrization, used in earlier shape optimization, is applied and it is shown that
an almost uniform stress state can be obtained along the boundary of the hole. The
shape of the hole is parameterized by the super-elliptic equation
(x1/a) η +(x2/b) η = 1 (7.13)
where the x1-direction is the direction of the crack, as shown in figure 7.20. If we use
a quadratic design domain a = b (super-circle), then the only design parameter is η
(assuming that the area of the hole is kept constant). Simple
To make this shape optimization problem more clear, the results of a parameter
study are presented in figure 7.21, which also gives the possibility for explaining the
graphical display of the results from the study related to shape. We analyzed three
designs corresponding to η = 2 (circular), η =2.5 (optimized) and η = 6. Values for
relative maximum strain energy densities for the three designs are 1.00, 0.67 and 1.13
corresponding to relative maximum tangential stresses 1.00, 0.82 and 1.06. We note
that considerably better distributions of stresses are obtained with a non-circular
design. (Even more uniform fields along the hole boundary can be obtained when
more design parameters are included.) Figure 7.21 shows by isolines the resulting
fields of energy densities. In this figure it is, by the red (dark) areas added to the
boundary of the hole, illustrated how the energy density varies along the boundary
of the hole (the same technique is applied in other figures of the chapter). Note that
the illustrations are rotated 90 degrees relative to figure 7.20, i.e. the crack direction
parametrization
is now vertically downward. Illustrative
example
The objective of the optimal shape design in relation to cracks is not completely
clear. At first we may argue that the objective should be to minimize the stress
intensity factor as is done for drilled holes away from the crack tip. However, for
non-sharp crack tips the interpretation of the stress intensity factor is not clear. Thus
for this case we choose to minimize the maximum tangential stress at the boundary
of the drilled hole. In the case of plane stress this also corresponds to minimizing Objective
the maximum von Mises stress or the energy density. For the cases of non-isotropic
materials it seems very relevant to minimize the maximum energy density.
The optimization is performed by a simple parameter study, of the shape design
for non-circular holes at the crack tip. Influences of the size of the hole, of the relative
ellipse axes a/b, of the external load (stresses or forced displacements), of material
non-linearity, and of material non-isotropy are discussed and a number of specific
solutions are presented. The central aspect of the present study is to show that
circular holes by no means are optimal, as already illustrated in the introduction.
Restricting this study to holes at the crack tip we with a few (one or two) parameters
optimize the shape of the hole boundary.
The example shown in figure 7.20 is based on a number of assumptions and a

154 Copyright c○Pauli Pedersen: Optimal designs ...
Figure 7.21: The energy distribution around half the holes, corresponding to η =
2.0, 2.5, and 6.0. The red (dark) areas added to the boundary of the hole, illustrate
how the energy density varies along the boundary of the hole.
study of the sensitivity to these assumptions is needed. We discuss the influence from
the external load and size of the hole, from material power law non-linearity, from
material non-isotropy, and from the allowable design domain (square or rectangular)
for the hole.
7.9.1 Influence from the external load and size of the hole
Different external loads can be examined, either given stresses (forces) or given forced
displacements. As expected only little influence is seen as long as the crack is loaded
mostly in mode I. This follows from the fact that the near crack tip field, as a function
of the external load, only changes with a common factor (the stress intensity factor).
Optimization for cracks in mode II, mode III and combined modes needs further
studies.
In table 7.2 we show the relative concentration of the energy density, for different
sizes of the holes and for two alternative load cases. In all cases the best of the
analyzed designs corresponds to a super-elliptic power of η =2.5.
Relative to the size of the hole (0.5 , 1.0 , 1.5 and 2.0 mm), with the 1 mm size
as reference, the relative values of maximum energy density are 1.92, 1.00, 0.68, 0.52,
respectively. Thus, as expected, the larger holes give a more efficient stress release.
The size of the hole is assumed to be determined by alternative considerations, and
is not part of the optimization.

Shapes of minimum stress concentration 155
Shape parameter η 2.0 2.25 2.5 3.0 3.5 4.0 4.5
size stress load 1.0 0.857 0.840 0.876 0.934 0.994 1.05
0.5 mm displ. load 1.0 0.854 0.835 0.863 0.916 0.973 1.03
size stress load 1.0 0.857 0.843 0.894 0.970 1.05 1.13
1.0 mm displ. load 1.0 0.852 0.833 0.875 0.944 1.02 1.09
size stress load 1.0 0.856 0.842 0.895 0.974 1.06 1.14
1.5 mm displ. load 1.0 0.850 0.830 0.872 0.943 1.02 1.10
size stress load 1.0 0.854 0.841 0.894 0.973 1.06 1.15
2.0 mm displ. load 1.0 0.847 0.826 0.866 0.937 1.01 1.09
Table 7.2: Relative values of maximum energy density (for circle, η = 2 , the value is
set to 1.0). Corresponding values for stress are equal to the square root of the shown
values. Optimized values are shown in bold.
7.9.2 Influence from material non-isotropy
Optimal shape
independent of
load and size
It is expected that anisotropic material behaviour influences the optimal shape to a
large extent, see section 7.5. When the material is stiffer in the crack direction we
see little influence on the optimal shape, but when it gets more flexible in the crack
direction the influence is pronounced. We illustrate this in figure 7.22, where the ratio
of the two moduli is set to EC/ET =0.25.
We found that an energy concentration always appears for simple super-elliptic
designs. With one modification function to the shape of the hole, as described in
details in (Pedersen et al. 1992) and in chapter 6, we obtain designs with almost
uniform energy density along the hole boundary. A look at the stress distribution in
figure 7.23 for this anisotropic case, may give an understanding for the need of more
advanced designs for these cases. From the results, a two parameter description of
the boundary shape seems sufficient. Non-isotropy
influences
7.9.3 Influence from material power law non-linearity
Figure 7.24 shows results based on analysis with material non-linearity. As expected
from earlier results ((Pedersen 2001a)) the optimal shape of the hole is rather insensitive
to the power p (p ≤ 1) of the non-linearity.
We still obtain almost uniform energy density (here von Mises stress) along the
boundary of the hole. The isolines show equal levels of reduced stiffness, described

156 Copyright c○Pauli Pedersen: Optimal designs ...
Figure 7.22: Levels of energy density for EC =0.25ET corresponding to designs
with η =2.5, 4.0, and 4.0. The design to the right is a two parameter design.
Larger principal stresses
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0 20 40 60 80 100 120 140 160 180
Angle from crack direction
Isotropic
E(crack) = 0.25E(transverse)
Figure 7.23: Larger principal stresses as function of the angle relative to the crack
direction to illustrate the more complicated dependence when the crack direction is
more flexible.

Shapes of minimum stress concentration 157
by the factor (ɛe/ɛ0) p , where ɛe is effective strain and ɛ0 is the corresponding value
that gives the transition from linearity to non-linearity. The assumptions behind the
calculations leading to the results analyzed for η =2, 2.5, and 6 (as for the linear
material) correspond to deformation theory with a power law of p =0.1 Relative
values for these results are for the squared maximum von Mises stress 1.0, 0.98, 1.11
and for the minimum stiffness reduction factors 0.305, 0.336, 0.280. Non-linear
less influence
Figure 7.24: In the interior: isolines of stiffness reduction based on material power
law non-linearity. Along the boundary of the shapes: the variation of this stiffness
reduction. The three designs correspond to η =2.0, 2.5, and 6.0.
7.9.4 Influence from the allowable domain of the hole
By including the elliptic half-axes a, b as design parameters we may further improve
the levels of squared von Mises stress. The added condition of ab = constant practically
fixes the area of the hole (the parameter η has only slight influence).
In all cases we obtain almost uniform distribution along the highly stressed boundary.
Figure 7.25 shows the results corresponding to a/b =1.0, 0.9, 0.8, 0.7, 0.6, and
0.5, and give the resulting relative maximum values 1.0, 0.92, 0.84, 0.75, 0.66, and 0.59
for the resulting maximum energy densities, and thus the super-ellipse has distinct
advantages over the super-circle. The optimal super-elliptic power η changes with
the ratio a/b and for the solutions referred, we got η =2.5, 2.4, 2.3, 2.15, 2.10, 1.95,
respectively. Elongated
The same six designs were analyzed based on a strong material non-linearity
(σe/σ0) 10 and again almost constant von Mises stress is obtained along the boundary,
although now decreased with almost a factor of four, relative to the results given for
the linear solution. The values with the non-linear solution were 0.24, 0.23, 0.22, 0.21,
0.20, 0.18, respectively. The strong non-linearity levels out the difference, but still the
better designs

158 Copyright c○Pauli Pedersen: Optimal designs ...
possibility for smaller ratios a/b gives a better solution.
With illustrative examples we have shown that the stress field, at the boundary
of a stress releasing hole at the crack tip, can be significantly improved. To a large
extent the one parameter super-elliptic shape is able to return a field of constant
tangential stress along the boundary. A shape like this diminish the possibility for
further fatigue crack initiation, relative to a circular hole.
Figure 7.25: Levels of squared von Mises stress for optimized super-elliptic power,
when the ratio a/b are prescribed to 1.0, 0.9, 0.8 / 0.7, 0.6, and 0.5, respectively.

Chapter 8
Material design
Advanced materials are now used frequently in engineering design and have opened
the possibility of material design. A general characteristic of these materials is that
they are non-isotropic, which puts new demands on the analysis capabilities and on
the optimization methods. Also for materials we can classify the design parameters
in size, shape and topology. Size, shape
and topology
Let us imagine that a material point is described by a finite domain and then treat
this domain as a continuum or structure with periodic boundary conditions. This is
named a cell model. In a cell model of a material point, the size variables relate to
the density distribution, and also the orientation field within the material cell model,
belongs to this class.
For a material cell model with a hole, the boundary of the hole is classified as a
shape design parameter. The number of holes is a topology parameter, and so are the
selection of actual bar elements in a truss model, or the actual beam elements in a
frame model. Basically, the principal differences between designing a structure and
designing a material cell model are small.
Design of a material and of a structure are in principle identical problems. How- Material as
a structure
ever, for structures, we usually assume the external loads to be given, while in design
for material moduli the loads are evaluated from forced boundary displacements and
the loads are therefore design-dependent.
In the present study we also focus on the influence of these different load assumptions.
The influence of Poisson’s ratio and the influence of material non-linearity are
shown by examples, and more general insight is obtained by sensitivity analysis.
The objective can be the stiffest design, the strongest design or just a design of Different
uniform energy density, say along a boundary. In an energy formulation it is proven
in chapter 7 that these three objectives have the same solution, at least within the
159
objectives,
but the
same design

Theory in
chapter 15
Optimal
3D-modulus
matrix
160 Copyright c○Pauli Pedersen: Optimal designs ...
limits of geometrical constraints, including the parametrization. Without involving
stress/strain fields, this proof of the same solution holds for 3D-problems, for power
law non-linear elasticity and for non-isotropic elasticity. With this background knowledge,
the results of a number of specific influence studies are presented.
The chapter is divided into five sections, the first three of which are related to
size optimization. Firstly, the material design with the individual components of the
constitutive matrix as design parameters is presented, resulting in analytical solutions
for three–dimensional cases as well as for two-dimensional cases.
Secondly, the total local relative density is treated as the only design parameter.
For two-dimensional problems this is identical to the thickness optimization, treated
in chapter 4. The relation to material topology optimization is exemplified.
Thirdly and mainly for illustration, an analytical solution for a two parameter,
orthotropic case is derived, based on general knowledge from orientational design in
chapter 17 and a necessary optimality criterion, derived in chapter 14.
Section 8.4 is concentrated on shape optimal design as applied to design of materials
with a single hole in each cell of a periodic material model. Finally, in section
8.5 comments on the active research area of material design with multi-physics constraints
in addition to an objective are given. As an example such a problem could
be to optimize the bulk modulus with a lower bound on the shear modulus or bounds
on the thermal-elastic properties.
8.1 Free material design
In chapter 15 with reference to the papers (Bendsøe, Guedes, Haber, Pedersen and
Taylor 1994) and (Pedersen 1998), we have obtained analytical results for the individual
design of material constitutive components with the objective of maximizing
the stiffness of a material (minimize the stored elastic energy). The single constraint
for the optimization problem is a given total size of the constitutive matrix, measured
by the Frobenius norm or by the trace norm, see also (Taylor 2000).
For 3D-problems the non-dimensional part of the optimal constitutive matrix is
presented in (15.10) and this result also holds for non-linear elasticity modelled by
power law non-linearity. We give the results here with a dimensional factor L
[L]optimal =
⎡
L
(ɛ1 + ɛ2 + ɛ3) 2
⎢
⎣
ɛ 2 1 ɛ1ɛ2 ɛ1ɛ3 0 0 0
ɛ1ɛ2 ɛ 2 2 ɛ2ɛ3 0 0 0
ɛ1ɛ3 ɛ2ɛ3 ɛ 2 3 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
⎤
⎥
⎦
(8.1)

Material design 161
For 2D-problems the corresponding optimal constitutive matrix is
C
[C]optimal =
(ɛ1 + ɛ2) 2
⎡
⎣ ɛ21 ɛ1ɛ2
ɛ1ɛ2
ɛ
0
2 0
2
0
⎤
0 ⎦
0
(8.2)
The matrix [L] presented in (8.1) and the matrix [C] presented in (8.2) both have
only one non-zero eigenvalue, and thus only stiffness in relation to the specific strain
condition for which they are designed. We can obtain the same effective strain and
strain energy density with an isotropic, zero Poisson’s ratio material, but then the
corresponding material cost is six times greater for the 3D-problem with [L] =L[I],
and three times greater for the 2D-problem with [C] =C[I]. As shown in (Bendsøe
et al. 1994), the zero Poisson’s ratio material is valuable in numerical calculation,
Optimal
2D-modulus
matrix
because of the degeneracy of the ultimate optimal material. Examples in
chapter 4
In chapter 4 a number of designs are based on zero Poisson’s ratio, and thus may
serve as illustrations of ultimate optimal fields for specific load conditions.
8.2 Design of density distribution
Design of density distribution in a material cell with the objective of minimizing
the stored elastic energy, subjected only to the single constraint of given amount
of cell material can be solved with the use of an optimality criterion method, and
then the number of design variables is not a practical problem for the solution. Nonisotropic
and non-linear elasticity do not add further to the complexity of the solution
procedure, because the optimality criterion is in all cases uniform energy density, as Theory in
proven in chapter 12. To be specific our objective is to
chapter 12
Minimize Uɛ =
(uɛ)eρe
(8.3)
where Uɛ is the total strain energy, (uɛ)e is the specific strain energy that multiplied
with the density ρe gives the strain energy (Uɛ)e in design domain e. The total amount
of density (resource) ¯ R is given, i.e. the optimization constraint is
R :=
ρe = ¯ R (8.4)
e
We notice directly, that the optimality criterion of constant specific strain energy
(uɛ)e =ūɛ with the results of chapter 12 imply that the necessary condition of sta-
e
tionarity dUɛ = 0 is fulfilled with the constraint dR = 0. To actually find a density Procedure
distribution that satisfies the optimality criterion (uɛ)e =ūɛ we can use the heuristic
and examples
in chapter 4

Material,
yes or no ?
Modified
optimality
criterion
List of
practical
problems
162 Copyright c○Pauli Pedersen: Optimal designs ...
procedure, described in section 4.11 and used in chapter 4, again illustrating that design
of structural thickness distribution and design of material cell density distribution
are almost identical problems.
8.2.1 ”Black and white” designs
This book does not cover in detail the very active research area of the topology
optimization method, where topology design parameters are converted into size design
parameters; reference is given to the book of (Bendsøe and Sigmund 2003).
However, a comment in relation to the present description is given. To obtain a
”black and white” design, the intermediate densities must be penalized. A very used
technique to obtain this is to modify the objective (8.3) to
Minimize Uɛ =
(uɛ)e(ρe) q
(8.5)
where q is some power (often q = 3 is chosen). The sensitivity of this objective with
the results from chapter 12 is
dUɛ
= −
dρe
1
dUɛ
= −
p dρe
1
q−1
(uɛ)eq(ρe) (8.6)
p
e
fixed strains
(p = 1 for linear elasticity, but the final result is not influenced by p). We find that
if (uɛ)e(ρe) q−1 is constant, then the stationarity condition is fulfilled. This follows
directly from the condition of proportional gradients (proven in chapter 14), because
from the unchanged resource condition (8.4) follows dR/dρe =1.
The literature with methods to solve this modified problem is very rich, both with
respect to necessary specific methods and obtained designs. In fact for solution, the
methods of mathematical programming are used more than the optimality criteria
methods. A number of practical problems occur in relation to the search for these
”black and white” designs, and in fact some of these are not only related to design
for ”black and white”. We mention:
• non-unique solutions and local optimal solutions
• solutions with checkerboard patterns in the finite element models
• solutions depending on mesh
• solutions that violate limits of the actual model
• slow convergence in objective and/or design parameters
• solutions depending on stopping criterion for design iterations

Material design 163
In addition to these difficulties in obtaining a specific solution, it should be noted
that an optimized design may be strongly depending on the following aspects:
• total relative volume density
• modelling of loads and boundary conditions, especially with relation to rather
concentrated loads and reactions
• finite element modelling, i.e. element type chosen, mesh quality, and number
of degrees of freedom
• penalizing power and possible continuation approach, i.e. the strategy of penalizing
• approach for smoothing (filtering) of design and/or sensitivities
• approach for allowable design changes, such as move-limits
• approach for redesign, such as optimality criterion or mathematical programming
with specific choice of linear programming (LP) or method of moving
asymptotes (MMA) by (Svanberg 1987)
• initial design
• neglected modelling
• lower bound on density
In relation to all these aspects, see (Bendsøe and Sigmund 2003) and the references
in there as well as the two theses (Bendsøe 1995) and (Sigmund 2001).
8.2.2 An illustrative example
Without going into the discussion of all the mentioned aspects of the topology optimization
methods, we solve in a very simple manner with an optimality criterion, a
specific example and discuss the resulting designs with and without penalized intermediate
densities.
Figures 8.1, 8.2, 8.3, and 8.4 and table 8.1 show the result of the specific Quarter cell
example
problem, solved with a simple heuristic approach, based on the optimality criterion
(uɛ)e(ρe) q−1 = constant, comparing solutions with q = 3 and q = 1, i.e. with and
without penalizing. To be specific we use in all cases 16 iterations with q = 1, followed
by 16 iterations with q = 3, and the solutions are rather stable after these 16
iterations.
We find a strong influence of the penalization, and the ”black and white” designs
are obtained on the cost of lower stiffness and higher stress concentration. It must be

Penalized and
not-penalized
164 Copyright c○Pauli Pedersen: Optimal designs ...
noted that the total volume is kept the same in all the examples , and furthermore
the stiffness analyses are based on the actual densities, not the penalized densities
which are only used for the redesign procedure. (This is not the case in most topology
optimization programs). Also it should be mentioned that densities are not scaled to
the range of zero to one, but a common scaling factor does not change the penalization.
Figure 8.1: Results for 80% material usage. Left: square root of energy densities. Right:
Optimal designs. Upper not penalized with 0.001 ≤ ρe ≤ 1.25 (corresponding to 80% with
1.25·0.8 = 1), and lower penalized to ”black and white”. The colour code is red for minimum,
green for medium, blue/violet for maximum, and white indicates ”no” material.

Material design 165
Figure 8.2: Results for 50% material usage. Left: square root of energy densities. Right:
Optimal designs. Upper not penalized with 0.001 ≤ ρe ≤ 2 (50%) , and lower penalized to
”black and white”. The colour code is red for minimum, green for medium, blue/violet for
maximum, and white indicates ”no” material.

166 Copyright c○Pauli Pedersen: Optimal designs ...
Figure 8.3: Results for 25% material usage. Left: square root of energy densities. Right:
Optimal designs. Upper not penalized with 0.001 ≤ ρe ≤ 4 (25%) , and lower penalized to
”black and white”. The colour code is red for minimum, green for medium, blue/violet for
maximum, and white indicates ”no” material.

Material design 167
Figure 8.4: Results for 12.5% material usage. Left: square root of energy densities. Right:
Optimal designs. Upper not penalized with 0.001 ≤ ρe ≤ 8 (12.5%) , and lower penalized to
”black and white”. The colour code is red for minimum, green for medium, blue/violet for
maximum, and white indicates ”no” material.

168 Copyright c○Pauli Pedersen: Optimal designs ...
Constraints Design umean umax Figures Note
Uniform 1.00 22.7 all upper left a)
0.001 ≤ ρe ≤ 1.25 Optimized 0.86 14.5 8.1 upper right b)
q =1 not penalized
0.001 ≤ ρe ≤ 1.25 Optimized 0.90 14.5 8.1 lower c)
q =3and 80% penalized
0.001 ≤ ρe ≤ 2.00 Optimized 0.71 5.43 8.2 upper right d)
q =1 not penalized
0.001 ≤ ρe ≤ 2.00 Optimized 0.84 12.3 8.2 lower e)
q =3and 50% penalized
0.001 ≤ ρe ≤ 4.00 Optimized 0.65 1.28 8.3 upper right f)
q =1 not penalized
0.001 ≤ ρe ≤ 4.00 Optimized 1.02 36.2 8.3 lower g)
q =3and 25% penalized
0.001 ≤ ρe ≤ 8.00 Optimized 0.65 0.68 8.4 upper right h)
q =1 not penalized
0.001 ≤ ρe ≤ 8.00 Optimized 2.44 129.3 8.4 lower i)
q =3and 12.5% penalized
Table 8.1: Table of relative values for the solutions shown in figures 8.1, 8.2, 8.3, and
8.4.

Material design 169
Figures 8.1, 8.2, 8.3, and 8.4 show the results of optimal design with given boundary
stresses, and in each figure a chosen relative maximum density 1.25, 2, 4, 8,
respectively, corresponding for the penalized cases to total relative density of 80%,
50%, 25%, and 12.5%. Table 8.1 gives the relative values for compliance (measured
by umean) and for stress concentration (measured by umax). We comment the results,
the figures and the table by the following notes:
The finite element model has 19600 constant stress triangular elements and totally
19882 degrees of freedom with a bandwidth of 284. Discussion
a) The uniform design is used as reference for all the cases, and for this we find a
stress concentration factor equal to 22.7, active in the loaded corner as seen in the upper
left picture of all the figures 8.1, 8.2, 8.3, and 8.4, with a colour scale proportional
to the square root of element energy density, (i.e. proportional to stresses).
b) For maximum density only 25% higher than the mean densitiy not much can
be obtained by optimal design. Figure 8.1 shows this. Accepting intermediate densities
(”gray” design) we improve the compliance from 1.00 to 0.86 and the stress
concentration from 22.7 to 14.5.
c) The same case as in b) but now penalized to obtain ”black and white” design.
The lower part of figure 8.1 shows this design to the right, with its resulting distribution
of energy density to the left. The numbers are almost as for the ”gray” design
with compliance improved to 0.90 and stress concentration equal to 14.5.
d) For maximum density being the double of the mean density the optimal designs
are presented in figure 8.2. Accepting intermediate densities (”gray” design) we
improve the compliance from 1.00 to 0.71 and especially the stress concentration is
improved from 22.7 to 5.43.
e) The same case as in d) but now penalized to obtain ”black and white” design.
The lower part of figure 8.2 shows this design to the right, with its resulting distribution
of energy density to the left. The results are now different from those for the
”gray” design with compliance only improved to 0.84 and stress concentration still
rather high and equal to 12.3.
f) For maximum density equal to four times the mean density the optimal designs
are presented in figure 8.3. Accepting intermediate densities (”gray” design)
we improve the compliance almost to the limit, from 1.00 to 0.65 and the stress
concentration is then controlled from 22.7 to 1.28, but still active.
g) The same case as in f) but now penalized to obtain ”black and white” design.
The lower part of figure 8.3 shows this design to the right, with its resulting
distribution of energy density to the left. The compliance is no longer improved, but
almost unchanged from 1.00 to 1.02, and the stress concentration is now increased
to 36.2. However, this is partly due to the finite element modelling of the resulting
complicated design and boundary smoothing could be very effective. So the design in
8.3 lower right must be post-processed.
of results

170 Copyright c○Pauli Pedersen: Optimal designs ...
h) For maximum density equal to eight times the mean density the optimal designs
are presented in figure 8.4. Accepting intermediate densities (”gray” design)
we improve the compliance almost to the limit, from 1.00 to 0.65 and the stress concentration
is then almost completely eliminated from 22.7 to 0.68. We know from
chapter 14 that without size constraints the optimal design has uniform energy density,
and see that this is almost obtained. The practical question then is related to
the modelling accuracy with high density gradients at the loaded corner, as seen in
figure 8.4 upper right.
i) The same case as in h) but now penalized to obtain ”black and white” design.
The lower part of figure 8.4 shows this design to the right, with its resulting distribution
of energy density to the left. The compliance is now changed from 1.00 to 2.44,
and the calculated stress concentration is 129.3. However, this is partly due to the
finite element modelling of the resulting complicated design.
Thus, for the examples with ”black and white” solutions we need post-processing
with e.g. shape design variables. We return to this problem in section 8.4.
8.3 Design of a two parameter cell model
The design of material with total freedom in section 8.1 and the design with only
a single parameter (the density) in section 8.2 may be seen as extreme cases. For
illustration we go through the two-parameter case, for which an analytical solution is
Orthotropic possible to obtain, but is much more difficult to derive.
material Let us focus only on the objective of maximized stiffness (minimized strain energy)
based on orthotropic materials classified as low shear modulus materials. In reality
we go directly to the material model from (Bendsøe 1991), as illustrated in figure 8.5.
The total, relative volume densities (0

Material design 171
Figure 8.5: A material cell model with two directional densities.
From (Bendsøe 1991) we take the orthotropic constitutive matrix, expressed in
known modulus E and Poisson’s ratio ν and in the directional parameters µ, ξ as
[C] = Eξ
α−β2
1
β
β
α
⇒ [C] −1 = 1
Eξ
α
− β
−β
1
(8.9)
with the definitions α := ξ2 + ξ(1 − ξ)/µ and β := ξν.
The shear modulus is not involved, and we assume the major principal axis chosen
by
Constitutive
matrix
αξ/(1 + ξ)
The conditions of positive definite and limited [C] are satisfied for
(8.10)
0

172 Copyright c○Pauli Pedersen: Optimal designs ...
This is the result from the orientational optimization. Next, in the same way as
in the thickness optimization, we have uniform energy density on the global level to
determine the global ρ field. Also on the local level we must have the same energy
density in the µ and ξ ”fibers”. The statement of this local optimization problem for
given total density ρ is:
Determine the directional densities µ, ξ that
Minimize Uɛ = uɛ ˜ Vρ or
Minimize Uσ = uσ ˜ Vρ
subject to the constraint : ρ = µ + ξ − µξ (8.14)
The condition of proportional gradients from chapter 14 gives (after extended calculations
for the uɛ formulation) the analytical result
µ =
ρ
1+|ζ|
ξ =
|ζ|ρ
1+|ζ|−ρ
in terms of the total relative density ρ and stress ratio ζ := σ2/σ1.
8.4 Design of material with a single hole
(8.15)
The problem of designing the shape of a single hole in order to maximize the homog-
Optimize
enized bulk modulus is given special attention and a very simple parametrization
bulk modulus is chosen. The same parametrization is used in chapter 7. Agreement with earlier
results of (Vigdergauz 1986) and with Hashin-Shtrikman bounds, see (Grabovsky and
Kohn 1995), is obtained. Independence of Poisson’s ratio on the optimal shapes is
numerically found for these square-symmetric problems. As is shown in section 8.4.3,
this independence is generally not the case.
A very simple optimality criterion, derived in section 14.5, gives the background
for an efficient optimization procedure. Using this tool the goal of the present section
Influence
is to study the influence of the following aspects:
studies • Influence from volume constraint.
• Influence from boundary conditions.
• Influence from Poisson’s ratio.
• Influence from material non-linearity.
Without involving optimization it is useful first to study solutions where the shape
of the model in figure 8.6 is determined only by the parameter η (power of the superelliptic
function). We assume a symmetric model A = B, which with a constant solid
area gives the values α = β for a chosen parameter η.

Material design 173
Figure 8.6: The simplified shape parametrization used also for material design.
In summary, the main conclusions of the parameter study in this section are:
• The compliance of a material cell is rather insensitive to a variation
of the shape.
• In contrast the energy density concentration is very sensitive to a variation of
the shape.
• If a shape of constant energy density along the shape boundary is found, then
the compliance is stationary. Results and
conclusions
• A design for stationary compliance may return high energy density concentration
if geometrical constraints (parametrization) are a controlling parameter.
• In a finite domain problem, load applied by stress and load applied by forced
displacements give different optimal shapes.
• Especially for dense structures with given forced displacements, different Poisson’s
ratios of an isotropic material give in genera different optimal shapes.
However, for given applied stresses the optimal shape is proven to be independent.
• Also the non-linearity of a material is reflected in the optimal design, but the
influence is not very strong.
8.4.1 Influence from the volume constraint
Before the detailed results from the study of influence of boundary conditions, Poisson’s
ratio and material non-linearity are presented, we show the influence from the

Hole designs
with resulting
bulk modulus
Designs
independent of
Poisson’s ratio
174 Copyright c○Pauli Pedersen: Optimal designs ...
volume constraint, i.e. the total relative density. With reference to figure 8.6 we
define the relative density as the ratio of the area of the solid part (hatched) and the
total area. The asymptotic case of ρ → 1 is a small hole for which classical analytical
design results are available. The other asymptotic case of ρ → 0 gives a frame
structure and we concentrate on the intermediate cases of say 0.2

Material design 175
which with a constant solid area gives the values α = β for a chosen parameter η.
Three different load cases are analyzed:
• forced boundary displacements at x1 = A, x2 = B, corresponding to pure mean
dilatation ¯ɛx2 =¯ɛx1 =0.001 Different
load cases
• forced boundary stresses at x1 = A, x2 = B, corresponding to pure mean hydrostatic
pressure ¯σx2 =¯σx1 =0.0006 (modulus of elasticity C1111 =1)
• a mixed problem with forced boundary displacements at x1 = A, corresponding
to ¯ɛx1 =0.001, and forced boundary stresses at x2 = B, corresponding ¯σx2 =
0.0006
Note that for an infinitesimally small hole, these three problems are identical. Equal cases for
small holes
With zero Poisson’s ratio the Hashin-Shtrikman bound for the bulk modulus with
relative solid area ρ =0.75 is 0.3 and thus ¯σ/¯ɛ = σx1 /2ɛx1 =0.3 gives ¯σx1 =0.6ɛx1 ,
see (Pedersen 2001a).
The results of the analyses are shown in figure 8.8 in terms of the mean strain
energy densities in the solid areas (macroscopic bulk modulus = (1/2)(uɛ)mean · ρ =
(3/8)(uɛ)mean). Note, how insensitive the compliance (proportional to (uɛ)mean) is
in this rather large design domain 1.7

176 Copyright c○Pauli Pedersen: Optimal designs ...
Figure 8.8: Results of one-parameter studies, with mean strain energy density
umean = umean(η) as a function of the shape parameter η. The value from the
Hashin-Shtrikman bound is umean =0.8.

Material design 177
Figure 8.9: Starting designs in upper row with η = 2 and optimal solutions in
lower row with the left column: showing forced displacements with η =2.26, middle
column: showing forced stresses with η =1.84, right column: showing mixed loads
with η =1.96 and α =0.92β. Energy densities along the boundaries in black and
isolines for the larger principal stresses in the interiors.

Influence
on designs
Cases of no
influence
Sensitivity
analysis
178 Copyright c○Pauli Pedersen: Optimal designs ...
8.4.3 Isotropic cases with influence from Poisson’s ratio
The studies of section 8.4.2 were based on isotropic, zero Poisson’s ratio linear elasticity.
This was done to put the focus on the different boundary conditions. In
earlier optimization it was found that the optimal shape depends strongly on the
non-isotropy, see chapter 7.
However, weak dependence on Poisson’s ratio and on the non-linearity was found.
In the present section we go deeper into the influence of Poisson’s ratio and focus
on the influence on the optimal shape design, rather than on the resulting stresses,
strains and energies. It is seen that for non-symmetric problems ¯ɛx1
= ¯ɛx2 the
influence cannot always be neglected.
In the three papers (Thorpe and Jasiuk 1992), (Cherkaev, Lurie and Milton 1992)
and (Christensen 1993) the homogenized modulus of elasticity is proven to be in-
dependent of the Poisson’s ration ν for the basic material of an isotropic composite,
with critical comments found in (Zheng and Hwang 1997). In papers by Vigdergauz,
as referred in (Grabovsky and Kohn 1995), the shape for maximum bulk modulus
is also found independent of ν although the bulk modulus itself, as shown in figure
8.7, depends strongly on ν, which means that we have to be careful with the term
independent.
For more general problems with given forced displacements the objective and also
the optimal shape depend on ν, and the intention of this section is to show this
by an example. Before doing this we through sensitivity analysis get more detailed
information and separate this influence of ν into the local influence when the stress or
strain fields are kept unchanged and the more global influence from a redistribution
of these fields. From this it follows that the three sources of sensitivity are:
• a general scaling factor
• a further local change for fixed strains or fixed stresses
• a global redistribution of displacements, strains and stresses
For an isotropic material the energy densities can be evaluated from the principal
stresses σ1,σ2 and/or the principal strains ɛ1,ɛ2.
constitutive relations
Thus, we only need part of the
σ1
C1111
=
C1122
ɛ1
(8.16)
σ2
C1122 C1111
With the same C1111 all over the cell model this is only a scaling factor and the
Material
ratio
parameter µ := C1122/C1111 with − 1

Material design 179
is thus the important parameter. With a plane stress assumption we have µ = ν
(Poisson’s ratio) and with a plane strain assumption we have µ = ν/(1 − ν). Also the
ratio of principal stresses and the ratio of principal strains are important parameters
and we define as done in relation to (8.12) and (8.13) Stress/strain
parameters
γ := ɛ2/ɛ1 with − 1 ≤ γ ≤ 1, and ζ := σ2/σ1 with − 1 ≤ ζ ≤ 1 (8.18)
thus assuming both |ɛ1| ≥|ɛ2| and |σ1| ≥|σ2| (ordering). The constitutive behaviour
(14.3) is rewritten to
1
1 µ 1
σ1 = C1111ɛ1
(8.19)
ζ
µ 1 γ
It follows from (8.19) that we can express ζ in µ and λ or γ in µ and ζ
ζ =(γ + µ)/(1 + µγ), γ =(ζ − µ)/(1 − µζ) (8.20)
which is always valid as µγ < 1 and µζ < 1. The conditions in (8.18) are satisfied,
i.e. the numerically larger ɛ1 and σ1 are aligned. Note that γ = 0 implies ζ = µ and
ζ = 0 implies γ = −µ. Aligned larger
stress/strain
For linear elasticity the strain energy density uɛ and the stress energy density uσ
have the same value, which is determined by
uɛ = uσ = 1
2 (σ1ɛ1 + σ2ɛ2) = 1
2 σ1ɛ1(1 + γζ) (8.21)
Using relations (8.19) and (8.20) we can express this as
and as
uσ =
uɛ = C1111ɛ 2 1(1 + 2µγ + γ 2 ) (8.22)
σ 2 1
2C1111(1 − µ 2 ) (1 − 2µζ + ζ2 ) (8.23)
We have the global scaling factors C1111 and (1 − µ 2 ). In fact, for plane stress
models we have C1111(1 − µ 2 )=(E/(1 − ν 2 ))(1 − ν 2 )=E and therefore no scaling.
The main conclusion from (8.22) and (8.23) is that even in a fixed strain field or a
fixed stress field the energy density depends, in addition to the global scaling factor,
on the ratio µ = C1122/C1111. We have no influence with unidirectional strain, γ =0
or with unidirectional stress, ζ =0.
At the free boundary to be designed we have unidirectional stress. Thus if we
can prove the stress field to be unchanged, then the optimal shape is independent of
µ, i.e. independent of Poisson’s ratio. With given boundary stresses in equilibrium Unchanged
with
stress loads

Sensitivity
analysis
Changed
designs
with forced
displacements
180 Copyright c○Pauli Pedersen: Optimal designs ...
this is true for 2D linear elastic problems, see (Muskhelishvili 1953), pp. 160-161.
Numerical results for our actual problems confirm this.
However, with given boundary displacements the optimal design depends on µ. To
get some information also for this case we set up a global sensitivity analysis, primarily
with respect to the nodal displacements in a finite element model. In addition to the
ratio µ = C1122/C1111 we have for isotropic models C1212/C1111 =(1− µ)/2. With
this being the case for the total model, the total stiffness matrix is linear in µ and we
have
[S]{D} =([S0]+µ[S1]){D} = {A} (8.24)
where {D} are the nodal displacements and {A} the nodal loads obtained by either
prescribed boundary stresses or by prescribed boundary displacements. For {A} not
independent on µ we must solve
[S] ∂{D}
∂µ = −[S1]{D} (8.25)
and for {A} = −([S0]+µ[S1]){Df }, where {Df } is the vector of forced displacements,
we must solve
[S] ∂{D}
∂µ = −[S1]({D} + {Df }) (8.26)
to get ∂{D}/∂µ and from this the sensitivities of the strain and the stress fields.
For the symmetric problems in section 8.4.2 we have seen independence on µ,
but for non-symmetric problems this is not the case. Figure 8.10 shows rather strong
influence for an actual problem with only weak non-symmetry described by ¯ɛ1 =1.2¯ɛ2.
8.4.4 Influence from non-linear elasticity
In the sensitivity analysis relative to Poisson’s ratio we have identified three sources
of possible influence from the ratio µ := C1122/C1111. On the local level the scaling
influence through C1111 and (1 − µ 2 ) and the further influence depending on the ratio
of principal stresses or strains according to (8.22) and (8.23). On the global level we
have the influence through a redistribution of the stress/strain fields. We perform a
somewhat similar analysis for non-linear elasticity described by σeff = Eɛ p
, where
σeff , ɛeff are the effective stress, effective strain defined relative to energy densities,
see chapter 12. In general we have also for non-linear elasticity
eff
uɛ = uσ =(σ1ɛ1 + σ2ɛ2) (8.27)
and with the relation uσ = puɛ, proven in chapter 12 for power law non-linearity, we
get the generalization of (8.22) and (8.23) to be
uɛ = 1
1+p C1111ɛ 2 1(1 + 2µγ + γ 2 ) (8.28)

Material design 181
Figure 8.10: Influence of Poisson’s ratio for isotropic material, non-symmetric displaced
¯ɛ1 =1.2¯ɛ2. Strain energy densities at boundaries in black and isolines for the
larger principal stresses in the interiors.

Sensitivity
analysis
Unchanged with
size design
Changed with
shape design
182 Copyright c○Pauli Pedersen: Optimal designs ...
and
σ 2 1
uσ = p
1+p 2C1111(1 − µ 2 ) (1 − 2µζ + ζ2 ) (8.29)
On the local level we have directly from (8.28) and (8.29) that in a fixed strain field
or in a fixed stress field we get proportional changes with respect to the non-linearity
parameter p
∂uɛ
= −
∂p fixed strains
1 ∂uσ
uɛ, =
1+p ∂p fixed stresses
1
p(1 + p) uσ
(8.30)
Thus, this local influence from the non-linearity parameter p do not change a characteristic
of constant energy density and the optimality criterion is still satisfied.
We then perform a sensitivity analysis to get information about the global change
of the displacement field d{D}/dp, strain field and stress field. The finite element
formulation for this is
[S] d{D}
dµ
= d{A}
dµ
d[S]
− {D} (8.31)
dµ
Let us assume a constant stress, strain, energy density element and that the strain
state in element e is of magnitude ɛeff that relates to the actual strain energy density
uɛ by
uɛ = 1
1+p Eɛ1+p
eff
(8.32)
where E is a fixed modulus of elasticity, see chapter 12 for discussion of the definition
of ɛeff . The element stiffness matrix [Se] can be written with relative strain ˜ɛ :=
ɛeff /ɛref, where ɛref is a pre-defined reference strain
[Se] =˜ɛ p−1 [ ˜ Se] (8.33)
with [ ˜ Se] being independent of p, and then we get
d[Se]
dp = ln(˜ɛ)˜ɛp−1 [ ˜ Se] =ln(˜ɛ)[Se] (8.34)
Thus, for elements of equal energy density (equal effective strains ɛeff )weget
proportional change of the stiffness matrices. From this follows the result, seen in
(Pedersen and Taylor 1993), that in thickness design the optimal design does not
depend on the non-linearity p.
However, for shape design the effective strain ɛeff is only constant along the
boundary and thus a redistribution of displacements, strains and stresses takes place.
As we see in the example in figure 8.11, the optimal shape only changes slightly,
probably due to the fact that the change in ɛeff is only in the direction orthogonal
to the shape.

Material design 183
Figure 8.11: Change in optimal design for increasing non-linearity p =
0.75, 0.50, 0.25. Energy densities at boundaries in black and isolines for the larger
principal stresses in the interiors.

184 Copyright c○Pauli Pedersen: Optimal designs ...
8.5 Design with multi-physics constraints
See (Bendsøe and Sigmund 2003) for interesting examples and references.

Chapter 9
Bone mechanics and damage
evolution
9.1 Introduction
The structure of bone and of the bone material itself is complicated. Furthermore,
the bone material evolves and adapts to the actual load conditions. Much insight may
be gained by the mathematical modelling of bone mechanics even though it includes
rough approximations. In this chapter, results related to the research on optimal
structural and material design are communicated, although like in general for this
book, mostly limited to an ”in house” review.
9.1.1 Points of view
The complexity is very well described in (Huiskes and Kaastad 2000), a paper which
describes formulations, computations, and experiments together with expressing a
number of points of view. Also the present chapter starts with some points of view
of the author:
• Bone-mechanics is a very important subject which must be subjected to intensive
research for many years to come. We should attack the problems with all
the up-to-date tools available, i.e. extensive FEA (finite element analysis), advanced
experimental techniques at the micro-level, optimization formulations
and tools, identification techniques, and long term observations at the macrolevel.
There is no meaning in abandoning some of these tools.
185
Material and
structure

3D analysis
Identification
Formulation
186 Copyright c○Pauli Pedersen: Optimal designs ...
• The modelling for FEA must be three dimensional, due to the fact that two
dimensional models are too restrictive. In the light of improved computer
facilities extensive 3D modelling is possible.
• Static FEA are necessary to obtain homogenized material quantities and study
the influence of micro-structures.
• Long term simulations by FEA are necessary to model the re-modelling of bone,
to be tested against experimental observations. With this tool and improved
modelling, the response to change in loads or reactions to implants might then
be estimated.
• In relation to these simulations and observations, interesting identification (estimation)
of parameters can be solved using optimization procedures. Find the
parameters (the model) that minimize the difference between observations and
simulations.
in time • Like 3D is a must, so is a formulation in time. Loads and bone are time
dependent, and it make little sense to deal with combinations of loads. At any
time there is one and only one load situation. The re-modelling at any time
may however also depend on the history (recent history) of loads. In a similar
Energy density
Phenomenological
manner the sensing may also be non-local. However, models for time history
influence as well as space non-local influence can be formulated.
• With non-isotropy, a local measure of energy density must be more reasonable
than a measure of strain or stress. Note, that stiffness and strength are independent
quantities, although sometimes good stiffness also give good strength.
However, in certain cases good strength can be obtained by actually removing
material and decreasing the stiffness.
• Optimization formulations in bone-mechanics has a broad range of applications,
in particular in relation to re-modelling. So at any time step in a simulation, an
optimization model is formulated and solved. Note, that advanced optimization
models must include a number of constraints, and learning by doing the models
must be improved.
re-modelling • These phenomenological models, that are tested against observations, will also
be influenced by the increasing understandings obtained on the micro levels,
Shape and size
from FEA analysis and experiments.
optimization • Optimal re-modelling can be formulated as shape optimization (surface) as well
as size optimization (density), and combination of these. In addition the orientational
re-modelling is important, either in a macro sense controlled by
principal stresses or in a micro sense by changes in micro-structure, which give
changes in homogenization results.

Bone mechanics and damage evolution 187
9.1.2 Contents of the chapter
Analysis, evolution and layout in bone mechanics are described as seen from an optimal
design and solid mechanics perspective. Non-isotropic behaviour and three
dimensional modelling are necessary and focus is therefore put on the aspects of dealing
with a large number of material parameters. The tools of homogenization, inverse
homogenization and some basic results from optimal structural design are described. Non-isotropic
Bone structures/materials are not isotropic, and thus the non-isotropic nature
should be properly modelled. This holds for the bone stiffness as well as for the
bone strength and the bone evolution. There is also evidence that three dimensional
modelling is necessary, so just for stiffness we need 21 material parameters. The
homogenization method is a method to obtain these parameters, the method is based
on a detailed description of a substructure or a material point model.
Normally, an actual bone shows a very specific layout in terms of distribution
of densities and material orientations. Can we postulate an objective that leads to
these layouts, such as the postulate of maximum stiffness for minimum material? In
general the answer is no, but we can learn from such a postulate. The example of
a femur bone structure from the thesis by (Bagge 1999) and the paper by (Bagge A thesis
2000) is discussed. It is found that the uncertainty in knowledge about actual loads
(force transfer) is also a critical issue in bone mechanics.
The load conditions on a bone change with time due to the different activities
of the total body. The bone adapts to these changes, and we need a model for this Adaptation
adaptation. Several models are suggested, and we will her describe a model based
on stimulus from an energy density situation. The energy density situation is time
dependent and the modelling may include some form of memory.
With reliable tools for the simulation of bone evolution, we can then consider
the important problems related to design of the implants that replace bones, e.g. Implants
a hip prosthesis. Traditionally, these implants are made of homogeneous, isotropic
materials, in large contrast to the bone structure which they substitute. Mathematical
simulations can show the advantages of alternative implant design, although from a
practical point of view, it may be difficult to introduce such implants. A more feasible
design may be obtained by pure shape optimization of the implants already in use.
9.2 Bone layouts from an optimization process
In a discretized model of a bone, we assume that each element in a finite element
model has stiffness and strength given by a restricted number of parameters, e.g. the
density µe and the orientation(s). The principal (extreme) material directions align Aligned
directly with the principal stress directions, and we can then obtain the stiffness as a
function (interpolation) of the density µe. When this information is available, we can
orientations

188 Copyright c○Pauli Pedersen: Optimal designs ...
iteratively determine the material distribution that will maximize the total stiffness
(minimize the stored elastic energy) for a given total amount of material. For such a
procedure to be effective, the experience obtained through the extensive research in
Proximal
the area of topology optimization is needed. See (Bendsøe and Sigmund 2003).
femur For a proximal femur, it is shown in (Bagge 1999) and (Bagge 2000) that realistic
layouts in agreement with observed layouts are obtained. Much attention has been
paid to the loads on the femoral head due to compressive body weight forces and
to tension forces from the muscles. From this research it has been concluded that
3D-modelling is necessary.
The extension of homogenization based on frame models to continuum models
leads to plate-like structures with closed walls. It is concluded in (Sigmund 1999) that
to obtain the open wall micro-structures seen in reality, the optimization objective
of maximum stiffness must be modified, or appropriate constraints must be added to
the formulation.
Let us now assume an initial bone structure corresponding to a certain load level
is given, and then we change the load level, the load distribution or the bone structure.
The evolution of µe as a function of time can then be based on the tendency to reach
Memory
function
a new optimal solution, but a maximum rate of change dµe/dt must be included. In
(Bagge 1999) a memory function is also included in the formulation, and different
simulations over a range of 10,000 days are performed and shown. This procedure
is also used for modelling the influence of implants, and it may be a tool for design
of implants or training procedures. However, a much closer cooperation with actual
observations is needed before the role of these simulations can be clarified.
The modelling behind (Bagge 1999) and (Bagge 2000) is restricted to cubic symmetry
for the bone material model, and thus a one parameter description µe is possible.
In (Bendsøe et al. 1994) a procedure for free material evolution is described.
9.3 Finite element analysis and load modelling
The essential tool for mechanical analysis of bone is the finite element method with
nodal displacements {D} obtained from solving the linear system of force equilibrium
[S]{D} = {A}, where [S] is the total stiffness matrix and {A} the total load vector.
The total stiffness matrix, often of the order 10,000 to 100,000, is obtained by accumulating
element stiffness matrices [Se] of the order 10. The constitutive matrix [Le],
that for linear elasticity {σe} =[Le]{ɛe} relates element stresses {σe} and element
strains {ɛe}, is the essential basis for determining the element stiffness matrix [Se].
With reference to the thesis (Bagge 1999) we know that it is necessary to model
three dimensional and non-isotropic behaviour of bone. This means that the constitutive
matrix [Le] is determined from up to 21 material parameters. This does not

Bone mechanics and damage evolution 189
complicate the finite element analysis itself, but it is almost impossible to measure the
many necessary parameters and even more difficult to model the evolution of these.
A major part of this chapter is related to this problem. Load estimation
A further complication is to obtain knowledge about the actual load (in reality
load cases), as here described by the nodal loads {A}. A literature review and measures
of the hip joint forces, force directions and moments during walking and running
can be found in (Bergmann, Graichen and Rohlmann 1993). Muscle force magnitudes
during gait are reported in (Pedersen, Brand and Davy 1997). For more extensive
reference in relation to the proximal femur, see (Bagge 2000). Also it should be kept
in mind that the loads are by no means stationary.
The discretization into elements (or regions, or cells) should be commented. Although
stresses, strains and energy densities are defined with reference to a point, we
here assume constant stress, strain and energy density in an element e. Dealing with
models containing 10,000 to 100,000 elements, this will not restrict the generality.
The finite element analysis will not be further commented, but a comment on a
non-traditional formula for the elastic energy density is needed. We write the elastic
energy density ue as a scalar product { ˆɛe} T {Le} by defining the vector of constitutive
components {Le} and the vector of second order strain components { ˆɛe} Constitutive
vector
and
{Le} T := {{L1111,L2222,L3333}, √ 2{L1122,L1133,L2233},
2{L1112,L1113,L1123,L2212,L2213,L2223,L3312,L3313,L3323},
2{L1212,L1313,L2323}, 2 √ 2{L1213,L1223,L1323}} (9.1)
{ ˆɛe} T := {{ɛ 2 11,ɛ 2 22,ɛ 2 33}, √ 2{ɛ11ɛ22,ɛ11ɛ33,ɛ22ɛ33},
2{ɛ11ɛ12,ɛ11ɛ13,ɛ11ɛ23,ɛ22ɛ12,ɛ22ɛ13,ɛ22ɛ23,ɛ33ɛ12,ɛ33ɛ13,ɛ33ɛ23},
{ɛ 2 12,ɛ 2 13,ɛ 2 23}, 2 √ 2{ɛ12ɛ13,ɛ12ɛ23,ɛ13ɛ23}} (9.2)
where we have omitted the index e on the individual components. This writing Energy density
hopefully makes the following sections more clear. It is directly verified that the
result ue = { ˆɛe} T {Le} of these definitions agrees with alternative forms like ue =
{ɛe} T [Le]{ɛe}. In reality {Le} is just the vector notation for the fourth order elasticity
tensor defined with invariant length, see (Pedersen 1995). Note, that the energy
density in this chapter is the sum of the strain energy density and the stress energy
density.

190 Copyright c○Pauli Pedersen: Optimal designs ...
9.4 Homogenization and inverse homogenization
For calculating effective material properties we use the homogenization method and
assume the existence of a base cell that in a periodic distribution represents the bone
material. Subjecting this base cell to any test strain field that satisfies the periodicity,
we demand the elastic energy of the homogenized base field to equal the elastic energy
Homogenization of the detailed base cell (the micro-structure).
analysis For a homogeneous micro-structure with volume V and material parameters {LH }
the elastic energy U is directly given by U = V {ˆɛ} T {LH } when the strain field
{ˆɛ} is constant. With only six strain states we can relate the 21 material parameters
in {LH } to combinations of elastic energy densities (assuming linear elasticity).
For the first state we choose ɛ11 = 1, and others ɛij = 0; then similar for
ɛ22,ɛ33, √ 2ɛ12, √ 2ɛ13, √ 2ɛ23, i.e. only one strain component not equal to zero in each
state. The resulting formulas are listed in (Pedersen and Tortorelli 1998), see also
chapter 11.
For the inhomogeneous micro-structure, the bone micro-structural model is subjected
to boundary displacements corresponding to the six strain states and we perform
a finite element analysis. Except for the periodicity of the boundary conditions
this analysis is exactly the same as for any macro-structure with loads only from
forced displacements. With elastic energy from the six different load cases we have
the homogenized constitutive parameters.
Frame model
with cubic
symmetry
Micro-structure
optimization
In the thesis by (Bagge 1999) a micro-structural three dimensional frame model
was applied as idealized in (Guedes and Kikuchi 1990). This chosen micro-structure
gives cubic symmetric material properties, and calculations for discrete values of the
relative volume fraction µ are found using the program PREMAT3D by (Guedes
1995). Finally the homogenized material parameters are fitted to polynomials, thus
making derivatives available.
As an alternative to postulate a micro-structural model and then by homogenization
find the constitutive parameters as a function of e.g. a density parameter, we
may find the micro-structure as a result of an optimization procedure. This local
optimization problem is similar to the global problem presented in the next section
and results obtained with the techniques of topology optimization are given in (Sigmund
1999). This paper discusses the relation between closed and open wall cells and
conclude that closed wall cells have optimal stiffness properties whereas open wall
cells are non-optimal. Specific stress states are considered with focus on optimized
bulk modulus.
In a more abstract setting the paper by (Bendsøe et al. 1994) treats the ”free
material” with direct design of the individual constitutive parameters. Analytical
results are obtained and ultimate bounds are thereby available. Alternative formulations
are found in (Rodrigues, Jacobs, Guedes and Bendsøe 1999) and in (Jacobs,

Bone mechanics and damage evolution 191
Simo, Beaupre and Carter 1997).
The problem of micro-mechanical design is understood as the problem of finding a
distribution in a base cell of homogenization, in order to obtain prescribed constitutive
parameters or to satisfy constraints on these. For obvious reasons the problem Inverse
is also termed the inverse homogenization problem or identification of the micromechanical
structure. These problems can as most optimization problems be solved
by a number of different iterative methods, where each iteration includes analysis of
a micro-mechanical structure, sensitivity analysis and decision of an improved design.
For the sensitivity analysis we need the derivatives of the constitutive parameters,
which can be found from derivatives of elastic energies.
9.5 Bone layout based on
one parameter optimization
homogenization
If we for a frame-like bone structure postulate that the density distribution and the
field of orientations in a bone structure are controlled by an extremum principle, like
maximum stiffness for a given amount of material, then we should be apple to find the
bone layout by solving an optimization problem. Maximum stiffness corresponds to
minimum of elastic energy U, and the available amount of material ¯ V we accumulate
over the elements, each with relative volume fraction µe, i.e. the constraint of the
problem is µeVe − ¯ V = 0. Omitting at first the orientationel problem, then the
optimization problem is stated Optimization
formulation
F ind µe which Minimize U subject to g = µeVe − ¯ V = 0 (9.3)
The elastic energy is accumulated over element elastic energies Ue
U = Ue = Veue = Ve{ ˆɛe} T {Le} (9.4)
Using the results from (Pedersen 1998), see chapter 13, the derivative of U with respect
to the local constitutive parameters (without influence on the loads) simplifies to Sensitivity
dU ∂U
= −(
d{Le} ∂{Le} )¯ɛ = −( ∂Ue
∂{Le} )¯ɛ = −Ve( ∂ue
∂{Le} )¯ɛ = −Ve{ ˆɛe} T
(9.5)
where sub-index ¯ɛ indicates the fixed strain field.
For an optimization problem with only one constraint the optimality criterion is
that the ratio between the derivative of the objective (U) and the derivative of the
constraint (g = 0) should be the same for all optimization variables, see chapter 14.
of energy

Optimality
criterion
Special case
Incremental
formulation
Optimal
re-modelling
192 Copyright c○Pauli Pedersen: Optimal designs ...
These derivatives are with the result (9.5)
dU
dµe
=
dU d{Le}
T d{Le}
= −Ve{ ˆɛe}
d{Le} dµe
dµe
(9.6)
dg/dµe = Ve (9.7)
and thus the actual numerical problem is to find a distribution µe that satisfies the
optimality criterion
{ ˆɛe} T (d{Le}/dµe) =constant (same for all e) (9.8)
As mentioned earlier the relation between the vector of constitutive parameters {Le}
and the relative volume fraction µe follows from the basic homogenization of the
material model. It might be advantageous to fit the results of the homogenization
with functions that then gives analytical derivatives d{Le}/dµe. The special case of
a homogeneous relation {Le} = µ q e ˜ {Le}, where {Le} ˜ do not depend on µe, results in
ue/µe =a constant, i.e. constant elastic energy density per optimization variable.
Different methods are available for the numerical solution, see (Bagge 1999). For
a proximal femur, it is shown in (Bagge 2000) and (Bagge 1999) that realistic layouts
in agreement with observed layouts are obtained. For these solutions the principal
material directions are aligned directly with the principle stress directions, as by the
law of Wolff and in general proven for low shear stiffness materials in (Pedersen 1989),
see chapter 17.
With time varying loads the optimization problem is formulated as finding the
optimal redesign within a time-step where the load case is fixed. Constraints based on
maximum rate of change of the relative volume fractions as well as the maximum reorientations
are taken into account. The specific numerical procedures are described
in (Bagge 2000).
9.6 Multi-parameter evolution
Based on the conclusion that it will not be possible to measure individually the
simultaneously evolution of 21 material parameters, a procedure using an optimization
principle was suggested in (Hammer and Pedersen 1999) for laminated materials and
in (Tortorelli and Pedersen 1999) for bone materials.
Although experimental verification is still missing the procedure will be shortly
described here. We postulate that in a time-step of a simulation the change in {Le},i.e.
{∆Le}, will be such that the change in total elastic energy ∆U is extremized. From
(9.5) we get
∆U =
−Ve{ ˆɛe} T {∆Le} (9.9)
e

Bone mechanics and damage evolution 193
We see directly that extreme ∆U is obtained when {∆Le} and { ˆɛe} are aligned, i.e.
{∆Le} = Ce{ ˆɛe} and only the constant Ce needs to be determined.
The vector {Le} contains the 21 independent material parameters for an element
constitutive behaviour. The length Fe of this vector (F 2 e := {Le} T {Le}) is known as
the Frobenius norm of the constitutive matrix [Le], and we assume that the magnitude
of evolution of stiffness is described by the rate of change ˙
Fe = dFe/dt. Figure 9.1 Rate of
shows such phenomenological(postulated) bone evolution. Several different domains
are identified: degradation due to low energy density, quiescence due to ”normal
energy density”, growth due to high energy density, and damage due to extensive
energy density.
f
ud
(urd,frd)
ug
(urg,frg)
(uf ,ff )
0 2 4 6 8 10
u
degradation quiescence growth failure
Figure 9.1: Possible evolution function for the velocity factor f in an element as a
function of the actual element energy density u. The postulated function is specified
by the 8 parameters shown in the figure with an addition of three power parameters,
for this figure: nd = −1.5,ng =2.0, and nf =4.0.
The actual function ˙ F = ˙ F (u) is only illustrated by a non-dimensional factor
f and must actually be identified from observations. The present description by
constitutive
norm

Example of
rate function
Constitutive
increments
Constitutive
constraints
Non-bone
materials
194 Copyright c○Pauli Pedersen: Optimal designs ...
functions is
f = fd
nd ud−u
ud
0 ≤ u ≤ ud
f = 0 ud ≤ u ≤ ug
f = frg
u−ug
f = ff +(frg − ff )
ng
urg−ug
nf u−uf
urg−uf
ug ≤ u ≤ urg
ug ≤ u (9.10)
but alternatively a pure numerical tabulation may be chosen.
By differentiation of F 2 e we get 2Fe ˙ Fe =2{Le} T { ˙ Le} and with { ˙ Le}∆t = {∆Le} =
Ce{ ˆɛe} we obtain
Fe ˙ Fe∆t = {Le} T Ce{ ˆɛe} (9.11)
In (9.11) we know Fe, Fe, ˙ ∆t, {Le}, { ˆɛe} , then the unknown constant Ce is determined
by
Ce =(Fe ˙ Fe∆t)/({Le} T { ˆɛe}) (9.12)
and with ue = {Le} T { ˆɛe} we get the result
FeFe∆t ˙
{∆Le} = { ˆɛe} (9.13)
ue
During a simulation we can also put constraints on { ˙
Le} = {∆Le}/∆t and control
that the matrix [Le] behind the definition of the vector {Le} stays positive definite.
For details and example of evolution of a beam problem, see (Tortorelli and Pedersen
1999). For laminate problems a slightly modified procedure is described in (Hammer
and Pedersen 1999).
For metal a damage limit σd may define the initiation of the stiffness reduction
with increasing rate for increasing effective stress σeff . Figure 9.2 shows such a
phenomenological(postulated) curve for damage. This might seem as a rather unusual
alternative to keeping track of strength surfaces and their evolution. In (Hammer and
Pedersen 1999) another approach to evolution of strength surfaces is presented and
applied to laminate damage.

Bone mechanics and damage evolution 195
f
σd
0 2 4 6 8 10
no damage damage
(σrd,frd)
σeff
Figure 9.2: Possible evolution function for the velocity factor f in an element as a
function of the actual element effective stress σeff . The postulated damage function
is specified by the 3 parameters shown in the figure with an addition of the power
n =1.5, for the damage part.

Note
196 Copyright c○Pauli Pedersen: Optimal designs ...
9.7 Conclusion
The present chapter informs about methods and formulations that hopefully can be
helpful in further understanding the complicated behaviour of bone. Simultaneously
analysis on the micro-structural level as well as on the macro-structural level is now
possible. Large computer simulations in 3D-formulations with non-isotropic behaviour
are possible, but further efforts on clarifying the experimental possibilities are needed.
Close relation between research in bone mechanics and in optimal material design
has been the focus of international meetings, like an informal workshop on ”Comparing
structural optimization and tissue adaptation models”, arranged by B. Kohn and
S. Cowin in New York January 1997, an the IUTAM symposium in May 1998 with
proceedings in (Pedersen and Bendsøe 1999), and an AIMETA mini-symposium in
September 2001 with a journal special issue (Prendergast and Contro 2002). It seems
clear that many years of research is still needed within this interesting and important
area of science.
It must be stressed that the suggested evolution models illustrated in figures 9.1
and 9.2 are only an idea, and further research with experimental backup is very much
needed.

Chapter 10
Identification (estimation)
and inverse problems
This chapter describes solutions to two problems related closely to laminate analysis
and laminate design, but not with the objective of optimal design as in chapter Not optimal
design
6. The first part could be termed estimation of material parameters for composite
laminates and is closely related to the note (Pedersen 1999a) and the references in
this note, especially the papers by P. S. Frederiksen. The solution approach belongs
to the experimental-numerical methods for which a much wider area of application
is possible, as seen in estimation of stiffness models in (Pedersen 1986a), estimation
of applied loads, estimation from a hardness test (Pedersen, Mortensen and Larsen
1998), and estimation from a deep drawing process (a project in progress).
The second part could be termed inverse problems for laminates and materials, Laminate
where the goal is to find a design that returns specified specifications, say the bending
stiffnesses of a laminate or the constitutive parameters of a material. A recursive
iteration technique based on an optimality criterion is described, and may serve as
an illustration for the many other problems within the class of inverse problems in
general.
10.1 Estimation of orthotropic material parameters
inverse
problems
A combined experimental-numerical method is presented with the goal of obtaining Experimental
the material stiffness for composite materials. The identification is based on eigenfre- - numerical
quencies for a free rectangular plate, because excellent agreement between measured
and calculated eigenfrequencies can be obtained. The numerical identification prob-
197

Complicated
laboratory
tests
Simple
laboratory
tests
198 Copyright c○Pauli Pedersen: Optimal designs ...
lem is formulated as an optimization problem, and in one experiment we obtain all
the involved moduli.
The total approach consists of experiment, numerical , eigenvalue sensitivity analysis,
optimization, and error estimation. Results from these different parts of the
approach are commented and a general conclusion is that the method has several
significant advantages compared to traditional determination of material moduli. An
overview of the state of the art is intended.
Traditional laboratory tests are designed to give the desired quantities in a direct
manner. As an example, determination of material parameters of constitutive models
leads to special design of test samples and choices of applied load in an attempt to
obtain homogeneous stress-strain fields. These idealized tests are not easy to perform,
and especially with composite systems problems often arise. Furthermore, the tests
have a local nature, e.g. the point of the strain gauge. So for more general material
information a series of tests is required.
The identification approach is an experimental strategy from an opposite point
of view. The experiment is made as simple as possible to give reliable results.
On the other hand, this often leads to complex inhomogeneous stress-strain fields
where the desired quantities are not among the directly measured quantities. This
causes a complicated interpretation, but nowadays this is comfortably taken care of
by computer calculations. Although our aim is to find static material data, we decide
to measure eigenfrequencies of a free rectangular plate because excellent agreement
between measured and calculated eigenfrequencies can be obtained. A structural
Research
groups
eigenfrequency is an integrated quantity and we thus obtain material quantities that
are valid in the mean for the entire laminate.
In a pilot project by (Markworth and Petersen 1987) this technique was found
most promising (some of the results can be found in (Pedersen and Frederiksen 1992).
The same conclusion was reported in the doctoral thesis by (Sol 1986), and the research
groups in Brussels, Belgium and in Eindhoven, The Netherlands have continuously
refined the method and obtained impressive results. The proceedings from an
Euromech colloquium, (Sol and Oomens 1997), give a picture of the state of the art.
A doctoral thesis by (Kuttenkeuler 1998) shows the importance of these techniques
for composite design.
In the author’s home department the extended work of P. S. Frederiksen reports
to a great depth about the method. Especially the work related to thick plate formulations,
(Frederiksen 1995), (Frederiksen 1997b), should be noted and also the work
related to parameter uncertainty (Frederiksen 1998) deserves special attention. The
research group in Portugal has concentrated on the combination with the finite element
method and has also applied thick plate formulation, see (Soares, de Freitas,
Araujo and Pedersen 1993) and (Araujo, Soares, Freitas, Pedersen and Herskovits
2000).

Identification (estimation) and inverse problems 199
Here we first formulate the identification problem from an optimization point of
view. Then the experimental set-up is described, followed by numerical eigenvalue
analysis and sensitivity analysis. The optimization algorithms are shortly commented,
and then results from the literature are commented. Among these are the study of
material moduli in different environments.
10.1.1 Identification formulation
A structure can be described in many ways, directly or indirectly. The direct approach
can be by parameters, like length, width, height, weight, topology and material parameters.
Alternatively, we can describe it by its response like static displacements
or dynamic eigenfrequencies. Combination approaches are also possible. A laminated
plate is a structure where the ply stacking is an important aspect. The thickness
of the plies and their positions in addition to ply orientations are the parameters
that describe this stacking. Also boundary conditions are of vital importance for the
response of the plate.
The laminate membrane stiffnesses [A], as well as the laminate bending stiffnesses
[D] are linearly depending on the material stiffness [C] for each ply, i.e. upon the
constitutive matrix. The chosen identification problem we describe as a problem where
we do not have information enough about [C] , but we have information about the
structural response, say in terms of measured eigenfrequencies ωi. Let us be specific
and formulate the identification problem as an optimization problem. A number of
parameters pk are unknown for the determination of the material stiffnesses
[C] =[C(pk)] (10.1)
and therefore a calculated eigenfrequency response also depends on these parameters
ωi = ωi(pk) (10.2)
The functional relations (10.1) and (10.2) can be found in textbooks on laminates,
say in (Jones 1975) and in (Pedersen 1987b).
The optimization problem is to find best possible agreement between measured
eigenfrequencies ¯ωi and calculated eigenfrequencies ωi. An objective could be to Objective
Minimize Φ= (ωi − ¯ωi) 2
i
¯ω 2 i
(10.3)
i.e., a weighted least-squares estimator, but naturally other choices are possible. For
example, more weight could be given to more accurate measurements, and a priori

Move-limits
Non-dimensional
parameters
Constraint on
definiteness
200 Copyright c○Pauli Pedersen: Optimal designs ...
expectation on parameter values can be included. The solution must be obtained in
an iterative process, where the parameters are simultaneously improved
(pk)n+1 =(pk)n +(∆pk)n for n =1, 2,... (10.4)
and constraints can be put on the absolute values of the parameters (side constraints)
as well as on the changes of the parameters (move-limits)
(pk)min ≤ (pk) ≤ (pk)max and (∆pk)min ≤ (∆pk) ≤ (∆pk)max (10.5)
With this kind of constraints we do not need in advance to distinguish very clearly
between known and unknown parameters. For orthotropic two-dimensional models,
two parameters are especially important. The level of non-isotropy described by the
parameter α2
α2 := (1 − ET /EL)/2 (10.6)
where ET ,EL are the engineering moduli in transverse and fiber direction, respectively.
The relative shear stiffness is described by the parameter α3
α3 := (1 + (1 − 2νLT )ET /EL − 4α0GLT /EL)/8 (10.7)
where νLT is the major Poisson’s ratio and GLT is the shear modulus. The nonimportant
parameter α0 is given by α0 = 1 − ν 2 LT ET /EL, and a further not so
important parameter is
α4 = α3 + νLT (1 − 2α2) (10.8)
i.e. only new information from the Poisson’s ratio. Solving the problem (10.3) in the
formulation
Minimize Φ=Φ(α2,α3,α4) ≥ 0 (10.9)
subject to the constraint of a positive definite constitutive matrix. Written in the
non-dimensional parameters we find
1 − 2α2 > 0; 1 − α2 − 3α3 − α4 > 0; 1 − 2α2 − (α4 − α3) 2 > 0 (10.10)
When α2,α3,α4 are determined we can invert the relations (10.6), (10.7) and (10.8)
and identify EL, ET , GLT and νLT .
10.1.2 The experimental setup
A detailed description of the experimental procedure and the critical aspects of this
can be found in (Frederiksen 1997a). Here we describe only the main aspects. As
shown in figure 10.1 the setup consists of six main components:
1) the test specimen, i.e. the laminated plate

Identification (estimation) and inverse problems 201
2) the excitation source, i.e. an impact hammer or a loudspeaker
3) the response detector, i.e. an accelerometer, a microphone or
a laser vibrometer
4) a power amplifier
5) a transient recorder and frequency analyzer
6) a personal computer, which includes the identification program(s)
Figure 10.1: Scheme of the experimental setup.
The test specimens in the works of Frederiksen were chosen as rectangular, orthotropic
and symmetric laminates. In the work by (Kuttenkeuler 1998) also non-
rectangular test specimens were used, but then the numerical analysis needs to be Test
based on finite element modelling. In the section on uncertainties we discuss the other
physical aspects for the test specimens.
specimens
As excitation source we mostly use a small impact hammer, which imparts a force Excitation
with a broad frequency range to the plate. This simultaneously excites all the modes
of vibration. The pulse frequency range can be altered by adjusting the tip-hardness
and the hammer weight. In the identification of complex moduli by (de Vissher, Sol,
de Wiede and Vantomme 1997) the excitation by a loudspeaker or a disconnected
shaker is also described. The joke is: ”ask the plate for its moduli”, but do it loudly
(in any language).
In the early research a light weight accelerometer with a mass of 2.2 gram was
source
used as response detector, and the concentrated mass was taken into account in the Response
eigenfrequency calculations. In more recent research, especially for the study of small
and light plates, a microphone is used as response detector. In (de Vissher et al. 1997)
detector

Further
tools
Rayleigh-Ritz
or FEM
Higher order
plate theory
Subspace
iterations
202 Copyright c○Pauli Pedersen: Optimal designs ...
a laser vibrometer is shown as response detector. In general we can state that good
experimental repeat-ability is obtained and reported by the different authors.
The power amplifier, the transient recorder and the PC equipment are standard
experimental tools. The software separation between the transient recorder and the
PC is not very specific. Thus, the fast Fourier transformation could be contained in
either of them.
10.1.3 Plate eigenfrequency analysis and sensitivity analysis
Plate eigenfrequencies can be calculated with different methods. For the numerical
modelling as well as for the numerical procedure we have many possibilities and in
this section we comment on the ones used for the actual identification problem. For
detail see (Frederiksen 1995) and (Frederiksen 1997b).
The numerical modelling is performed either by the Rayleigh-Ritz method with
global expansion functions or by the finite element method with local expansion functions.
Both methods are applicable to rectangular plates, but for more complicated
boundary shapes only the finite element method is practical. The main issue of the numerical
modelling is to choose the relevant plate theory, and it is generally agreed that
higher order plate theories are often required. In the studies of (Frederiksen 1997b)
an estimation of the errors that result from the use of the classical plate theory is
included. The higher order shear deformation theory by (Reddy 1984), (Levinson
1981) is a good compromise between accuracy and complexity.
When the numerical modelling is chosen we end up with a generalized, linear
eigenvalue problem
[S]{Di} = ω 2 i [M]{Di} (10.11)
where [S] is a ”stiffness” matrix, [M] a ”mass” matrix, and ω 2 i , {Di} a squared natural
frequency and its corresponding eigenvector. In a Rayleigh-Ritz model the order of
the matrices is say 500 , while in a finite element model the order may be say 20.000 .
(Comparisons of results from the two different methods can be found in (Frederiksen
1997b)). The assumption of linearity is justified by very small strains during the
vibrations (1-10 micro-strains).
An effective numerical procedure to solve the matrix eigenvalue problem (10.11)
is the subspace iteration method, see (Bathe 1982). By iteration this method de-
termines the eigenvectors and then evaluates the eigenvalues in a selected domain of
the spectrum, say the lowest eigenfrequencies. The method is capable of dealing with
multiple eigenfrequencies, and in general computer time is no problem.
For the identification we also need the partial derivatives of the eigenfrequencies
with respect to the material moduli, here symbolized by a parameter pk. The general

Identification (estimation) and inverse problems 203
result for distinct eigenfrequencies is
when the normalization
dω 2 i
dpk
= {Di} T (−ω 2 i
d[M]
dpk
+ d[S]
){Di} (10.12)
dpk
{Di} T [M]{Di} = 1 (10.13)
is assumed. Detailed description of this sensitivity analysis is presented in chapter
18.
For the actual problem where d[M]/dpk = [0], because the material moduli have Eigenvalue
no influence on the mass matrix, we get
sensitivities
dω 2 i
T d[S]
= {Di} {Di} (10.14)
dpk dpk
For multiple eigenfrequencies a special calculation is necessary to obtain the eigenvectors
that uncouple the sensitivities. With {D}i, {D}j being two eigenvectors corresponding
to the same eigenfrequency ωi, we determine these vectors uniquely by the
two orthogonality conditions:
where δij is Kronecker’s delta.
{D} T i [S]{D}j = δijω 2 i and {D} T i
10.1.4 Example results
d[S]
dpk
dω
{D}j = δij
2
dpk
(10.15)
In the different studies of identification techniques, different objectives and different
optimization algorithms have been applied. Different algorithms for the different steps
have been necessary to remove the necessity of good initial guesses. As local minima
are an actual problem, (Frederiksen 1992) suggested initial steps based on rather
rough estimation to get good initial guesses before the more advanced optimization
algorithms are applied. The thesis (Frederiksen 1992) should be consulted for more
detailed advice on the optimization algorithms.
Let us comment on results for weakly, moderately and strongly non-isotropic Weakly
materials. A rolled aluminum plate is identified in the early work by (Markworth and
Petersen 1987) which based on ten frequencies found EL =68.9GP a, ET =66.0GP a,
GLT =24.8GP a, νLT =0.343 and fairly good agreement with traditional uniaxial
tests. (Frederiksen 1997a) for another aluminum plate identified EL =71.1GP a,
ET =70.7GP a, GLT =25.9GP a, νLT =0.336, GLZ =27.9GP a, GTZ =28.0GP a,
thus showing the possibility for identifying the out-of-plane shear moduli GLZ,GTZ.
non-isotropic

Strongly
non-isotropic
Citation from
Frederiksen
204 Copyright c○Pauli Pedersen: Optimal designs ...
For glass-epoxy also the early studies report good agreement with static strain Moderately
gauge test results in the identification of the in-plane moduli. Based on 14 frequencies,
(Frederiksen 1997a) identified the following parameters EL = 42.4GP a,
ET =11.6GP a, GLT =4.68GP a, νLT =0.305, GLZ =4.55GP a, GTZ =4.07GP a,
with details on iteration history and the extremely small residuals from experimental
and calculated frequencies.
Carbon-epoxy identification is properly of most interest and extended results are
available, in early as well as recent studies. Let us here again comment the results of
(Frederiksen 1997a), who reports EL = 113.0GP a, ET =8.50GP a, GLT =4.45GP a,
νLT = 0.323, GLZ = 4.43GP a, GTZ = 2.97GP a. We find almost transversely
isotropic stiffness for this unidirectional laminate and the ratio GTZ/GLZ =0.666
is in other studies reported between 0.548 and 0.624. Note that the determination
of out-of-plane shear moduli is very difficult to perform by other experimental techniques.
Finally, we show in 10.2 the results from identification of the temperature dependence
for glass-epoxy moduli; mainly to illustrate the possibilities when knowledge
on environmental influence is important.
10.1.5 Uncertainties and optimal experiments
In the early thesis of (Sol 1986) and also in the recent thesis of (Kuttenkeuler 1998)
assessment of uncertainties was given priority. In the extended study of (Frederiksen
1998) these questions are analysed based on statistics, and from the abstract in this
paper we cite ”This paper investigates an inverse technique for the identification of
orthotropic elastic constants from measured plate natural frequencies. In general, the
accuracy of the identified parameters depends on the method of estimation, modelling
errors and measurement errors. The paper addresses the parameter uncertainty due
to errors in the measurements. Based on assumptions of the measurement errors,
second-order statistics of parameters are approximated by linearization schemes. The
main focus is on the possibility of designing the experiment to minimize the uncertainty
of the estimated parameters. The uncertainty of each estimate as function of
the experimental design variables is investigated. Also the overall optimality of the
experimental design defined as the hyper-volume of the confidence region is considered.
The results show that not all parameters are estimated with a sufficient precision in
the general case, but by carefully designing the experiment, the parameter uncertainties
can be greatly reduced. Both thin and thick plates are considered with focus on
single-layer plates, but the results for laminates plates are also discussed.”
The individual errors can be grouped into
• Experimental errors
• Physical modelling errors
non-isotropic

Identification (estimation) and inverse problems 205
Figure 10.2: Temperature dependence for glass-epoxy ply obtained by identification.

206 Copyright c○Pauli Pedersen: Optimal designs ...
• Numerical modelling errors
• Plate defects
Experimental The experimental errors are reported as limitations of the instruments, and are
errors generally low. (Frederiksen 1998) states the standard deviations of the frequency
values to be less than 0.2% . Measuring the thickness of a laminated plate may be
Modelling
errors
Numerical
errors
Plate
errors
Alternative
materials
non-unique and a method based on volume measure in water is therefore suggested.
The physical modelling errors include the assumptions related to linearity, damping,
surrounding air, gravity, suspension and frequency dependence. Most critical of
these sources might be the suspension, but with a free plate only hanged in rubberbands
the boundary conditions are well established. Other boundary conditions normally
cause severe errors. Lowest and highest frequency in the identification often
deviate by one order, and if the material moduli are frequency dependent we should
conclude that averaged properties in some sense are found.
The numerical modelling errors have been investigated in (Frederiksen 1995).
For the first 15 frequencies, the maximum error is found to be 0.4% for a moderately
thick orthotropic plate. For most of the frequencies the error is considerably smaller.
Through many experiments the most essential problems were related to plate defects,
such as non-rectangular shape, non-uniform thickness, non-zero plate curvature,
and most important a non-homogeneity on the macro level. This last aspect has different
influence on the individual eigenmodes, eigenvalues and therefore often makes
identification impossible, which in a way is better than an artificial adjustment of the
moduli.
10.1.6 Comments
The identification technique is a fascinating approach involving very close cooperation
between experimental and numerical work. The studies have shown that reliable
values of the six most important moduli of orthotropic laminates can be determined
from one experiment, measuring plate frequencies. Because of the simplicity of the
free isolated specimens, environmental influence on material moduli can easily be
determined. Results from temperature dependence are shown to illustrate these potentials.
Naturally also other materials than composite laminates can be studied with
the method. With thick plate modelling these potentials are greatly enlarged. As ex-
ample we mention bone material (see early study by (Thomsen 1990)), ceramics, and
sandwich combinations.
The design of optimal experiments for identification of specific parameters is an
important new aspect that follows from the statistical analysis for determination of
parameter uncertainties. Bounds related to the identified parameters on the laminate
level as well as on the ply (lamina) level are important additional information to
obtain.

Identification (estimation) and inverse problems 207
10.2 Laminate inverse problems
From searching the very detailed information in constitutive parameters, we here
focus on problems at another level. Given the constitutive description of the plies
(layers, laminas) in a laminate, how to choose the stacking (thicknesses, orientations,
positions) to obtain a laminate with specified stiffnesses (if possible) ? This laminate
inverse problem is not simple to solve without the tools of optimization. Laminate
In two-dimensional formulation of laminates we deal with a number of 3 × 3
matrices: the constitutive (material stiffness) matrix [C], the laminate membrane
stiffness matrix [A], the laminate bending stiffness matrix [D], the laminate coupling
stiffness matrix [B], the laminate stress strength matrix [F ], and the laminate strain
strength matrix [G]. In relation to the present inverse problem they are all treated
analogously and as example we formulate the problem of finding a laminate that
results in a given bending matrix [ ¯ D]. This example is described more detailed in
(Pedersen 1999b).
10.2.1 Laminate with given bending stiffness
design
With the four index notation from the tensor formulations the matrix [D] of laminate Bending
bending stiffnesses is defined as
⎡
[D] = ⎣
D1111
D1122 √
2D1112
√
D1122 2D1112 √
D2222 2D2212
√
2D2212 2D1212
or with matrix [D] in contracted vector notation {D}
{D} T = {D1111 D2222 2D1212
⎤
⎦ with [D] positive definite (10.16)
√ 2D1122 2D1112 2D2212} (10.17)
The factors √ 2 and 2 in (10.17 ) are related to a moment vector definition {M} T √ =
2κ12} that
stiffnesses
√
{M11 M22 2M12} and to a curvature vector definition {κ} T = {κ11 κ22
imply orthogonal rotation transformations and then, as an example, invariant trace
of [D] equal to (D1111 + D2222 +2D1212).
The matrix [D] = [D(pk)] depends on the design parameters pk (thicknesses,
orientations, positions), and our goal is to find parameters that give [D] =[ ¯ D]orat
least obtain as close as possible agreement. This closeness is measured by the residual
matrix ([D] − [ ¯ D]). To get an objective that is invariant we define an error function Objective
Φ as the squared Frobenius norm of this matrix, i.e.
Φ=({D} T −{ ¯ D} T )({D}−{ ¯ D}) (10.18)

Optimization
problem
Iteration
strategy
208 Copyright c○Pauli Pedersen: Optimal designs ...
It follows that Φ ≥ 0, and if Φ = 0, we have located an exact solution [D] =[ ¯ D]. A
solution approach is then by iterations to minimize Φ and in such an approach we
need the gradients
dΦ
=2({D}
dpk
T −{ ¯ D} T ) d{D}
(10.19)
dpk
where formulas for d{D}/dpk can be found in (Pedersen 1987b).
10.2.2 A redesign procedure
A redesign described by
then gives
(pk)next = pk +∆pk
(Φ)next =Φ+
(dΦ/dpk)∆pk
i
(10.20)
(10.21)
and since (dΦ/dpk) is only correct for small ∆pk a possible approach would be in each
iteration to
Minimize (Φ)next subjected to g =
(∆pk) 2 − R = 0 (10.22)
where R is a redesign limit. Problem (10.22) can be solved analytically by proportionality
between the gradient of the constraint and the gradient of the objective (see
chapter 14) and we get
∆pk = a dΦ
= a2({D}
dpk
T −{ ¯ D} T ) d{D}
dpk
where the constant a is determined by the condition in (10.22)
a
2
i
i
(10.23)
(dΦ/dpk) 2 = R (10.24)
A practical approach is to start the redesign iterations with rather large resource for
change R, and ending up with a redesign based on smaller resources. Thus, a strategy
for R as a function of the iteration history is needed, very much like a strategy for
move-limits.

Identification (estimation) and inverse problems 209
10.3 Material identification and inverse homogenization
We now turn our attention to design of micro-structures, using optimization techniques,
not necessary to get optimal material but just to get a material that satisfies Inverse
certain specifications.
In chapter 8 we deal more specific with optimal material design, that for the
completely free material design is also covered in chapter 15. In the present chapter
only inverse problems related to material design are included.
10.3.1 Homogenization
material
problems
For calculating effective material properties we use homogenization and assume the Base cell
homogenization
existence of a base cell that, in a periodic distribution, represents the material. Subjecting
this base cell to any test strain field that satisfies the periodicity, we demand
that the strain energy of the homogenized base field be equal to the strain energy of
the detailed base cell (the micro-structure).
For a homogeneous micro-structure no calculations are needed and we can simply
set up the relations between total strain energies U for the cell subjected to elementary
strain states and the homogenized constitutive parameters C H ijkl in 2D or LH ijkl
in 3D
as specified in detail in chapter 11.
For the inhomogeneous micro-structure we have to perform a detailed analysis
(often a finite element analysis) to obtain the strain energy. In 2D three different
boundary conditions are prescribed (in 3D six), corresponding to the corner condition Elementary
for the homogeneous micro-structure when evaluating the elementary strain states.
Note that we assumed periodicity (shape not known) in the boundary displacements,
which, for the homogeneous micro-structure, are straight lines.
Except for the forced periodicity of the boundary conditions, the analysis of the
inhomogeneous micro-structure is exactly the same as for any macro-structure with
load only from forced displacements. From a finite element analysis with the microstructural
stiffness matrix, we get the displacement, which gives the strain energy.
We have established the direct relation between the strain energies as obtained in
a traditional finite element analysis of the inhomogeneous micro-structure and the
homogenized constitutive parameters, listed in chapter 11.
10.3.2 Inverse homogenization
The problem of micro-mechanical design is here understood as the problem of finding
a distribution of a given material (materials) in a base cell of homogenization in order
to obtain prescribed material parameters. For obvious reasons, the problem is also
strain cases

Multiple
solutions
References
to results
210 Copyright c○Pauli Pedersen: Optimal designs ...
termed the inverse homogenization problem or identification of the micro-mechanical
structure. As in the case of most identification problems, we cannot expect a unique
solution, but, in fact, this is an advantage and we can introduce penalties and/or
priorities as exemplified by a minimum mass micro-mechanical structure. Even so
there may be several possible solutions.
Like most optimal design problems, the inverse homogenization problem can be
solved by a number of different methods. Most of them are based on an iterative
scheme in which each iteration includes analysis of a micro-mechanical structure,
sensitivity analysis and determination of an improved design. For the sensitivity
analysis we need, for each design variable pe, (related to element e of the micromechanical
structure), the change in the homogenized constitutive parameter, i.e. in
3D dL H ijkl /dpe. Using the results of chapter 12 we can determine this in a localized
calculation.
In the Ph.D.-thesis by (Sigmund 1994a) and in the papers (Sigmund 1994b),
(Sigmund 1995), as well as in the dr.techn.-thesis (Sigmund 2001), we find 2D and 3Dsolutions
to inverse homogenization problems, often with minimum micro-structural
volume as a solution priority. Ultimate optimal materials as described in chapter 15
are also obtained, as well as materials with negative Poisson’s ratio. In papers by
(Sigmund and Torquato 1996), (Sigmund and Torquato 1997) the inverse homogenization
problem is formulated and solved for the thermoelastic problem. Properties
of thermal expansion cannot be designed independently of the constitutive elastic behaviour.
Very interesting results for the bounds that limit our possibilities are given
in these papers. Many examples of interesting base cells are shown in (Sigmund and
Torquato 1997), where one of the more interesting situations is a homogenized material
with negative thermal expansion, obtained by mixing two phases both of positive
thermal expansion together with void.

Chapter 11
Alternative Descriptions of
Constitutive Parameter
11.1 Stress/strain relations
The constitutive parameters relate stresses to strains, and we describe them in many
different forms, say as fourth order tensors, as squared matrices, as vectors, or by Many
invariants of these quantities. In a three-dimensional formulation the parameters
are pure material parameters, but in two-dimensional formulations the parameters
incorporate the modelling from three to two dimensions.
Even for one-dimensional formulation with the modulus of elasticity E as constitutive
parameter, giving normal stress σ from normal strain ɛ by σ = Eɛ, it is not
always simple. For non-isotropic materials, E depends on the direction we choose. Reference
Therefore, to be more specific, the constitutive parameters might be added an index
that specifies the actual reference coordinate system.
11.2 Two-dimensional formulations
possibilities
coordinates
In tensor formulation the constitutive relation is Tensors
σij = Cijklɛkl
(11.1)
and due to symmetry of the stress tensor σji = σij and of the strain tensor ɛlk = ɛkl
we may contract these tensors to vectors {σ} and {ɛ} with each having only three Vectors
components. The constitutive parameters are then described by the matrix [C] giving
Matrices
211

212 Copyright c○Pauli Pedersen: Optimal designs ...
{σ} =[C]{ɛ} and {ɛ} =[C] −1 {σ} (11.2)
The square matrix [C] of order three is symmetric and positive definite. To obtain
vectors of invariant length (the same in mutually rotated coordinate systems), we
define the row vectors {σ} T and {ɛ} T by
{σ} T √
:= {σ11 σ22 2σ12} and {ɛ} T √
:= {ɛ11 ɛ22 2ɛ12} (11.3)
knowing that σ2 11 +σ2 22 +2σ2 12 and ɛ2 11 +ɛ2 22 +2ɛ2 √Definition of
12 are not changed by coordinate rota-
2 contraction tion. (This follows from the invariant trace (σ11 + σ22) and the invariant determinant
Constitutive
trace
Constitutive
Frobenius
(σ11σ22 −σ 2 12) of the stress matrix, so (σ11 +σ22) 2 −2(σ11σ22 −σ 2 12) =σ 2 11 +σ 2 22 +2σ 2 12
is an invariant).
In order to be specific we use the four-index notation from the tensor formulation
and then define the matrix [C] by
⎧
⎨
⎩
σ11
σ22
√ 2σ12
⎫
⎬
⎭ =
⎡
⎣
C1111
C1122 √
2C1112
Among the invariants of [C] we have the trace
√
C1122 2C1112 √
C2222 2C2212
√
2C2212 2C1212
Tr([C]) = C1111 + C2222 +2C1212
⎤ ⎧
⎨
⎦
⎩
ɛ11
ɛ22
√ 2ɛ12
⎫
⎬
⎭
(11.4)
(11.5)
The second order norm, named the Frobenius norm F ([C]), is the ”length of the
matrix”, i.e. F 2 is equal to the sum of all the squared matrix components
(length) F 2 = C 2 1111 + C 2 2222 +4C 2 1212 +2C 2 1122 +4C 2 1112 +4C 2 2212 (11.6)
Orthonormal
rotational
Although rather unusual, in relation to optimal material design, it is very convenient
also to contract further the constitutive matrix [C] into a vector {C} with six
component
√
2C1122 2C1112 2C2212} (11.7)
{C} = {C1111 C2222 2C1212
that has invariant length, equal to the Frobenius norm of the matrix [C]
F 2 = {C} T {C} (11.8)
as seen by direct comparison with (11.6).
Furthermore, the vector description of not only stress {σ} and strain {ɛ} but also
of the constitutive parameters {C} makes rotational transformations from a coordinate
system y to another rotated coordinate system x rather simple
transformations {σ}x =[T ]{σ}y , {ɛ}x =[T ]{ɛ}y , {C}x =[R]{C}y (11.9)

Alternative Descriptions of Constitutive Parameter 213
where the components of the matrix [T ] of order three and the components of the
matrix [R] of order six are given by trigonometric combinations, for details see (Pedersen
1995). The matrices [T ] and [R] are both orthonormal matrices so the inverse
transformations to (11.9) are easily found
{σ}y =[T ] T {σ}x , {ɛ}y =[T ] T {ɛ}x , {C}y =[R] T {C}x
(11.10)
Like for other squared matrices, the eigenvalues of [C] give important information.
With [C] defined as in (11.4) these positive invariants can easily be determined
numerically, but the analytical expressions for the general case are rather lengthy. So,
we only give the results for the orthotropic case, i.e. for C1112 = C2212 =0 Constitutive
λ = 1
2 (C1111
+ C2222 ± (C1111 − C2222) 2 +4C2 1122 ) ,λ=2C1212 (11.11)
Note that the sum of the three eigenvalues is equal to the trace of [C]. The
minimum eigenvalue is the smallest energy density u that any normed state of strain
(ɛ 2 11 + ɛ 2 22 +2ɛ 2 12 = 1) can give, and the maximum eigenvalue gives the largest energy
density.
The total energy density u defined here as the sum of strain energy density uɛ
and stress energy density uσ (also named complimentary energy density) is
which we may write in a linear description
if the vector of second order strains {ˆɛ} is defined by
{ˆɛ} T := {ɛ 2 11 ɛ 2 22
The description (11.14) that for linear elasticity gives
u = uɛ + uσ := {ɛ} T [C]{ɛ} (11.12)
u = {C} T {ˆɛ} (11.13)
√ 2ɛ11ɛ22 2ɛ11ɛ12 2ɛ22ɛ12 2ɛ 2 12} (11.14)
uɛ = uσ = 1
2 {C}T {ˆɛ} (11.15)
is very useful in optimal material design and in modelling evolution of bone materials.
This is due to the fact that sensitivities can often be determined in a fixed strain field
where the change of {ˆɛ} is not involved.
eigenvalues

Analog
in 3D
Eigenvector
description
Elementary
strain cases
214 Copyright c○Pauli Pedersen: Optimal designs ...
11.3 Three-dimensional formulations
The analog and complete description of the three-dimensional case is not given in this
book, but can be found in (Pedersen and Tortorelli 1998). However, we add some
further aspects that are also of interest for the two-dimensional case.
For 3D-problems the constitutive matrix [L] is a square, symmetric, and semi-
positive definite matrix of order 6. The corresponding contracted vector description
{L} has 21 components. The strain vector {ɛ} has 6 components and the vector of
second order strains {ˆɛ} has 21 components. The total energy density u expressed in
these quantities is
u = {ɛ} T [L]{ɛ} = {L} T {ˆɛ} (11.16)
11.3.1 Spectral decomposition
The constitutive matrix [L] has six non-negative eigenvalues λi and corresponding
mutual orthogonal eigenvectors {Λi} for i =1, 2,...,6. We may choose to order and
normalize them by
0 ≤ λ1 ≤ λ2 ≤ λ3 ≤ λ4 ≤ λ5 ≤ λ6
{Λi} T {Λj} = δij, {Λi} T [L]{Λj} = δijλi (11.17)
where δij is the Kronecker delta. Again with general results from linear algebra we
can describe the constitutive matrix (vector) by the spectral decomposition
[L] =
6
λi{Λi}{Λi} T , {L} =
i=1
6
λi{ ˆ Λi} (11.18)
with {Λi}{Λi} T being the dyadic product of the eigenvector {Λi}. From these dyadic
products we may construct vectors { ˆ Λi} of order 21 in analog to the definition of the
vector {ˆɛ}.
11.3.2 Constitutive parameters by energy densities
Finally we show how constitutive parameters can be described in terms of energy
densities, very much in parallel to the tools used in homogenization theory, and for
the two-dimensional case listed in (Pedersen 1997).
We define only six cases of elementary strain states (here specified by the nonzero
components only) and the corresponding resulting mean strain energy densities
uɛ =ūstrain case
uɛ11 =ū(ɛ11 =1), uɛ22 =ū(ɛ22 =1), uɛ33 =ū(ɛ33 =1)
uɛ12 =ū( √ 2ɛ12 =1), uɛ13 =ū( √ 2ɛ13 =1), uɛ23 =ū( √ 2ɛ23 = 1) (11.19)
i=1

Alternative Descriptions of Constitutive Parameter 215
For linear elastic material the homogenized constitutive parameters are given by
L1111 = 2uɛ11, L2222 =2uɛ22, L3333 =2uɛ33
L1122 = uɛ11+ɛ22 − uɛ11 − uɛ22, L1133 = uɛ11+ɛ33 − uɛ11 − uɛ33
L2233 = uɛ22+ɛ33 − uɛ22 − uɛ33 (11.20)
for the relations between normal components,
2L1212 = 2uɛ12, 2L1313 =2uɛ13, 2L2323 =2uɛ23
2L1213 = uɛ12+ɛ13 − uɛ12 − uɛ13, 2L1223 = uɛ12+ɛ23 − uɛ12 − uɛ23
2L1323 = uɛ13+ɛ23 − uɛ13 − uɛ23 (11.21)
for the relations between shear components, and finally
√ 2L1112 = uɛ11+ɛ12 − uɛ11 − uɛ12, √ 2L1113 = uɛ11+ɛ13 − uɛ11 − uɛ13
√ 2L1123 = uɛ11+ɛ23 − uɛ11 − uɛ23
√ 2L2212 = uɛ22+ɛ12 − uɛ22 − uɛ12, √ 2L2213 = uɛ22+ɛ13 − uɛ22 − uɛ13
√ 2L2223 = uɛ22+ɛ23 − uɛ22 − uɛ23
√ 2L3312 = uɛ33+ɛ12 − uɛ33 − uɛ12, √ 2L2213 = uɛ33+ɛ13 − uɛ33 − uɛ13
√ 2L2223 = uɛ33+ɛ23 − uɛ33 − uɛ23 (11.22)
for the coupling relations between normal and shear components.
Note, that due to the superpositions in, say uɛ11+ɛ22, the relations (11.20), (11.21),
(11.22) are only valid for linear elasticity. Only linear
elasticity
For the material design in chapter 8 we need sensitivities of individual material
constitutive components. Using the relations (11.20), (11.21), (11.22) we can find
these sensitivities from sensitivities of elastic energy, using directly the results in
chapter 12 for the linear case.

3D-bulk
modulus
2D-bulk
modulus
216 Copyright c○Pauli Pedersen: Optimal designs ...
11.4 Bulk modulus and alternative isotropic parameters
In this section, concentrated on isotropic materials, we primarily derive the relations
between strain energy density, bulk modulus and eigenvalues for constitutive matrices.
11.4.1 Three-dimensional case
In the three-dimensional case, only for the principal components, we have for {σ} =
[L]{ɛ} the 3 × 3 constitutive matrix [L] defined by
[L] =
⎡
E(1 − ν)
⎣
(1 + ν)(1 − 2ν)
1 ˜ν ˜ν
˜ν 1 ˜ν
˜ν ˜ν 1
⎤
⎦ , with ˜ν := ν
1 − ν
(11.23)
and for a pure hydrostatic stress state and a pure dilatational strain state we have
the stress/strain vectors
{σ} T = {¯σ ¯σ ¯σ}, {ɛ} T = {¯ɛ ¯ɛ ¯ɛ}/3 (11.24)
because dilatation ∆V/V is determined by (ɛ11 + ɛ22 + ɛ33) =¯ɛ.
The bulk modulus K is physically defined as the ratio ¯σ/¯ɛ, i.e. from (11.23) and
(11.24)
E(1 − ν)
E
¯σ =
(1 + 2˜ν)1 ¯ɛ ⇒ K =
(11.25)
(1 + ν)(1 − 2ν) 3 3(1 − 2ν)
The strain energy density (uɛ)1 corresponding to the normalized strain state gives
(uɛ)1 = 1
2 {ɛ}T E
[L]{ɛ} =
2(1 − 2ν)
(11.26)
The largest eigenvalue λmax of the constitutive matrix [L] is3K, and thus for the
3D-case we have
K = 2
3 (uɛ)1 = 1
3 λmax
E
=
3(1 − 2ν)
(11.27)
(The eigenvector corresponding to λmax is actually pure dilatation {ɛ} T = {1 11}/ √ 3).
11.4.2 Two-dimensional case
The ”bulk” modulus K2 in the two-dimensional case is depending on the modelling
from 3D to 2D and thus has a less physical definition. With a plane stress assumption

Alternative Descriptions of Constitutive Parameter 217
we have for {σ} =[C]{ɛ} the constitutive matrix [C] and the stress/strain vectors
defined by
E
[C] =
(1 − ν2
1 ν
) ν 1
and thus with the definition K2 =¯σ/¯ɛ we have
gives
{σ} T = {¯σ ¯σ}, {ɛ} T = {¯ɛ ¯ɛ}/2 (11.28)
E
K2 =
(11.29)
2(1 − ν)
The strain energy density (uɛ)1 for the normalized strain state {ɛ} T = {1 1}/ √ 2
(uɛ)1 = 1
2 {ɛ}T E
[C]{ɛ} =
2(1 − ν)
and the largest eigenvalue λmax (with the eigenvector {ɛ} T = {1 1}/ √ 2) is
λmax =
Thus, the relations for the 2D-case are
E
(1 − ν)
K2 =(uɛ)1 = 1
2 λmax =
E
2(1 − ν)
(11.30)
(11.31)
(11.32)
11.4.3 Relations between other alternative moduli
For isotropic materials, two parameters are enough to describe the constitutive behaviour,
even for the 3D-case. Different combinations of these two parameters are
used for different purposes, and we here relate them to the minimum and the maximum
eigenvalues of the 6 × 6 constitutive matrix [L].
The minimum eigenvalue λmin is twice the shear modulus G that can also be 3D-isotropic
expressed by the longitudinal modulus E and the Poisson’s ratio ν
modulus
relations
λmin =2G = E
=2µ (11.33)
1+ν
where the Lame’ constant µ (describing distortion) is equal to G.
The maximum eigenvalue λmax is (as derived in section 11.4.1) three times the
bulk modulus K that can also be expressed by the longitudinal modulus E and the
Poisson’s ratio ν, or in the Lame’ constant λ and the Poisson’s ratio ν
λmax =3K = E ν
= λ (11.34)
1 − 2ν 1+ν

Collected
results
218 Copyright c○Pauli Pedersen: Optimal designs ...
The Lame’ constant λ describes the dilatation.
Alternative descriptions for the non-dimensional Poisson’s ratio ν are
ν =
λ
2(λ + µ)
and an alternative description for the Lame’ constant λ is
ν
λ = E
(1 + ν)(1 − 2ν)
1 E E
= (1 − )= − 1 (11.35)
2 3K 2G
(11.36)
as seen from (11.34). In most textbooks on solid mechanics we also find the longitudinal
modulus E expressed in the Lame’ constants
11.5 Summing up
3λ +2µ
E = µ
λ + µ
In this chapter the important results to focus on are:
(11.37)
• Many different descriptions of constitutive relations are possible, and we may
choose what is most simple for specific purposes
• Energy density described as a scalar product (11.13) and (11.16) is important
in sensitivity analysis that can be performed with fixed strain fields
• The formulation with square root definitions as in (11.4) is important for obtaining
invariant lengths of strain vector, stress vectors, and constitutive matrices
or vectors
• The bulk modulus is derived for the 3D-case as well as for the 2D-case, because
this information is needed for the material optimization

Chapter 12
Effective stress/strain and
energy densities
The goal of the present chapter is to put forward some results that can be used more General
generally. We deal with non-linear problems but with the simplest possible extension results
from linear elasticity, which is the power law non-linear elasticity.
Because this class of problems can also be used to describe plasticity theory in the
deformation plasticity formulation and stationary creep, this simple extension covers
a broad range of practical important problems. Non-isotropic description is included Non-linear,
although a number of specific results are given only for the important orthotropic
case.
12.1 Analysis by secant formulation
non-isotropic
The analysis of non-isotropic, non-linear elastic structures/continua are presented in
the secant formulation, as described in (Pedersen and Taylor 1993) and in (Pedersen
1995). We firstly concentrate on the compliance matrix and the definition of effective
stress. The preferred effective stress is defined in an energy-consistent way and the
relation to the von Mises stress is pointed out.
Then an important relation between strain and stress energy densities is established.
This relation is not well known, but was already mentioned by (Hill 1956). We
use the following notation: for the stress/strain vectors {σ} and {ɛ} (for 3D-problems, Notation
each having 6 elements); for the scalar effective stress/strain the notation σeff and
ɛeff ; and for the constant reference modulus of elasticity E. The non-linearity is
described by the powers n or p , where n =1/p.
219

Compliance
power law
description
Sub-vectors,
sub-matrices
General
non-isotropic
Orthotropic
220 Copyright c○Pauli Pedersen: Optimal designs ...
12.1.1 Non-dimensional compliance matrix
The one-dimensional power law model is often written ɛ =(σ/E) n but with proper account
for signs it should be written ɛ =(|σ| n−1 E −n )σ. In two- and three-dimensional
models we take |σ| = σeff , where σeff ≥ 0 is a scalar measure. Based on this we
take the following compliance description (see end of this chapter for an alternative
derivation of this postulate)
{ɛ} = σ n−1
eff E−n [β]{σ} and σ 2 eff := {σ} T [β]{σ} (12.1)
the non-dimensional matrix [β] describes the non-isotropy, and the only restriction
on this matrix is that it must be symmetric and positive semi-definite, i.e.
[β] T =[β] and σ 2 eff ≥ 0 (12.2)
Now, separating the stress vector into normal terms with index N and shear
terms with index S, we have
√
2σ23}} (12.3)
{σ} T = {{σ} T N {σ} T S } = {{σ11 σ22 σ33}{ √ √
2σ12 2σ13
and, accordingly, we have the following separated [β] matrix
[β]NN
[β] =
[β]
[β]NS
T NS [β]SS
with the sub-matrices containing at most 21 different parameters
[β]NN =
⎡
β1111
⎣ β1122
β1122
β2222
β1133
β2233
⎤
⎦
[β]NS = √ 2
[β]SS
=2
⎡
⎣
⎡
⎣
β1133 β2233 β3333
β1112 β1113 β1123
β2212 β2213 β2223
β3312 β3313 β3323
β1212 β1213 β1223
β1213 β1313 β1323
β1223 β1323 β2323
⎤
⎦
(12.4)
⎤
⎦ (12.5)
In the orthotropic directions, the orthotropic case is characterized by the simpler form
of two of the matrices in (12.5)
⎡
⎤
[β]SS =2⎣
⎦ , [β]NS = [0] (12.6)
β1212 0 0
0 β1313 0
0 0 β2323
and is thus described by at most only 9 different parameters.

Effective stress/strain and energy densities 221
The isotropic case is described by only 2 parameters with Isotropic
⎡
⎤
[β]NN =
⎦ , [β]NS = [0]
[β]SS =
⎣ β1111 β1122 β1122
β1122 β1111 β1122
β1122 β1122 β1111
⎡
⎣ β1111 − β1122
0
0
β1111 − β1122
0
0
⎤
⎦ (12.7)
0 0 β1111 − β1122
For the following analysis we list the conditions of incompressibility of an orthotropic
description for any stress state, which is General
incompressible
β1111 + β1122 + β1133 = 0
β1122 + β2222 + β2233 = 0
β1133 + β2233 + β3333 = 0 (12.8)
while incompressibility in relation to hydrostatic pressure is obtained by the single
condition Specific
β1111 + β2222 + β3333 +2(β1122 + β1133 + β2233) = 0 (12.9) incompressible
12.1.2 The von Mises effective stress
In traditional plasticity theory the effective stress is not defined as shown in (12.1)
but instead by means of the von Mises stress σMises defined by Deviatoric
σ 2 Mises := 3
2 {s}T {s} (12.10)
stresses
where {s} is the vector of deviatoric stresses, i.e. the hydrostatic pressure is eliminated.
In (Pedersen 1987a) it is shown that this deviatoric stress vector can be
obtained by a projection with the projection matrix [P ] (i.e. [P ] T =[P ],
[P ], [P ] singular), and we have
[P ][P ]=
Projection
{s} =[P ]{σ} with [P ]:= 1
3
⎡
⎢
⎣
2 −1 −1 0 0 0
− 1 2 −1 0 0 0
− 1 −1 2 0 0 0
0 0 0 3 0 0
0 0 0 0 3 0
0 0 0 0 0 3
⎤
⎥
⎦
(12.11)
matrix

Matrix of
von Mises
Condition for
proportionality
Matrix of
Hill
222 Copyright c○Pauli Pedersen: Optimal designs ...
Inserting (12.11) in (12.10), we get the von Mises stress in terms of the total
stresses
σ 2 Mises = 3
2 {σ}T [P ]{σ} (12.12)
and comparing this to the definition of σ2 eff in (12.1), we find that σMises is proportional
to σeff only for the specific compliance matrix that corresponds to an isotropic
and incompressible material
[β]isotropic and incompressible proportional to [P ] (12.13)
For other materials there is in general a difference between the von Mises stress σMises
and the effective stress σeff based on total energy density. In optimal designs which
take strength constraints into account the solution naturally depends on the chosen
effective measure. Therefore, with σMises not proportional to σeff , we get solutions
related to the specific choice. Note, that the more general results in sensitivity analysis
are based on the energy related definition σeff .
12.1.3 The Hill strength measure
The Hill (Hill 1950) strength reference σHill, for non-isotropic materials is
or in matrix notation
σ 2 Hill = F (σ22 − σ33) 2 + G(σ33 − σ11) 2 + H(σ11 − σ22) 2 +
2Lσ 2 23 +2Mσ 2 13 +2Nσ 2 12
σ 2 Hill = {σ} T
⎡
⎢
⎣
G + H −H −G 0 0 0
− H F + H −F 0 0 0
− G −F F + G 0 0 0
0 0 0 N 0 0
0 0 0 0 M 0
0 0 0 0 0 L
(12.14)
⎤
⎥ {σ} (12.15)
⎥
⎦
We see that for hydrostatic stress {σ} T =¯σ{1 11000} we obtain σHill = 0 so this
measure is insensitive to hydrostatic pressure. Note also that the Hill measure has
only 6 parameters (plus one constraint for hydrostatic pressure), while the orthotropic
[β] matrix has 9 parameters.

Effective stress/strain and energy densities 223
12.2 Strain energy density and
stress energy density
As used in (Pedersen and Taylor 1993), the secant description (12.1) can also be Modulus
stated,
{σ} = ɛ p−1
eff E[α]{ɛ} and ɛ2 eff := {ɛ} T [α]{ɛ} (12.16)
where the non-dimensional constitutive matrix [α] is just the inverse of the compliance
matrix [β] (here assumed positive definite)
[α] =[β] −1
with sub-matrix definitions in complete analogy to (12.5), (12.6) and (12.7).
The integrated strain energy density uɛ based on (12.16) is
(12.17)
power law
description
uɛ = E 1
p +1 ɛp+1
eff
(12.18)
and the stress energy density uσ based on (12.1) is Energy
uσ = 1
En 1
n +1 σn+1
eff
(12.19)
densities
where n =1/p. It follows (most easily from uɛ + uσ = σeff ɛeff ) that we have the
following important relation Energy
uσ = puɛ
(12.20)
and this relation gives rise to simplified sensitivity analysis as well as rather general
optimality criteria, see chapters 13, 14 and 15.
An alternative to start with (12.1) and then derive (12.19) is to start with (12.19)
and then derive (12.1). From
{ɛ} = duσ 1
=
d{σ} T En and from the definition of σ 2 eff
1
n +1 (n +1)σn dσeff
eff
d{σ} T
(12.21)
density
RELATION
in (12.1) Derivation
dσeff
2σeff
=2[β]{σ} (12.22)
d{σ} T
we get directly (12.1) as an alternative to starting with this as a postulate.
from elastic
potential

224 Copyright c○Pauli Pedersen: Optimal designs ...
12.3 Summing up
In this chapter the important results to focus on are: Collected
results
• Secant relations (12.1) and (12.16) give the energy densities (12.18) and (12.19).
Alternatively, the energy densities give the secant relations.
• Sub-matrix descriptions are valuable to clarify the difference between different
definitions of effective stress/strain.
• For power law, non-linear and non-isotropic elastic materials, the proportionality
(12.20) between strain energy density and stress energy density holds
independently of the level of energy density.

Chapter 13
Elastic potentials, relations
and sensitivities
13.1 Elastic potentials
The general equation of energy equilibrium is Energy
equilibrium
Uɛ + Uσ + Uext = 0 (13.1)
with elastic strain energy Uɛ and elastic stress energy Uσ (elastic complementary en-
ergy) from the corresponding densities uɛ, uσ integrated over the structure/continuum
volume V
Uɛ = uɛdV and Uσ = uσdV (13.2)
and the external potential Uext defined by Potential
Uext := −( TividA + pividV ) (13.3)
with surface tractions Ti, volume forces pi , corresponding displacements vi, and area
A surrounding the volume V .
definitions
For materials resulting in the density relation uσ = puɛ, as proven by (12.20) Basic
in chapter 12, to hold everywhere in the continuum or structure, we have in total
Uσ = pUɛ and the equilibrium (13.1) is simplified to
(1 + p)Uɛ = −Uext
with 0

Total
potential
Potential
relations
Extremum
relations
Using
virtual work
principle
Non-changed
loads
Important
RESULT
226 Copyright c○Pauli Pedersen: Optimal designs ...
Defining the total potential Π by
we have the relations
Π:=Uɛ + Uext
(13.5)
Uɛ = Uσ/p = −Π/p = −Uext/(1 + p) (13.6)
and by p>0 and Uɛ > 0 we have Uσ > 0, Π < 0 and Uext < 0. From this follows
that design for a number of differently stated extremum problems are equivalent and
that their values at the solutions are related as
min Uɛ = min Uσ/p = −max Π/p = −max Uext/(1 + p) (13.7)
with p = 1 for the specific case of linear elasticity.
13.2 Derivatives of elastic potentials
The derivative of the total potential Π with respect to an arbitrary parameter, say a
design parameter h, is
dΠ
dh =
∂Π
+
∂h fixed strains
∂Π dɛ
∂ɛ dh =
∂Π
(13.8)
∂h fixed strains
because of stationary total potential ∂Π/∂ɛ = 0 (virtual work principle) with respect
to kinematically admissible strain variations.
For design-independent external loads, (∂Uext/∂h)fixed strains = 0, the definition
(13.5) gives
∂Π
∂Uɛ
=
(13.9)
∂h fixed strains ∂h fixed strains
and then from (13.6), (13.8) and (13.9) we get the result that is frequently used
dUɛ ∂Uɛ
= −1
(13.10)
dh p ∂h
fixed strains
For a local design parameter he that only changes the design in the domain e of
the structure/continuum this gives the possibility of a localized determination of the
sensitivity for the total elastic strain energy
dUɛ
dhe
= − 1
p
∂((ūɛ)eVe)
∂he
fixed strains
(13.11)

Elastic potentials, relations and sensitivities 227
where (ūɛ)e is the mean strain energy density in the domain of he and where Ve is the
corresponding volume. We note that the only difference between linear (p = 1) and
non-linear material is the factor 1/p, and for a condition on stationarity dUɛ/dhe =0,
p has no influence.
Note, that the sensitivity is not physically localized, but still we can without approximation
determine the sensitivity localized. Two specific cases are exemplified in
more detail. Local
13.2.1 Constant constitutive matrix, a specific case
For the case of a constant constitutive matrix (non-changed material) in a fixed strain
field, the result (13.11) simplifies to
dUɛ
= −
dhe
(ūɛ)e ∂Ve
(13.12)
p ∂he
and often ∂Ve/∂he is simply equal to Ve/he (or proportional to), e.g. when he is a
thickness, a relative volume density, or an area.
13.2.2 Constant volume, a specific case
For the case of constant volume, the result (13.11) simplifies to
dUɛ
= −
dhe
Ve
∂(ūɛ)e
p ∂he
fixed strains
(13.13)
Two such even more specific cases should be mentioned: firstly the case of sensitivity
to material orientation θe, for which ∂(ūɛ)e/∂θe must be evaluated, and secondly
sensitivity to a specific constitutive parameter Lijkl, for which ∂(ūɛ)e/∂Lijkl must be
evaluated. Due to the fact that these sensitivities are determined in a fixed strain
field it is possible to obtain analytical expressions, as shown in chapters 15, 17 and
used in chapters 8, 9, 10.
design
parameter

Collected
results
228 Copyright c○Pauli Pedersen: Optimal designs ...
13.3 Summing up
In this chapter the important results to focus on are:
• Due to the simple relations (13.6) between potentials, alternative statements of
optimal design problems are possible.
• In an energy formulation, and thus with great generality, the possibility of
determining sensitivities in a fixed strain field (13.10) is proven.
• Local design parameters often give further simplifications by, more or less,
bringing calculation from system (total structure/continuum) level to element
(domain) level.

Chapter 14
Some necessary conditions for
optimality
In this chapter we primarily present two necessary conditions for optimal solutions in
general, i.e. not directly related to optimal design. The first condition is only valid General
for non-constrained problems and the second condition is valid for problems with only
a single constraint in addition to the objective.
After the two first sections we then determine, expressed in physical terms, the
condition for solution of the most simple optimal design problems, for which sizes (or
field of size) optimize stiffness as well as strength. The optimal designs to these two
different problems are shown to be the same. Design
In close relation to the analysis for size optimization, we treat optimization of
shape, again to optimize stiffness as well as strength. The shape solution to these two
different problems is shown in many cases also to be the same, very much in parallel
optimization
optimization
to the result for size optimization. Size and
shape
A simple parametrization for shape optimization is finally described, this approach
is used in the examples presented in chapter 7. The goal of the present chapter is to
obtain basic understanding of very simple optimal design problems, without involving
extended numerical calculations.
14.1 Non-constrained problems
A non-constrained optimization problem may be defined as
Extremize Φ=Φ(he) with variables he non − constrained (14.1)
229

Stationary
objective
Converted to
230 Copyright c○Pauli Pedersen: Optimal designs ...
and a necessary condition for this is that the objective Φ is stationary with respect
to all the independent variables he
dΦ/dhe =0for all e (14.2)
The optimization of material orientation is an important example of such a nonconstrained
problem. Unfortunately, this problem is not simple to solve, because
many local optima exist. Each variable (orientational angle θe in a domain e) has
several solutions to the stationarity condition (14.2), and this inherent problem is
described in detail in chapter 17.
14.2 Problems with a single constraint
An optimization problem with only a single constraint may be defined as
Extremize Φ = Φ(he) with variables he
constrained by g = g(he) = 0 (14.3)
To obtain an optimality condition we convert this problem to a non-constrained
non-constrained problem, using a Lagrangian function L =Φ− λg to be made stationary for arbitrary
value of λ (the Lagrangian multiplier). It follows from (14.2) that a necessary
optimality condition is
Proportional
gradients
dL/dhe = 0 for alle ⇒
dΦ/dhe = λdg/dhe (14.4)
This general result we read as proportionality between the gradient of the objective
and the gradient of the single constraint. The factor of proportionality, the Lagrangian
multiplier λ, is determined by the constraint condition g(he) = 0, that follows from
dL/dλ = −g =0.
The use of the optimality condition (14.4) for obtaining more general information
about optimal design is very important. It should be noted that behind this result is
the assumption that the constraint is active. Extensions to two and more constraints
are possible, but the uncertainty about the active constraints is then often a limiting
factor on the usefulness.
An alternative look at the problem is to postulate (14.4) and then see that dg =0
implies dΦ = 0, i.e.
dΦ = ∂Φ
∆he = λ
∂he
∂g
∆he = λdg = 0 (14.5)
∂he
e
e

Some necessary conditions for optimality 231
14.3 Size optimization for stiffness and strength
The theoretical results for size optimization are more developed than those for shape
optimization. Let us therefore start with some basis knowledge from size optimization,
as it can be found in (Pedersen 1998) for non-linear elasticity or in (Wasiutynski 1960)
for linear elasticity.
14.3.1 Size design with optimal stiffness
If the objective is to minimize compliance (minimize elastic energy) for given total
mass then we have (for optimal stiffness design with homogeneous assumptions and Homogeneous
design independent loads): the ratio between sub-domain energy and sub-domain mass mass (volume)
should be the same in all the design sub-domains.
dependence
Let the design parameters be he, then homogeneous mass relations are obtained
with M =
e Me =
e hme ¯ Me, where M is the total mass, Me is the mass in domain
e, m is a given positive value, and ¯ Me is independent of the design parameters. The
homogeneous energy relations are obtained with Uɛ =
Uɛe e = e hne Ūɛe , where Uɛ
is the total strain energy, Uɛe is the strain energy in domain e, n is a given positive
value, and Ūɛe is explicitly independent of the design parameters.
Restricted to problems with constant mass density we get, in all design domains,
the same mean strain energy density. Furthermore, if the model has constant energy
density within a design domain, then the result for the optimal design is uniform Stiffest
strain energy density u design
∗ ɛ , i.e.
u ∗ ɛe =ūɛ for all free design domains (14.6)
where lower and upper size constraints are not reached. The symbolism here is a
super-index ∗ related to the optimal design, and a overhead bar ¯indicating a constant
value for each domain e (mean value).
Assume now that the necessary condition (14.6) give a global minimum solution,
then for any other design the total strain energy Uɛ is larger (or equal to)
Uɛ =
uɛeVe ≥ U ∗ ɛ =
e
e
u ∗ ɛ V ∗
e =ūɛ
e
V ∗
e =ūɛ
Ve =
e
e
u ∗ ɛe Ve
(14.7)
where V ∗
e is the optimal volume of the design domain e. For an alternative design
with design volumes Ve we have the same total volume V V =
e Ve = ∗
e Ve . From
(14.7) we get
(uɛe − u∗ɛe )Ve ≥ 0 (14.8)
e

Also best
232 Copyright c○Pauli Pedersen: Optimal designs ...
14.3.2 Size design with optimal strength
strength With positive volumes Ve we read from (14.8), that at least one uɛe is not less than
u∗ ɛe . Thus if the strongest design is defined by minimum of maximum uɛe , then the
stiffest design characterized by the optimality condition (14.6) is also the strongest
design.
We note that the strength may also be defined in relation to the von Mises stress
or an alternative effective stress, and these measures are not always proportional to
the energy density. For a detailed discussion of these aspects see (Pedersen 1998).
General
knowledge
Localized
volume change
14.4 Shape optimization for stiffness and strength
In the following we use the same kind of reasoning to draw conclusions about shape
optimization, without involving a solution to the actual stress problem. Thus we gain
general knowledge, valuable for 3D and 2D-problems, for non-linear elastic as well as
for linear problems, for non-isotropic or isotropic problems, for any external, design
independent load. Also valid for non-homogeneous problems and independent of the
solution procedure.
In order to simplify the mathematics the design parametrization is chosen as
illustrated in figure 14.1. An alternative parametrization with expansion in terms of
shape design functions is formulated in (Dems and Mroz 1978), a paper closely related
to this presentation.
We assume a homogeneous state for the strain energy density uɛe
within the
volume Ve related to the shape parameter he, say a constant stress finite element.
Let us now subject the shape to variation using only two parameters hi and hj.
Furthermore, let the total volume V of the structure (continuum) be fixed, then
∆V = dV
∆hi +
dhi
dV
dhj
∆hj = dVi
dhi
∆hi + dVj
∆hj = 0 (14.9)
dhj
because we also assume the domain volumes to be depending only on one design
parameter and with a positive gradient (to be used later)
Ve = Ve(he) and dVe/dhe > 0 (14.10)
14.4.1 Shape design with optimal stiffness
In shape optimization for extremum elastic strain energy the increment of the objective
corresponding to increments ∆hi, ∆hj is
∆Uɛ = dUɛ
dhi
∆hi + dUɛ
∆hj
dhj
(14.11)

Some necessary conditions for optimality 233
Figure 14.1: Discretized design parametrization, showing two design domains i and
j.
which for power law non-linear elasticity σ = Eɛp can be written as Design
∆Uɛ = − 1
∂Uɛ
∆hi +
p ∂hi
∂Uɛ
∆hj
∂hj fixed strains
(14.12)
This is proven in (Pedersen 1998) for design independent loads, and follows from
(13.10) in chapter 13. Therefore only the local energies Uɛi = uɛiVi and Uɛj = uɛj Vj
are involved and the variations in the strain energy densities need not be determined,
because the constitutive relations are unchanged. We have Localized
∆Uɛ = − 1
p (uɛi
dVi dVj
∆hi + uɛj ∆hj)
dhi dhj
and inserting (14.9) in (14.13) we obtain
(14.13)
∆Uɛ = − 1
(uɛi − uɛj )dVi ∆hi
p dhi
(14.14)
A necessary condition for optimality ∆Uɛ = 0 with dVi/dhi > 0 is therefore
uɛi = uɛj .
With all design parameters, eq. (14.9) and (14.13) are written
∆V = dVe
∆he
dhe
e
independent
loads
energy change

Constant
energy density
Basic
assumption
Detail of
proof
234 Copyright c○Pauli Pedersen: Optimal designs ...
∆Uɛ = − 1
p
e
uɛe
dVe
∆he
dhe
(14.15)
and we conclude that a necessary condition for optimality ∆U = 0 with constraint
∆V = 0 is constant strain energy density uɛe . Thus for the stiffest design the energy
density along the shape(s) to be designed, here denoted uɛs , must be constant
14.4.2 Shape design with optimal strength
uɛs =ūɛ (14.16)
We now relate the stiffest design (minimum compliance) to the strongest design (minimum
maximum strain energy density). Let us assume that the highest strain energy
density is at the shape to be designed. With index s referring to shape design domains
and index n referring to domains not subjected to design changes, this means that
for the stiffest design we assume
uɛs =ūɛ >uɛn (14.17)
A design domain that depends on design parameter is given index s (hs) and a design
domain which is not subjected to design change is given index n (hn). For the total
design domain we use index S and for the total domain not subjected to design, index
N. The total elastic strain energy Uɛ is obtained from
Uɛ = UɛS + UɛN = Uɛs + Uɛn = uɛsVs +
uɛnVn, i.e.
Vs +
s
Uɛ = ūɛ
uɛn
s n
Vn
With unchanged domain N and for the stiffest design Uɛ >U∗ ɛ we obtain
as
s
uɛs
s
Vs +
uɛn
n
Vn >
ūɛV
s
∗
s +
(uɛs
n
− ūɛ)Vs >
(u ∗ (14.19)
ɛn − uɛn )Vn
s
n
n
s
n
u ∗ ɛn V ∗ n , i.e.
(14.18)
ūɛV ∗
s =
s ūɛVs due to given total volume, and furthermore individual un-
changed in the non-design domains V ∗ n = Vn.
The right hand side might be negative, so we can not directly draw conclusions
as from (14.8). However, in a complementary formulation with stress energies we can
prove that the right hand side is non-negative and then the proof holds.
The proof of increasing energy in the shape domain is as follows. We write the

Some necessary conditions for optimality 235
total stress energy Uσ as the sum of stress energy in the shape domain UσS and stress
energy in the non-shape domain UσN and obtain
Uσ = UσS
+ UσN ⇒ dUσ
dh
= dUσS
dh
From the principle of complementary virtual work follows
dUσ/dh =(∂Uσ/∂h)fixed stress field and we get
dUσ
dh =
∂UσS
∂h
+ ∂UσN
∂h
+ dUσN
dh
fixed stress field
(14.20)
(14.21)
where the last term is zero when h has no direct influence on the non-shape domain.
Finally for the stiffest design we have dUσ/dh > 0 and from this we conclude
∂UσS
=
∂h fixed stress field
dUσ
dh
1 dUɛS
=
p dh
> 0 (14.22)
Summarizing the theoretical results of this section; we have for the general threedimensional
case with non-isotropic, power law non-linear elastic material in an nonhomogeneous
structure, and for any design independent single load case that:
The minimum compliance shape design (stiffest shape design) has uniform energy
density along the designed shape, as far as the geometrical constraints make this
possible.
If we furthermore assume that the highest energy densities are found at the designed
shape, then the stiffest design is also the strongest design, as defined by a design
which minimizes the maximum energy density. Design for
Note that these results are obtained without calculating the stress/strain fields
and without specifying the constitutive behaviour. This behaviour need not be homogeneous
and thus we can also include the multi-material case. The many different
problems in chapter 7 support these results.
14.5 Conditions with a
simple shape parametrization
stiffness
and strength
In the final conclusions in section 14.4.2, we have added the note ”as far as the
geometrical constraints make this possible”. Also it was commented that normally
the shape parametrization implies such a geometrical constraint. In this section we
use a simple shape parametrization that makes a rather simple optimality condition
possible. The limitations of using this simple parametrization can be evaluated by
the possibility to obtain almost uniform energy density distribution along the shape Good
experience

Two or
only one
parameter(s)
236 Copyright c○Pauli Pedersen: Optimal designs ...
Figure 14.2: A three parameter (α, β, η) description of an internal hole in a rectangular
domain, specified by A, B.
to be designed. The examples in chapters 7 and 8 illustrate that the parametrization
is in fact able to describe optimal shapes in many cases.
Figure 14.2 shows a single inclusion hole, where the shape of the boundary is modelled
as a super-elliptic shape, described by only three non-dimensional parameters,
relative axes α, β and power η
x
η
η
y
+ = 1 (14.23)
αA βB
With known area of the hole we only have two parameters and if furthermore symmetry
is enforced, say αA = βB, we only have one free parameter, which might be
the power η. Figure 14.3 shows the great flexibility even for this one parameter
description. This parametrization naturally has its limitation, but several examples
show its usefulness, and furthermore it can easily be extended to 3D-problems by
x
η
η η y z
+ + = 1 (14.24)
αA βB γC
In the 2D-model (14.23) the area of the hole is
αA
4 βB 1 − (
0
x
αA )η 1/η ) dx =2αβABg(η) (14.25)
with the function g = g(η) defined by
1 η +1 2
g(η) :=Γ Γ /Γ
(14.26)
η η η

Some necessary conditions for optimality 237
Figure 14.3: Shapes giving equal area of the hole, with powers of the super-elliptic
shape being η = 0.75, 1.25. 1.75 and 3.00, respectively.
where Γ is the Gamma-function. With the rectangular area being 4AB the relative
area of the hole φ (relative to the area 4AB) and the relative area of the solid (relative Relative
density) ρ are
φ = 1
αβg(η) =1− ρ (14.27)
2
An optimal design problem is formulated in order to extremize the elastic energy
U for constant relative area
hole area
or density
Extremize U subject to φ(α, β, η) = ¯ φ (14.28)
Within the possibilities of the three parameters α, β, η this also minimizes energy Design
concentration and returns constant energy density along the boundary of the hole,
as discussed in section 14.4.2. Using the result (14.12) from sensitivity analysis we
determine the differential of the elastic energy (p = 1 for linear elasticity)
dU = − 1
∂U ∂U ∂U
dα + dβ +
p ∂α ∂β ∂η dη
(14.29)
fixed strains
and the differential of the constraint follows from (14.27) (using a formula manipulation
program to differentiate the Gamma-functions)
dα dβ p(η)dη
dφ = φ + +
α β η2
(14.30)
problem

Available
functions
Optimality
condition
For model
by the FEM
Strength or
stiffness
238 Copyright c○Pauli Pedersen: Optimal designs ...
with the function p = p(η) defined by
2 1 η +1
p(η) :=Ψ − Ψ − Ψ
(14.31)
η η η
where Ψ is the Psi-function. To illustrate that the functions g(η) and p(η) are wellbehaved
functions we show in figure 14.4 these functions and there derivatives. These
functions are available in many libraries of computer routines.
Figure 14.4: Left g-function and right the p-function with their derivatives as a
function of the shape power η.
The condition of dU = 0 when dφ = 0 is a necessary condition for optimality and
thus (as in general with only a single constraint) we from (14.29) and (14.30) get the
optimality condition by proportional gradients (14.4), i.e.
α ∂U
∂U η2 ∂U
= β = (14.32)
∂α ∂β p(η) ∂η fixed strains
In a fixed strain field the energy densities u are constant and only the volumes of
domains (elements) connected to the hole boundary change. Thus in a finite element
formulation the optimality condition (14.32) is written
α ∂Vs
∂Vs η2 ∂Vs
us = β us = us
(14.33)
∂α ∂β p(η) ∂η
s
s
s
where index s refers to an element connected to the hole boundary. The only information
needed in addition to the results from analysis is ∂Vs/∂α, ∂Vs/∂β, ∂Vs/∂η,
i.e. only information from geometry. We note, in agreement with section 14.4.2, that
if us is constant along the hole boundary then
s ∂Vs/∂α = ∂V/∂α = φ/α etc., and
the optimality criterion (14.33) is satisfied by usφ = usφ = usφ. Thus a constant
energy density along the boundary of the hole implies stationary total elastic energy.
However, we can have stationary energy without constant energy density, if the
possible designs are restricted. This is illustrated by the examples in chapter 7.

Some necessary conditions for optimality 239
14.5.1 Possible iterative procedure
The problem is how to find a boundary shape that satisfies (14.32) or in finite element
formulation (14.33). The heuristic approach of successive iterations could be to
estimate the Lagrange multiplier λ by the mean value
λestimated = 1
3
α ∂U ∂U
+ β
∂α ∂β
η2 ∂U
+
p(η) ∂η
(14.34)
and then redefine α, β, η by Estimated
αnew = λ/( ∂U
∂α )old, βnew = λ/( ∂U
∂β )old, ( η2
p(η) )new = λ/( ∂U
∂η )old
with iterations on λ to satisfy the constraint of (14.28)
14.5.2 Extended design space
(14.35)
φnew = 1
2 αnew βnew g(ηnew) = ¯ φ (14.36)
Let us assume that the three-parameter boundary shape does not in a satisfactory way
give constant energy density along the boundary shape, i.e. the energy concentration
must be made smaller. We can then add modification functions fk = fk(s) where s is
the natural coordinate along the super-elliptic shape. The functions fk (k =1, 2, ...)
are chosen as described in (Pedersen 1988) and used in (Pedersen et al. 1992). Figure
multiplier
6.9 in section 6.5.1 illustrates in detail this approach. The added design parameters Modification
are the amplitudes ck of the modification functions and gradients ∂φ/∂ck and ∂Vs/∂ck
must be evaluated, numerically or eventually analytically. Better solutions to problem
(14.28) can then be obtained. Alternatively we may formulate the problem directly
as minimum energy concentration problem
Minimize by α, β, η, ck
over all domains e
subject to φ(α, β, η, ck) = ¯ φ (14.37)
maximum uɛe
Solutions can be obtained with the integrated FEM-SLP approach, see (Pedersen
1981) and (Pedersen et al. 1992). With the modification functions we also have the
possibility to introduce discontinuities in the slope of the boundary shape.
functions

Collected
results
240 Copyright c○Pauli Pedersen: Optimal designs ...
14.6 Summing up
In this chapter the important results to focus on are:
• For non-constrained problems the necessary optimality condition is stationarity
of the objective with respect to all the independent variables.
• For problems with a single constraint the necessary optimality condition is
proportionality between the gradient of the objective and the gradient of the
single constraint.
• For size optimization of stiffness and strength the stiffest design is characterized
by the optimality condition of uniform energy density and this design is also
the strongest design.
• For shape optimization of stiffness, the minimum compliance shape design
(stiffest shape design) has uniform energy density along the designed shape,
as far as the geometrical constraints make this possible.
• For shape optimization of strength, if we assume that the highest energy densities
are found at the designed shape, then the strongest design, as defined by a
design which minimizes the maximum energy density, is also the stiffest design.
• For shape optimization a simple super-elliptic description makes it possible to
design a wide spectrum of shapes, and the analytical treatment of this case is
almost as simple as the classic elliptic case.

Chapter 15
The ultimate optimal material
15.1 The individual constitutive parameters
In ultimate optimal material design, also named free material design, we represent the
material properties in the most general form possible for an elastic continuum, namely Free
the unrestricted set of components in positive semi-definite constitutive matrices.
For a given material (given constitutive relations), we normally measure cost by
the amount of material, say by thickness or density. With the free material we need
a measure of the ”amount of a matrix”, and cost is then measured on the basis of
invariants of these matrices.
With reference to the paper by (Bendsøe et al. 1994), we extend the results obtained
in that paper to be valid also for power law non-linear elasticity, as done in
(Pedersen 1998). If we choose as cost constraint the Frobenius norm (length of a ma-
material
trix) of the constitutive matrix, then the analytical proof of the optimal constitutive Frobenius
matrix, even for 3D-problems, is rather direct.
norm
15.2 Sensitivity analysis
With localized sensitivity analysis as shown in chapter 13, and also given specifically
by (13.13), we have
dUɛ ∂Uɛ
1 ∂uɛ
= −1
= −Ve
(15.1)
dh p ∂h fixed strains p ∂he fixed strains
where Ve is the volume of the domain of the localized design variable he, (here a Localized
component of the constitutive matrix). Thus minimum total strain energy Uɛ implies
241
determined
sensitivities

Design
problem
Principal
strains only
Direct
conclusions
Positive
definite
242 Copyright c○Pauli Pedersen: Optimal designs ...
maximum strain energy density uɛ in a fixed strain field. (In domains of non-constant
strain energy density, the notion of mean value ūɛ should be used). The strain energy
density depends homogeneously on the squared effective strain ɛeff , see (12.18) in
chapter 12. The problem formulation can therefore be stated as
Maximize ɛ 2 eff := {ɛ} T [α]{ɛ} subject to F robenius([α]) = 1 (15.2)
where the matrix [α] describes the non-dimensional part of the constitutive matrix in
the secant formulation (12.16).
In the invariant formulation for [α] we can choose the coordinate system of principal
strains
{ɛ} T = {{ɛ1 ɛ2 ɛ3} {000}} (15.3)
and obtain
ɛ 2 eff = {ɛ} T [α]{ɛ} = {ɛ1 ɛ2 ɛ3}
⎡
⎣ α1111 α1122 α1133
α1122 α2222 α2233
α1133 α2233 α3333
⎤ ⎧
⎨
⎦
⎩
ɛ1
ɛ2
ɛ3
⎫
⎬
⎭
(15.4)
Now, the Frobenius norm of a matrix is defined as the square root of the sum
of the squares of all the elements of the matrix (equal to the squared length of the
contracted vector). It thus follows directly that for optimality, the matrix elements
not involved in (15.4) must be zero. This means directly that also for the non-linear,
power law materials we have:
• the optimal material is orthotropic
• principal directions of material, strain and stress are aligned
• there is no shear stiffness
This result for linear elastic material is proven in (Bendsøe et al. 1994), based
also on a constraint on the trace of the constitutive matrix. Here, the extension to
non-linear elastic material follows directly from the localized sensitivity result (15.1).
For simplicity of proof we have chosen the Frobenius norm as the constraint.
15.3 Final optimization
The further analysis relates only to the sub-matrix in (15.4). To fulfill the condition
of being positive definite, we have as necessary conditions
α1111 > 0, α2222 > 0, α3333 > 0
α1111α2222 >α 2 1122, α1111α3333 > α 2 1133, α2222α3333 >α 2 2233 (15.5)

The ultimate optimal material 243
The problem formulation (15.1) can now be written as
Maximize ɛ 2 eff = α1111ɛ 2 1 + α2222ɛ 2 2 + α3333ɛ 2 3 +
2α1122ɛ1ɛ2 +2α1133ɛ1ɛ3 +2α2233ɛ2ɛ3
(15.6)
constrained by (15.5) and by given Frobenius norm
F
New design
problem
2 − 1=α 2 1111 + α 2 2222 + α 2 3333 +2α 2 1122 +2α 2 1133 +2α 2 2233 − 1 = 0 (15.7)
The general necessary condition for optimality is proportional gradients
chapter 14), i.e. for this specific case
(see Optimality
condition
which gives the result
ɛ 2 1
α1111
d(ɛ 2 eff )/dαiijj = λd(F 2 )/dαiijj
= ɛ22 =
α2222
ɛ23 =
α3333
ɛ1ɛ2
=
α1122
ɛ1ɛ3
=
α1133
ɛ2ɛ3
α2233
(15.8)
(15.9)
and we can finally write the resulting constitutive matrix in the directions of principal
strains/stresses (evaluating λ to satisfy (15.7): Optimal
modulus
matrix
1
[α]optimal =
(ɛ1 + ɛ2 + ɛ3) 2
⎡
⎤
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥ (15.10)
⎢
⎥
⎣
⎦
ɛ 2 1 ɛ1ɛ2 ɛ1ɛ3 0 0 0
ɛ1ɛ2 ɛ 2 2 ɛ2ɛ3 0 0 0
ɛ1ɛ3 ɛ2ɛ3 ɛ 2 3 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
15.4 Numerical aspects and comparison with isotropic
material
The result (15.10) is valid also for power law, non-linear elastic materials. We note
that the matrix in (15.10) has only one non-zero eigenvalue and that the material
therefore only has stiffness in relation to the specified strain condition. For the ultimate
optimal material, the effective strain ɛeff , the strain energy density uɛ, and the
Frobenius norm F are
ɛ 2 eff = ɛ 2 1 + ɛ 2 2 + ɛ 2 3
uɛ = E 1
p +1 (ɛ2 1 + ɛ 2 2 + ɛ 2 3) (p+1)/2
F = 1 (15.11)

Collected
results
244 Copyright c○Pauli Pedersen: Optimal designs ...
We can obtain the same effective strain and strain energy density with an isotropic,
zero Poisson’s ratio material [α] =[I], but then the corresponding Frobenius norm is
F = 6, i.e. the material cost is six times greater. As shown in (Bendsøe et al. 1994),
the zero Poisson’s ratio material may be valuable in numerical calculation, because
of the degeneracy of the ultimate optimal material.
All the examples in chapter 4 are calculated with zero Poisson’s ratio material, and
therefore may be interpreted as examples of ultimate optimal material distributions.
15.5 Summing up
In this chapter the important results to focus on are:
• The ultimate optimal material is very degenerate and is only stable in relation
to the specific strain state for which it is designed.
• The obtained solution is also valid for power law non-linear elastic materials,
and simple arguments lead to the obtained analytical solution.
• The direct comparison with isotropic, zero Poisson’s ratio material is most interesting,
and can be used for obtaining numerical solutions to specific problems.
For specific problems this is demonstrated in chapter 4.

Chapter 16
Conditions for statically
determinate trusses
In the early papers on optimal design of truss topology by (Dorn et al. 1964) and
by (Fleron 1964), the optimization of truss topology was formulated as a linear programming
problem with bar forces as unknowns. From this follows that there exist
statically determinate optimal trusses. These formulations are based on design in- Early
dependent allowable stresses, possibly different for each bar and possibly different for
bars in tension and bars in compression, but given in advance.
Extension to include compressive allowable stresses that account for local stability
references
of the bars is presented in (Pedersen 1969) and (Pedersen 1970), and still statically Extensions:
determinate trusses resulted. In (Pedersen 1992) and (Pedersen 1993) unknown supports
with individual costs were included in the formulation, and the presentation
below very much follows these 92-93 proceedings.
In a statically determinate truss the force in each bar is determined only by
stability,
supports
equilibrium and is not dependent on the size of the bars. Therefore, the size of each Individual
bar can be designed individually and for a minimum cost solution each bar must be
designed to its minimum size, i.e. it must be fully stressed.
In the formulation below we assume that each bar carries two forces, a tensile
force and a compressive force which are both non-negative. In the minimum cost
truss either the tensile or the compressive force is zero, but in the formulation the
account for two independent forces is important. In general the presentation in this
chapter is closely related to the theory of linear programming, and therefore may
seem complicated if this theory is unknown to the reader. However, the chapter is
written to be be self-contained.
245
bar design

Assumptions
Statically
determinate
solution
3D
Equilibrium
at joints
Unknown
support
reactions
246 Copyright c○Pauli Pedersen: Optimal designs ...
16.1 Theorem for the single load case
If each bar n of the truss is designed with a minimum cost φn (fully
stressed), which as a function of the actual non-negative bar force Pn,
satisfies
dφn
dPn
then there exists at least one statically determinate truss that minimizes
the cost of the total truss.
≥ 0 and dφn
non − increasing with Pn, i.e.
dPn
d2φn dP 2 ≤ 0 (16.1)
n
16.2 Proof with supports as further unknowns
The proof follows from equilibrium, change in equilibrium by introducing an additional
bar and the corresponding change in the total cost. We directly deal with threedimensional
trusses, and thus a two-dimensional truss is just a special case.
16.2.1 Force equilibrium
At each joint j three force equilibriums must be satisfied, in all
[Rm]{Pm} = {L} (16.2)
where [Rm] is a matrix of direction cosine’s for the members (bars) in the truss (matrix
order 3J × 2N with totally J joints and totally 2N = J(J − 1) possible bar forces
(N bars)). The vector {L} is a vector of 3J external forces including the reactions.
Let us express the forces {L} in known external forces {B} added a number K of
possible support reactions {Bk} for k =1, 2,...,K (K

Conditions for statically determinate trusses 247
and the total vector (order 2N + K) of unknown member forces {Pm} plus reactions
{Ps} at the possible supports
{P } T := {{Pm} {Ps}} (16.5)
then with a completely known right hand side {B} the total equilibrium is written
[R]{P } = {B} (16.6)
To state it differently, we have treated the possible reactions as possible truss members
connected only to one joint. Note that it is possible to describe any support direction Supports
at a joint and that a joint with totally fixed displacements requires three columns in
the matrix [Rs].
In sub-matrix form we write the force equilibriums Basis
{PI}
[[RI] [RO]]
{PO}
= {B} (16.7)
where the squared matrix [RI] of order equal to the rank of the total matrix [[R] {B}]
is non-singular. There may be several possibilities for choosing [RI], but at least one
exists.
We solve system (16.7) in terms of {PI} and get
{PI} = {
Basis
solution
˜ PI}−[Z]{PO} = { ˜ PI}−
{Zo}Po
(16.8)
with the definitions
o
[Z] := [RI] −1 [RO] (16.9)
{ ˜ PI} := [RI] −1 {B} (16.10)
and where {Zo} is column o of the matrix [Z]. The force Po is force o of the set {PO},
and later we use forces Pi that belong to the other set of forces {PI}.
16.2.2 The total cost with all forces
as members
separation
When designed to its minimum allowable value, the cost to of carrying force Pn in a Cost
certain bar is given by a function φn
functions
φn = φn(Pn) (16.11)
satisfying the conditions stated in the theorem. The total cost Φ for the truss corresponding
to all the forces is then
Φ=
φi(Pi)+
φo(Po) (16.12)
i
o

Derivative
of cost
Cost nonincreasing
248 Copyright c○Pauli Pedersen: Optimal designs ...
which, with the solution (16.8), we write as
Φ=
16.2.3 The change in total cost
i
φi
˜Pi −
(Zi)oPo +
φo(Po) (16.13)
o
Assuming now that all the Po forces are zero, i.e. {PO} = {0}, a total cost ˜ ΦI is
defined
From (16.13) we also calculate dΦ/dPo to
i
o
˜ΦI =
φi( ˜ Pi) (16.14)
i
dΦ
=
dPo
dφi(Pi)
(−(Zi)o) +
dPi
dφo(Po)
dPo
16.2.4 Monotonous behaviour and new basis solution
(16.15)
We can prove dΦ/dPo to be a non-increasing function of Po: For the second term
in (16.15) that is directly the assumption of our theorem. In the summation term
of (16.15), then for (−(Zi)o) > 0 we have increasing Pi as seen from (16.8), thus
non-increasing dφi(Pi)/dPi and therefore also (dφi(Pi)/dPi)(−(Zi)o) non-increasing.
Finally, terms with (−(Zi)o) < 0 give decreasing Pi, thusdφi(Pi)/dPi non-decreasing
with Po and thus also for these terms (dφi(Pi)/dPi)(−(Zi)o) non-increasing.
All terms of dΦ/dPo are thus non-increasing with Po, and the change in total
cost is monotone. In order to minimize Φ we should therefore choose Po as large as
possible, remembering that all forces by definition are positive.
The maximum ∆Po is determined by the condition that all the values of Pi should
stay non-negative. This condition is by (16.8) stated
Pi = ˜ Pi −
(Zi)o∆Po ≥ 0 for alli (16.16)
o
which means that a new basis is obtained. The bar i that gives rise to the smallest ∆Po
in (16.16) leave basis and the new basis with Pi = 0 and ∆Po introduced corresponds
to a new statically determinate truss.

Conditions for statically determinate trusses 249
16.3 Optimality condition
In other words, our optimality condition is that all neighbouring basis solutions give
larger cost ∆Φ > 0 than the present basis solution. In mathematical terms with ∆Po
determined by (16.16), the optimality condition is Finite cost
increment
∆Φ = φo(∆Po)+
i
φi
˜Pi − (Zi)o∆Po
− ˜ ΦI > 0 for all o (16.17)
i.e. for all forces not in basis, tensile as well as compressive forces. The special case
of ∆Po = 0 is controlled by the gradient condition from (16.15), i.e.
dΦ/dPo > 0 for ∆Po = 0 (16.18)
16.4 Alternative proofs and solution procedures
In the literature on mathematical programming, as in (Gass 1964) and in (Hadley Convex
1964) we can find geometrical interpretations showing directly that the optimal solution
is at a vertex of the convex feasible space described by the linear constraints
(16.6). When the objective is a concave function and not just a linear function the
solution is still at a vertex, but local optima may exist if the concave functions are too
curved relative to the curvature of the feasible space. In the actually solved problems Concave
of topology design of trusses such local optima have not been seen.
cost
For the pure linear programming problem, the simplex procedure described in
most literature on mathematical programming and in (Press, Teukolsky, Vetterling
and Flannery 1992) can be applied. For the actual problem with a concave objective
function, the modified simplex procedure described in (Pedersen 1994) can be applied.
This modified procedure also illustrates how the doubling of unknowns (both tensile
and compressive forces in each bar) can be avoided and directly build into the solution
procedure.
16.5 A truss member model
The size of a truss member is determined directly by the force P transmitted through
the member, and the size is assumed to be unchanged through the length l of the
member. The member cross-sectional area A, moment of inertia I and, finally, the
mass/volume/cost φ are thus only dependent on the force P in addition to given
parameters.
The dependence on the force, however, is not very simple because practical design
codes, which account for local stability, divide the force domain into four cases:
feasible space

Member
classes
Slender
or short
250 Copyright c○Pauli Pedersen: Optimal designs ...
• Tensile bars, i.e. P>0
• Zero strings, i.e. P =0 +
• Slender columns, i.e. PL

Conditions for statically determinate trusses 251
16.5.3 Short columns, i.e. P 1 (16.26)
we write the linear cost function
φ = ηCT (PL − P )+CC |PL| ⇒ dφ/d|P | = ηCT > 0 ⇒
16.5.4 Zero strings, i.e. P =0 +
d 2 φ/d|P | 2 = 0 (16.27)
In trusses with only a single load case, the optimal topology is often be a mechanism.
To deal with this problem, we include tensile bars of vanishing area and cost No cost
assumed
A = 0 (16.28)
φ =0 ⇒ dφ/dP = CT > 0 ⇒ d 2 φ/dP 2 = 0 (16.29)
According to (16.24) the derivative corresponding to 0 − is infinite; thus, by definition,
the zero strings belong to the tensile domain.
16.5.5 Concave cost functions
We note that all the cost functions (16.22 ), (16.24), (16.27), (16.29) satisfy
φ(0)=0and dφ/dP > 0 and d 2 φ/dP 2 ≤ 0 (16.30)
and the only remaining condition for concave cost functions is then related to the
transition from a slender column to a short column. From (16.24) and (16.27) we get
the condition of non-increasing derivative for the transition to be Concave
CC
2 |PL| >ηCT
(16.31)
transition

Assumption
for cost of
supports
Collected
results
252 Copyright c○Pauli Pedersen: Optimal designs ...
Let us from (16.23 ,16.24) assume CC = l/(πα E/ξ) which with (16.19) gives
CC
2 √ |PL| = |PL|ξ/σL. Then, the left hand side of (16.31) is ξ/(2σL). With η := σT /σ0
and from (16.21, 16.22) CT =1/σT we can finally write the condition (16.31)
ξ ≥ 2σL/σ0
(16.32)
When the factor of safety has the value 2σL/σ0, the slopes have a continuous transition.
Totally, with (16.31) or specifically (16.32), we conclude that the cost functions
are all concave functions.
φ = φ(|P |) (16.33)
16.5.6 Cost functions for the supports
We assume that the support costs are also described by concave functions, with possibility
of a different function when the support reaction is either positive or negative.
This assumption is as an example fulfilled by simple linear functions.
Totally, we conclude that the cost functions for bar members as well as for supports
are all concave functions.
16.6 Summing up
In this chapter the important results to focus on are:
φ = φ(|P |) (16.34)
• The single load case assumption behind the results in this chapter is most
essential.
• Local stability of each bar in the truss is accounted for, treating both slender
columns and short columns.
• Support conditions need not be specifically given and need not be statically
determinate. Support reactions are found as part of the optimization.
• Specific optimal truss examples are presented in chapter 3, and the necessary
numerical procedure for the optimization is described in chapter 19.

Chapter 17
Orientation of orthotropic
material
A problem of major practical importance is related to the optimization procedure for
locating optimal orientation, i.e. how to find an optimal design, which satisfy the Many local
extremum
optimality criterion, stated in chapter 14 by (14.2). Procedures like steepest decent
are not reliable due to the inherent problem of many local extremum.
Recursive procedures are used extensively, but also here several possibilities exist.
We may redefine according to actual larger principal strain direction, according to
actual larger principal stress direction or according to some combination of these
directions. The relative effectiveness of these procedures is problem dependent, i.e. Recursive
dependent on the degree of non-isotropy and not at least on the degree of statically
indeterminacy of the problem.
In the present analysis a stress characteristic of the optimal solution is proven.
This states a condition for the principal stress ratio σ2/σ1 (where |σ1| ≥|σ2|), which
must be satisfied for the design with optimal orientation. To a large extent the
presentation in this chapter follows (Pedersen and Bendsøe 1995).
17.1 Inherent practical problems
In order to use a non-isotropic material effectively it should be oriented optimally
with respect to the actual stress strain condition. We are not here concerned with the
design of material as in chapter 8, but merely with the use of a given non-isotropic
material. Complete analytical results are obtained for orthotropic materials in two
procedures
dimensions, showing that point-wise local as well as global maxima and minima Gradient
result. Thus a pure gradient information is not sufficient and we need to obtain a
deeper understanding.
253
information
non-sufficient

Alignments
The match
problem
254 Copyright c○Pauli Pedersen: Optimal designs ...
In optimal orientation for maximum total stiffness (minimum compliance) it is
shown that for orthotropic materials we get alignment between principal strain directions,
principal stress directions and principal material directions, at least in most
cases, see (Seregin and Troitskii 1982), (Banichuk 1983), (Fedorov and Cherkaev
1983), (Pedersen 1989), (Rovati and Taliercio 1991), (Pedersen and Taylor 1993).
Materials with relative high shear stiffness (to be defined later) are not aligned
with the principal strains and principal stresses, but also for these materials are the
principal strain directions and the principal stress directions aligned when the criterion
for optimal orientation is satisfied, (Pedersen 1990).
However, alignment is not satisfactory information, because even for 2D-problems
we have two principal strains, two principal stresses and two orthotropic material
directions. Thus a match problem is identified. If the criterion for optimal orientation
is satisfied, then the numerically larger principal strain and the numerically larger
principal stress together with the numerically larger material modulus are all aligned.
For all relative low shear stiffness materials we obtain this result, but not in general
for all materials.
17.2 From global to local optimality criterion
The actual optimization problem is stated as orientational design for minimum compliance
(equal to the negative external potential Uext), for minimum elastic strain energy
Uɛ, for maximum stiffness (-Uɛ), or for maximum potential energy Π := Uɛ + Uext.
Different
These different statements are all the same with a constant ratio between elastic
objectives stress energy Uσ and elastic strain energy Uɛ. For power law elasticity (including
linear elasticity) this ratio is p (see chapter 12) with 0

Orientation of orthotropic material 255
independent of the power p of the non-linear elasticity model. In a point-wise (continuum)
formulation the mean strain energy density (ūɛ)e is substituted by the strain
energy density uɛ itself, and the angle θe of element e with the angle θ at the actual
point. In the following, for simplicity, we use this notation without index e and
without the mean value bar super-index¯.
17.3 Multiplicity of extremum
We restrict the discussion here and in the following to the case of linear, two-dimensional
elasticity. The conditions for optimal orientation for an orthotropic material are presented
here, following the developments in (Pedersen 1989), and (Cheng and Pedersen
1997). First consider the maximization or minimization of the point-wise strain energy
density for an orthotropic material. Using the axes of principal strains (ɛ1,ɛ2)
as the frame of reference, the strain energy density can be written as The θ function
for strain
uɛ = C2(ɛ energy density
2 1 − ɛ 2 2) cos 2θ + C3(ɛ1 − ɛ2) 2 cos 4θ (17.3)
with the definitions (used in the theory of laminates, see also chapter 6)
2C2 := C1111 − C2222, 8C3 := C1111 + C2222 − 2C1122 − 4C1212
(17.4)
where θ is the angle between the principal strain frame and the axes of orthotropy,
and where ⎡
⎤
Non-isotropic
parameters
[C] = ⎣
⎦ (17.5)
C1111 C1122 0
C1122 C2222 0
0 0 2C1212
is the constitutive matrix in the frame of orthotropy. Ordering so that |ɛ1| ≥|ɛ2| and
C1111 ≥ C2222, we see that all coefficients to the trigonometric functions in (17.3), Solution
except C3, are positive. The condition of stationarity (17.2) becomes
equation
sin 2θ(C2(ɛ 2 1 − ɛ 2 2)+4C3(ɛ1 − ɛ2) 2 cos 2θ) = 0 (17.6)
and from second order conditions or direct evaluation at the stationary points, we
obtain, with the definition of the strain related parameter γ for C3(ɛ1 − ɛ2) = 0,
γ := C2 ɛ1 + ɛ2
4C3 ɛ1 − ɛ2
(17.7)

Results in
strains
Relative
low or high
shear stiffness
The θ function
for stress
energy density
Results in
stresses
256 Copyright c○Pauli Pedersen: Optimal designs ...
C3 > 0: uɛ is max for θ = 0
uɛ is min for θ = ±(1/2) arccos (−γ) if |γ| ≤1
uɛ is min for θ = ±(π/2) if |γ| > 1 (17.8)
C3 < 0:uɛ is max for θ = ±(1/2) arccos (−γ) if |γ| < 1
uɛ is max for θ = 0 if |γ| > 1
uɛ is min for θ = ±(π/2) (17.9)
and in all cases (seen by inspection) are the axes of principal strains and principal
stresses (at the optimum) aligned. In line with (Pedersen 1989), we say that a material
with C3 > 0 has relative low shear stiffness, and relative high shear stiffness if C3 <
0. The match between larger principal stress directions and larger principal strain
directions is discussed in section 17.4.
We note here that the results above can be used directly to obtain similar results
for the dual problem of minimization or maximization of the stress energy density uσ
with the definitions
uσ = H2(σ 2 1 − σ 2 2) cos 2θ + H3(σ1 − σ2) 2 cos 4θ (17.10)
2H2 := H1111 − H2222, 8H3 := H1111 + H2222 − 2H1122 − 4H1212
Here [H] =[C] −1 in the frame of orthotropy is given as
[H] =
1
C1111C2222 − C 2 1122
⎡
⎣
C2222 −C1122 0
− C1122 C1111 0
0 0
C1111C2222−C 2
1122
2C1212
(17.11)
⎤
⎦ (17.12)
We get in analogy to (17.8, 17.9), with the definition of the stress related parameter
ζ for H3(σ1 − σ2) = 0,
ζ := H2 σ1 + σ2
(17.13)
4H3 σ1 − σ2
H3 > 0: uσ is max for θ = 0
uσ is min for θ = ±(1/2) arccos (−ζ) if |ζ| ≤1
uσ is min for θ = ±(π/2) if |ζ| ≥1 (17.14)

Orientation of orthotropic material 257
H3 < 0:uσ is max for θ = ±(1/2) arccos (−ζ) if |ζ| < 1
uσ is max for θ = 0 if |ζ| > 1
uσ is min for θ = ±(π/2) (17.15)
It can be shown, see (Cheng and Pedersen 1997), that C3 < 0 leads to H3 > 0, but if
C3 > 0 the sign of H3 can be positive as well as negative. Maximization of stiffness
corresponds to maximization of uɛ in a fixed strain field or minimization of uσ in a
fixed stress field (or vice versa for minimization of stiffness).
Taking for example a case with C3 < 0 and H3 > 0, we see from (17.9), (17.14) Example
that the strain and stress fields at the optimum are related by the condition γ = ζ.
Also, note that if a given (non-optimal) set of stress and strain fields do not satisfy
this relation, the optimum rotation angles read off from (17.8), (17.9), (17.14), (17.15)
may be quite different. The complete analysis for the various combinations of signs
of C3 and H3 can be found in (Cheng and Pedersen 1997). In the following we
concentrate on finding characteristic bounds for optimal stress fields, and consider
the effects these bounds have on computations.
17.4 The match problem
Even for 2D-problems we have two principal strain directions, two principal stress Principal
directions and two orthotropic material directions. Thus knowledge on alignment of
principal directions is not enough information, and in this section this problem is
analysed in more detail to end up with a specific strategy.
Firstly, we consider principal strains ɛ1,ɛ2 =1,η aligned with principal stresses
σa,σb and related by the constitutive equation
σa C1111 C1122 1
=
(17.16)
σb C1122 C2222 η
with constitutive parameters C1111,C2222,C1122. For well-ordered strains, i.e. ɛ2 = Relative
directions
ηɛ1 with −1 ≤ η ≤ 1, we now derive the size-ordering of the stresses, given in terms strains
of the constitutive parameters and the relative strain value η. First, note that the
condition for a positive definite constitutive matrix implies that Positive
definite
C1111C2222 − C 2 1122 > 0, C1111 > 0, C2222 > 0, (17.17)
With the stresses given by (17.16), we can obtain the necessary
investigation of the sign of a second order polynomial in η
information by an 2nd order
polynomial
σ 2 a − σ 2 b =(C 2 1122 − C 2 2222)η 2 +2C1122(C1111 − C2222)η +(C 2 1111 − C 2 1122) (17.18)

Stress
ordering
from strains
258 Copyright c○Pauli Pedersen: Optimal designs ...
over the interval −1 ≤ η ≤ 1. This function defines a parabola parameterized in the
variable η which at the interval end-points takes the values
Moreover, the two roots σ 2 a − σ 2 b
(C1111 − C2222)(C1111 + C2222 ± 2C1122) (17.19)
= 0 are found to be at
η± = ∓ C1111 ± C2222
C2222 ± C1122
which lie in the interval −1 ≤ η ≤ 1if(C2222 ± C1122) 2 > (C1111 ± C1122) 2 , i.e. if
(17.20)
−(C1111 − C2222)(C1111 + C2222 ± 2C1122) ≥ 0 (17.21)
The factor (C1111 + C2222 ± 2C1122) common to (17.19) and (17.21) is positive as we
have C1111C2222 − C2 1122 > 0. Thus if (C1111 − C2222) > 0, the end values by (17.19)
are positive and no roots fall in the interval −1 ≤ η ≤ 1 indicating that σ2 a − σ2 b > 0
for all −1 ≤ η ≤ 1. On the other hand, if (C1111 − C2222) < 0 the end values by
(17.19) are negative and the roots fall in the interval −1 ≤ η ≤ 1. Then σ2 a − σ2 b > 0
holds between the roots (17.20). In conclusion we have :
C1111 >C2222 : |σa| > |σb| for any − 1 ≤ η ≤ 1
C1111 |σb| for − C1111 + C1122
C2222 + C1122

Orientation of orthotropic material 259
The analysis (17.16) to (17.22) may be started from the assumption of stress
ordering |σ1| ≥|σ2| ending up with conditions like (17.22) for |ɛa| ≥|ɛb|. We here
only list this specific result that is needed for the further discussion
H1111 |ɛb| for − H1111 + H1122

Strain or
stress
directions
Collected
results
260 Copyright c○Pauli Pedersen: Optimal designs ...
optimization is given in (Pedersen 1991), and here we comment on the two alternatives
for orientational optimization, i.e. redefinition according to actual larger principal
strain direction or according to larger principal stress direction.
The experience from many solved problems shows that redefinition according
to larger principal stress gives the most stable optimization procedure. This can be
understood from the fact that the stress distribution is not strongly effected by the
design changes. In contrast the redefinition according to larger principal strain often
gives fluctuating results, and when this is the case we see that the characteristic
(17.24) is not satisfied. Thus, when (17.24) is violated the principal stress and strain
direction may be close to alignment, but the larger principal stress does not match
the larger principal strain, which then is very sensitive to design changes.
Should we redefine material orientation according to the principal strain field or
according to the principal stress field ? Normally, principal stresses is to be preferred
but it depends on the specific problem. For an almost statically determinate problem
the stress field changes very little, and if it does not change at all then only one
iteration is necessary. On the other hand in maximizing flexibility we may have
rather unchanged strain field and then a different strategy is needed.
17.6 Summing up
In this chapter the important results to focus on are:
• The elastic energy as a function of material orientation has multiple stationary
solutions, in most cases corresponding to alignment of principle stress, principle
strain and orthotropic material directions.
• The case of high relative shear stiffness material, should be treated with special
care. See (Cheng and Pedersen 1997) and (Pedersen 1989) for more detail.
• The optimal match of the specific principle directions for stress, strain and
material is to a large extent known for two-dimensional problems, and recursive
iterations work in most cases. The three-dimensional problems still need some
clarification.
• Especially for strongly non-isotropic materials and strongly statically indeterminate
cases, severe restrictions are put on the optimal stress state, as stated
by (17.24). Thus the existence of solutions that satisfy the optimality criterion
is not always guaranteed.
• Gradient based methods are not reliable, due to the mentioned multiplicity of
stationary solutions.

Chapter 18
Sensitivity analysis for
dynamic problems
18.1 Including non-conservative problems
The dynamic behaviour of systems subjected to non-conservative loads is often against
our physical intuition and can only be understood with a deep insight. As exam- Non-intuitive
ples: vibration frequency may increase with increasing load and damping may act behaviour
destabilizing. The non-conservative forces make it necessary to deal with complex
functionals in the description, and the intuitive understanding of the physical results
is not easy because the indirect effects may be the determining ones. However, in this
chapter high priority is given to the physical interpretation.
The usual (familiar) analysis for the displacements etc., could be based either
on the equations of equilibrium, set up directly, or on the stationarity of an energy
functional. For the non-selfadjoint problems such functionals are now also available in
the literature and we term them here the mutual energy functionals - mutual in the Mutual
sense that displacements of the physical as well as of the adjoint problem are involved. potentials
The sensitivity analysis may also be based directly on the equations of equilibrium,
but the simplicity is striking when we base it on the mutual energy, because then the
variations of displacements do not have to be considered.
The response analysis and sensitivity analysis can be formulated in either a continuous
or in a discrete form. As almost all problems are solved numerically with Finite
only finite degrees of freedom, the sensitivity analysis is presented in a matrix for- degrees of
mulation, which has the advantage that boundary conditions are an integrated part. freedom
Furthermore, this formulation relates directly to the practical methods of analysis,
such as the finite element method and the Galerkin method.
261

Linear
systems
Stationary
systems
Normalized
mass matrix
Conservative
systems
262 Copyright c○Pauli Pedersen: Optimal designs ...
18.2 Classification of dynamic systems and their behaviour
Restricting ourselves to linear, elastic system, we can in matrix notation write the
dynamic equilibrium by
[M]{ ¨ D} +[C]{ ˙ D} +[S]{D} = {A} (18.1)
where [M] is termed the mass matrix, being the coefficient matrix to the vector
{ ¨ D} of accelerations. Thus derivative with respect to time t is denoted by the ˙ (dot)
superscript. The matrix [C] is termed the damping matrix, being the coefficient
matrix to the vector { ˙ D} of velocities. The stiffness matrix [S] is the coefficient
matrix to the displacements {D}, and the external forces {A} then make the system
inhomogeneous.
18.2.1 System classification
The system classification follows the classic book by (Ziegler 1968), and we immediately
identify the class termed instationary systems. If only one element in one of the
matrices [M], [C], [S], {A} is explicitly a function of time t, then the system is termed
instationary. Sensitivity analysis for these systems is described in (Pedersen 2001b).
With stationary external forces {A} , we can concentrate on the homogeneous system
that describes the dynamics relative to the static equilibrium [S]{D} = {A}. Thus
we continue our classification based on system description by
[M]{ ¨ D} +[C]{ ˙ D} +[S]{D} = {0} (18.2)
and furthermore we assume this to be normalized in such a way that [M] is symmetric
and positive definite. Now, the remaining classification relates directly to the contents
of the matrices [C] and [S]. Writing these matrices as a sum of their symmetric and
skew-symmetric parts
[C] = 1
2 ([C]+[C]T )+ 1
2 ([C] − [C]T ), [S] = 1
2 ([S]+[S]T )+ 1
2 ([S] − [S]T ) (18.3)
we can directly define our remaining four classes of systems.
A conservative, non-gyroscopic system has no velocity depending terms and a
symmetric stiffness matrix, which in physics may describe simple, free, undamped
vibrations. The characteristics of the possible solutions are well known, and these
solutions often serve as expansions for more complicated systems. A conservative,
gyroscopic system may in addition include gyroscopic forces which are reflected in
the skew-symmetric part of the damping matrix which in physics often is a possible

Sensitivity analysis for dynamic problems 263
description for a rotor system. The non-conservative systems are systems with dissipative
forces and/or with circulatory forces. The dissipative forces are described by
the symmetric part of the damping matrix, and the circulatory forces are described
by the skew-symmetric part of the stiffness matrix. Thus for a dissipative system Non-conservative
we have ([C] +[C] T ) = [0] without further condition on our general system. For a
circulatory system we have ([S] − [S] T ) = [0] which means that non-potential forces
are included in the system.
18.2.2 Classification of the behaviour
We are concerned about the stability of the solution {D} = {0} of (18.2) to small
disturbances. Now to classify as far as possible the behaviour of the above classes of
systems, we assume the time separation by
{D} = {Φ} exp ((α + ıω)t) ={Φ} exp (αt)(cos ωt + ı sin ωt) (18.4)
and thus in this separation from the initial formulation uses a complex exponential
factor λ = α + ıω as time coefficient. With this formulation we can avoid the physical
systems
nonsense of complex frequency. Two distinct classes of instability follow from this Flutter
separation. The first one is flutter instability (also termed dynamic instability), it
corresponds to solutions α>0,ω = 0. The second one is divergent instability (also
- dynamic
termed static instability), it corresponds to solutions α>0,ω = 0. Stability is Divergence
defined by solutions having only negative α (real part), and a critical solution is the
terminology for the case of a solution with at least one α = 0 and negative α values
for the remaining solutions. For ω = 0 this corresponds to a harmonic vibration part
of the solution. To be more specific about stability limits we should include gradient
information. With p as a load parameter we define a flutter load pF by the conditions
that at least for one solution we have
ωF = 0, αF =0, (∂α/∂p)F > 0 (18.5)
and non-positive values for the remaining α solutions. In parallel a divergence load
pD is obtained for
ωD =0, αD =0, (∂α/∂p)D > 0, (∂ω/∂p)D = 0 (18.6)
Now, one of the most important theorems in stability tells that a conservative, nongyroscopic
system can only lose its stability by a divergent instability. This means
that the definiteness of the stiffness matrix gives complete information about stability,
but naturally not about the dynamic behaviour in general. Thus, such a system is
stable if and only if [S] is positive definite.
- quasi-static

Different
sensitivities
System
matrix
Adjoint
problem
264 Copyright c○Pauli Pedersen: Optimal designs ...
18.3 Sensitivity analysis
The notion of sensitivity analysis is a very wide one, even when we restrict it to
response sensitivity for dynamic systems. It has been argued that sensitivity is just a
new word for partial derivative, and to some extent this is correct. When our response
is measured by a scalar like an eigenvalue and we ask for the sensitivity with respect
to another scalar then it is a partial derivative. However, we may also ask for the
sensitivity with respect to a function, and then the notion of gradient function is often
used. Furthermore, the response may be a function like an eigenfunction or another
displacement description, and then we need a notion for the change of a function with
respect to another function. So sensitivity analysis is a notion that covers the work
of determining all this important information. The description in this chapter follows
closely to (Pedersen 1984).
18.3.1 General variational analysis
We are interested in studying the dynamic behaviour, and with the separation (18.4)
the spatial problem is described by the homogeneous matrix equation
[L]{Φ} = {0} with [L] =[L(α + ıω,p,h)] (18.7)
where the system matrix [L] for the present analysis depends on the complex eigenvalue
λ with α as a stability measure and ω as frequency according to the separation
by {D} = {Φ} exp (α + ıωt). Furthermore [L] depends on the load level described
by the real parameter p and on design, load distribution, damping etc., all of which
we symbolize by the real quantity h, which - in individual cases - may be a scalar
parameter or a spatial parameter function. In addition to the physical system we also
analyze the adjoint problem
[L] T {Ψ} = {0} or {Ψ} T [L] ={0} T
(18.8)
i.e. we need both right {Φ} and left {Ψ} eigenvectors, for the same eigenvalue. We
also remember that with a symmetric system matrix (self-adjoint problem) we have
{Φ} = {Ψ}. The results we obtain in this section are expressed as ratios between
complex functionals, and in order to make it less abstract, an interpretation in terms
of specific, mutual energies is presented. The term specific means that the energy may
be per area or per volume and a (for the sensitivity analysis) non-important factor
may be omitted. The term mutual relates to the fact that two different eigenvectors
determine the energy. The functional, termed the total, specific, mutual energy Π is
defined by
Π:={Ψ} T [L]{Φ} = 0 (18.9)

Sensitivity analysis for dynamic problems 265
where the zero follows from (18.7) as well as from (18.8). Taking general variations
of Π, which is also zero with the variations, we get from δΠ =0 Total mutual
potential
{δΨ} T [L]{Φ} + {Ψ} T [δL]{Φ} + {Ψ} T [L]{δΦ} = 0 (18.10)
We find that the variations of the eigenvectors {Φ}, {Ψ} disappear, which is naturally
the reason for introducing the adjoint eigenvector and for taking variations of the
mutual energy, δΠ = 0, instead of variations relative to the equilibrium (18.7).
In relation to the sensitivity analysis where [δL] = [0] we write (18.10) more
specifically in the variations of the involved independent parameters δp and δh. To
have a shorter notation we define the important functionals (normally complex) Complex
functionals
A = {Ψ} T (∂[L]/∂λ){Φ}, B = {Ψ} T (∂[L]/∂p){Φ}, C = {Ψ} T (∂[L]/∂h){Φ} (18.11)
and then get
A(δα + ıδω)+Bδp + Cδh = 0 (18.12)
Direct interpretation of this equation (18.12) gives the results needed.
Sensitivity with respect to load level. With a fixed ”design” δh = 0 the
sensitivity from (18.12) makes possible a more rigorous definition of terms like critical
load and flutter load. Assuming A = 0 we read from (18.12) with δh =0
∂α/∂p = −Re(B/A), ∂ω/∂p = −Im(B/A) (18.13)
where the notation for real part is Re() and the notation for imaginary part is Im().
To clarify the term instability initiation, we must in addition to α = 0 require ∂α/∂p >
0. By (18.13) this instability condition is Well defined
instability
α =0 and Re(B/A) < 0 (18.14) criterion
In the case of ω = 0 condition (18.14) corresponds to initiation of flutter and in the
case of ω = 0 together with Im(B/A) = 0 condition (18.14) corresponds to initiation
of divergence.
Sensitivity with respect to design. With a fixed load level δp = 0 we read
from (18.12), again assuming A = 0
∂α/∂h = −Re(C/A), ∂ω/∂h = −Im(C/A) (18.15)
We are often interested in knowing whether a certain parameter h acts in a stabilizing Stabilizing or
or destabilizing manner, which, from (18.15), we find to
destabilizing
Re(C/A) > 0 ⇒ h stabilizing, Re(C/A) < 0 ⇒ h destabilizing (18.16)

Flutter load
sensitivity
Flutter
frequency
sensitivity
Kinetic,
dissipative,
elastic and
external
potentials
Sensitivity
functionals
266 Copyright c○Pauli Pedersen: Optimal designs ...
Flutter load sensitivity to design. A main question is related to the change
in instability level as a function of design, i.e. all variations in (18.12) are involved.
Since we focus on a load level of instability initialization, the variations are restricted
by
α =0 and δα = 0 (18.17)
Again for A = 0 we from (18.12) divided by A with (18.17) and using that ıδω is
purely imaginary, get
∂pF /∂h = −Re(C/A)/Re(B/A) (18.18)
Note that by (18.18) the sign of ∂pF /∂h is equal to the sign of Re(C/A). This simply
states the natural fact from (18.12) that if h is stabilizing, then pF increases with h.
If ω = δω = 0 the load pF , here termed flutter load, is as a special case a divergence
load. No specific analysis for this is therefore needed. The change in flutter frequency
is in a similar way obtained with δα = 0 and we get
∂ωF /∂h = −Im(C/B)/Re(A/B) (18.19)
because δpF
from (18.14).
is a real quantity. In (18.19) we have used that B = 0 which follows
18.3.2 Results with finite element formulation
We now restrict the description to system matrices [L] given by
[L] =λ 2 [M]+λ[C]+[S]+p[K] (18.20)
with λ as the only complex quantity. In order to get convenient short notation we
define specific, mutual energies: kinetic T , dissipative D, elastic U and external W
by
T := {Ψ} T [M]{Φ}, D := {Ψ} T [C]{Φ},
U := {Ψ} T [S]{Φ}, W := −{Ψ} T [K]{Φ} (18.21)
and with (18.20) we may express W by
W =(λ 2 T + λD + U)/p (18.22)
The important functionals A and B
energies
can then directly be expressed in the mutual
A =2λT + D, B = −W = −(λ 2 T + λD + U)/p (18.23)
and thus the results for pure load change (18.13) are directly available
∂λ/∂p = ∂α/∂p + ı∂ω/∂p =(λ 2 T + λD + U)/(p(2λT + D)) (18.24)

Sensitivity analysis for dynamic problems 267
To obtain the functional C we need more specific information, and even for the case
of [K] independent of h , the evaluation of
C = {Ψ} T (λ 2 ∂[M]/∂h + λ∂[C]/∂h + ∂[S]/∂h){Φ} (18.25)
seems rather complicated. For a finite element model we have
[M] = [Me], [C] = [Ce], [S] = [Se] (18.26)
This has to be read symbolically because [M], [C] and [S] are matrices of order say
10 3 while [Me], [Ce] and [Se] are element matrices of order 10. Often a parameter he
only influences a specific element and then (18.25) is brought down to
C = {Ψe} T (λ 2 ∂[Me]/∂he + λ∂[Ce]/∂he + ∂[Se]/∂he){Φe} (18.27)
where the components of the lower order vectors {Ψe}, {Φe} are contained in the
higher order vectors {Ψ}, {Φ}. Thus the sensitivity analysis is brought down to the
element level. Furthermore, the he dependence is often homogeneous, such that Element
level FEM
∂[Me]/∂he = m[Me]/he, ∂[Ce]/∂he = l[Ce]/he, ∂[Se]/∂he = k[Se]/he (18.28)
where m, l, k are given factors. Then (18.27) can be written directly in element mutual
energies defined in parallel to (18.21)
C =(mλ 2 Te + lλDe + kUe)/he
(18.29)
With the assumption behind (18.29), localized sensitivities are available , and the
general results (18.15), (18.18), and (18.19) simplify to
and
∂ωF
∂pF
∂he
= Re((mλ2 Te + lλDe + kUe)/(2λT + D))
heRe((λ 2 T + λD + U)/(2λT + D))
= −Im((mλ2 Te + lλDe + kUe)/(λ 2 T + λD + U))
heRe((2λT + D)/(λ 2 T + λD + U))
∂he
These formulas are directly suited for evaluation or programming.
(18.30)
(18.31)
18.3.3 Results with Galerkin modelling (global expansion)
An alternative to the finite element method, which we can interpret as a Galerkin Expansion
method with local expansion functions, we have methods with global expansion func- functions
tion often named Galerkin, Ritz and weighted residual. The advantage of these analysis
methods is the moderate number of degrees of freedom, and the disadvantage is

Galerkin
discretization
Gradient
function
268 Copyright c○Pauli Pedersen: Optimal designs ...
that expansion function valid over the whole structural domain V must be obtainable.
With x as the space parameter in V , we assume given expansion functions uj = uj(x)
for the physical eigenfunction and vj = vj(x) for the adjoint eigenfunction. Let us
rewrite (18.27) without matrix notation
C = (λ 2 ∂mij/∂h + λ∂cij/∂h + ∂sij/∂h)ΨiΦj
(18.32)
where mij,cij,sij are matrix elements of [M], [C], [S], respectively. These elements
are for the global Galerkin discretization determined by
mij = mH(ui,vj)dx, cij = cG(ui,vj)dx, sij = sF (ui,vj)dx, (18.33)
with F, G, H as given expressions in the functions u, v and their spatial derivatives.
The factors m = m(x),c = c(x),s = s(x), depend on the design function h = h(x).
By the partial derivative with respect to h(x) we mean change of h at position x in
space. Assuming only local influence we get
∂mij
∂h(x)
∂m(x)
∂cij
= H(ui,vj),
∂h(x) ∂h(x)
Inserting this in (18.32) we have
∂c(x)
∂sij
= G(ui,vj),
∂h(x) ∂h(x)
∂s(x)
= F (ui,vj)
∂h(x)
(18.34)
C(x) = 2 ∂m
∂c
∂s
(λ H(ui,vj)+λ G(ui,vj)+
∂h(x) ∂h(x) ∂h(x) F (ui,vj))ΨiΦj (18.35)
The sensitivities are then a function of space and we term them gradient functions.
For example in relation to the most important sensitivity (18.18) we have
g(x) :=∂pF /∂h(x) =−Re(C(x)/A)/Re(B/A) (18.36)
and the resulting flutter load variation is obtained by integration
δpF = g(x)δh(x)dx (18.37)
The following example has results evaluated by these formulas.
18.3.4 Example of a cantilever beam
In figure 18.1 (left) we have shown a cantilever beam of non-uniform mass m(x), bending
stiffness s(x), Kelvin-Voigt internal damping with coefficient γ, external damping
with coefficient β, and loaded according to the distribution pq(x) as follower forces.

Sensitivity analysis for dynamic problems 269
Figure 18.1: Left: extended Leipholz and Hauger columns including external and
internal damping. Right: characteristic curves for the Becks column for a uniform
well as for a linearly tapered column.
With eigenvalue λ and eigenfunction u(x) , the continuous eigenvalue problem is
((su ′′ ) ′′ + pQu ′′ )+λ(βu + γ(su ′′ ) ′′ )+λ 2
mu ≡ 0, Q := pq(ξ)dξ, (18.38)
with the boundary conditions Continuous
problem
u(0) = u ′ (0)=0, (1 + λγ)(su ′′ )x=1 =(1+λγ)(su ′′ ) ′ x=1 = 0 (18.39)
Special cases are known as Leipholz’s column and Hauger’s column, and with an
end load Beck’s column. Discretizing this problem by Galerkin expansions as given
in details in Pedersen and Seyranian (1983) we obtain the analysis results shown in

270 Copyright c○Pauli Pedersen: Optimal designs ...
figure 18.1 (right) for the Beck column and with very similar results for the Leipholz
and Hauger columns. The sensitivity analysis follows the theory described and the
specific functional for this problem is
C(x) = ((1 + λγ)2mv ′′
i u ′′
j + λ 2 viuj)ΨiΦj
(18.40)
and figure 18.2 shows results from sensitivity analysis. Note, that at positions (horizontal
axes) where the curves are below the zero level we have negative flutter load
gradient, and at these beam positions it is advantageous to decrease the beam area.
18.3.5 Multiple eigenvalues
The case of multiple eigenvalues is often excluded in sensitivity formulations, but in
reality it is a physical important example that must be properly dealt with. The most
simple case of a conservative, non-gyroscopic system for which
[L] =−ω 2 [M]+[S] (18.41)
With [M] and [S] symmetric we always have a complete set of eigenvectors, even
when some of them have the same eigenvalue. All eigenvalues and eigenvectors are
real with left and right eigenvectors being identical (self-adjoint). A straight forward
sensitivity analysis as derived from (18.25) gives for the specific solution (ω2 Distinct
, {Φ})
eigenvalues ∂ω2 ∂h = {Φ}T 2 ∂[M] ∂[S]
(−ω + ){Φ} (18.42)
∂h ∂h
when the normalization
{Φ} T [M]{Φ} = T = 1 (18.43)
Eigenmodes
identification
is assumed. This result can be found in (Wittrick 1962) and in many later references.
The problem is that when the eigenvector {Φ} is not unique, the sensitivity is only
defined as a directional derivative (h positive scalar coefficient to a specified design
direction), and furthermore {Φ} can be any linear combination of the set of eigenvectors
all with the same eigenvalue ω 2 . Only for some specific { ˆ Φ} the sensitivity
(18.42) has a physical meaning. To find the specific { ˆ Φ} vectors let us deal only with
the bimodal case, where with {Φ1}, {Φ2} being two arbitrary, but mutual orthogonal
eigenvectors. We have
{Φ} = c1{Φ1} + c2{Φ2}
{Φ1}[M]{Φ2} = 0, c 2 1 + c 2 2 =1 ⇒ {Φ} T [M]{Φ} = 1 (18.44)
Inserting this expansion for {Φ} in (18.42) we get
∂ω 2 /∂h = c 2 1g11 + c 2 2g22 +2c1c2g12 with
gmn := {Φm} T (−ω 2 ∂[M]/∂h + ∂[S]/∂h){Φn} (18.45)

Sensitivity analysis for dynamic problems 271
and thus depending on c1,c2. Extreme values of ∂ω2 /∂h can be found by solving the
eigenvalue problem constituted by these sensitivities Additional
eigenvalue
problem
g11
− g g12
g22 − g = 0 (18.46)
g12
The eigenvalues of this matrix are the true sensitivities, with corresponding eigenvectors
{ ˆ Φ}. When the eigenvalues ga,gb to (18.46) are found, we also have the
corresponding eigenvectors {Φa}, {Φb} that return gab = gba = 0, which means that
the mutual sensitivities are zero.
Note, that this case learns us the primary importance of eigenmodes as compared
to eigenvalues. A recent, detailed discussion of the multiple modal problem is given by
(Seyranian, Lund and Olhoff 1994). This reference includes several specific examples.
18.3.6 Double eigenvalue with only a single eigenvector
For non-selfadjoint problems the sensitivity analysis may be further complicated by
the fact that we do not have a complete set of eigenvectors. Thus the case of a double
eigenvalue with the same eigenvector has to be dealt with. Often this is necessary
to describe important physical problems like flutter initialization and interaction of
eigenvalue branches. The paper (Seyranian and Pedersen 1995) treats this in detail.
18.4 Summing up
In this chapter the important results to focus on are: Collected
results
• For non-conservative problems we often find non-intuitive response. This implies
that the necessary sensitivity analysis requires special attention.
• However, with definition of mutual potentials (complex potentials) a major part
of the simplicities are recovered. Especially the possibility to determine eigenvalue
sensitivities without involving eigenvector change is of great importance.
• As for static problems, localized design parameters lead to localized determination
of sensitivities, although physically not localized. Especially for homogeneous
design dependence of mass and stiffness, the resulting expressions are
simple and may be expressed directly in localized potentials.
• Special attention to multiple eigenvalues must be given, and very special attention
to the cases with non-complete eigenvector spaces. For details see (Pedersen
2001b) with an extended list of the references.

272 Copyright c○Pauli Pedersen: Optimal designs ...
Figure 18.2: Gradient functions g(x) for the Hauger columns. Left: Uniform beam.
Right: Linearly tapered beam. 1st row: both external and internal damping. 2nd
row: only internal damping. 3rd row: only external damping. 4th row: no damping.
For more detail see (Pedersen and Seyranian 1983).

Chapter 19
Simplex and a modified
simplex procedure
A modified simplex procedure is developed for solution of truss topology optimization.
The procedure is presented here from a more general point of view, and in (Pedersen
1994) it is documented by the involved Fortran subroutines.
A simple, yet storage-effective ’linear’ programming code is available, and the
assumption of non-negative variables is bypassed without increasing the size of the
problem. Furthermore, the objective is allowed to be summed over not just linear,
but also concave functions.
The procedure differs in three ways from the classic simplex procedure: LP
modifications
• reformulation to non-negative variables is not necessary
• the objective is not restricted to the linear case, but extended to concave functions
• full pivoting and change of basis is obtained without reordering the constraint
coefficient matrix.
Some or all of these aspects can probably be found in other linear programming
procedures. However, linear programming is to a large extent used as a ’black box’,
and the main idea of the present chapter is to show the simplicity of the necessary
programs and thereby make it easier to perform individual modifications. We start
by describing the standard linear programming problem.
273

Separated
objective
Separated
constraints
Regular
sub-matrix
Basis
solution
Feasible
solution
Global
optimal
274 Copyright c○Pauli Pedersen: Optimal designs ...
19.1 Standard linear programming form
A linear programming problem in a standard form is defined by a linear objective Φ,
here assumed to be minimized
Minimize Φ={C} T {X} = {CI} T {XI} + {CO} T {XO} (19.1)
where the vector {C} contains given cost coefficients and the vector {X} ≥ {0}
contains the unknown non-negative quantities to be found. For later use we have
immediately divided these unknowns into two groups with index I referring to In and
index O referring to Out. This separation is not known in advance, and in essence
constitutes the solution to be found.
The constraints in addition to the constraint of a non-negative solution are described
by the linear set of equations
[A]{X} = {B} or [AI]{XI} +[AO]{XO} = {B} (19.2)
where the coefficient matrices are of order M ×N for matrix [A], order M ×M for the
sub-matrix matrix [AI] and order M × (N − M) for the sub-matrix [AO]. We assume
N>M and [AI] regular, i.e. the inverse matrix [AI] −1 exists. Normally with more
unknowns N than the number of constraints M we have many feasible solutions, and
we must find the one that minimizes the objective Φ.
In the literature on mathematical programming, such as (Gass 1964) and (Hadley
1964), it is proven that the solution is a basis feasible solution (if any), i.e. a solution
with {XO} = {0} from which follows the index notation of O and o for out of basis.
An actual feasible basis solution is found by solving (19.2) in terms of { ˜ XI}, with a
super-index˜ to remember the assumption of {XO} =0,
{ ˜ XI} =[AI] −1 {B} ≥{0} (19.3)
We assume that a set of M unknowns (the selected {XI}) makes a solution like
(19.3) possible, and our goal is then to determine the set which minimizes the objective
(19.1).
19.2 Effect of changing the basis set
Based on the theoretical knowledge that the optimal solution is a global optimal
solution, we in one iteration, search for a better set among the better neighbour solutions.
When this is not possible the limited number of linear programming iterations
are ended and we have found a global optimal (basis feasible) solution.
If several neighbour solutions are better, we have to decide the change, i.e. what
unknown Xo from the set {XO} to be introduced. Among the possible procedures

Simplex and a modified simplex procedure 275
for this we have: here listed with increasing complexity of calculation but also with
increasing effect: Deciding
• The first found unknown with the effect of dΦ/dXo < 0
• The unknown with minimum gradient, i.e. minimum of dΦ/dXo < 0
• The unknown with the best effect, i.e. minimum of ∆Φ
For the criteria based on dΦ/dXo, we determine this gradient, and for this we write
the equilibrium (19.2), with only one variable from the set {XO} not zero, say Xo
[AI]{XI} + {Ao}Xo = {B} or
{XI} = { ˜ XI}−{Zo}Xo with {Zo} := [AI] −1 {Ao} (19.4)
The objective for this solution as seen from (19.1) with (19.4) is
neighbour
solution
Change in
equilibrium
Φ={CI} T ({ ˜ XI}−{Zo}Xo)+CoXo
(19.5)
with a resulting gradient Objective
dΦ/dXo = −{CI} T {Zo} + Co
(19.6)
derivative
This is information enough for the first two procedures.
However, to carry through the change of basis we also need to know the unknown
actually in basis, Xi from the set {XI}, that should be eliminated, i.e. transferred to
the set out of basis {XO}. This is determined by the condition of only non-negative
solution. From (19.4) this condition reads Eliminated
19.3 Non-linear objective
˜Xi − (Zi)oXo > 0 for all i giving
Xo = Minimum ˜ Xi/(Zi)o for (Zi)o > 0
and this also locates the actual i (19.7)
variable
from basis
With a non-linear objective we may still be interested in locating a solution at a
vertex, i.e. a set {XO} = {0}, for which all neighbour vertex solutions are not better.
If the objective function is concave, then this locates an optimal solution, possible
local but if the non-linearity is not very strong, then even a global optimal solution,
see (Hadley 1964) and chapter 16 of this book.
The procedure for obtaining such a solution is very closely related to a normal Best vertex
solution

Feasible
basis
solution
276 Copyright c○Pauli Pedersen: Optimal designs ...
simplex procedure, except that ∆Φ is used for deciding change of basis and not just
dΦ/dXo. Thus for all the unknowns at the present not in basis we first determine Xo
from (19.7). Then the changed solution follows from (19.4) and the changed objective
follows from ∆Φ = ˜ Φ − Φ with ˜ Φ as well as Φ non-linear depending on {XI} and Xo.
It is thus possible to locate the unknown Xo with the best effect, giving minimum
of ∆Φ. If this is negative we change, eliminating the unknown Xi determined from
(19.7). If minimum ∆Φ is positive a vertex solution which is better than all neighbour
vertices solutions is found. This solution is obtained in a finite number of iteration
steps.
19.4 Positive and negative variables
When the unknowns are not necessarily non-negative, a common trick is to use twice
the number of variables {X} = {X + }−{X − } with {X + }≥{0} and {X − }≥{0},
and we then reformulate the constraints (19.2) to
[A]{X + }−[A]{X − } = {B}
[AI]({X +
I }−{X−
+
I })+[AO]({X O }−{X− O })={B} (19.8)
The total coefficient matrix [[A] (−[A])] is now twice the size, which may be critical
for solving a large problem.
However, it should be clear that we do not need to store the information in [A]
twice. But how should we modify the simplex procedure to account for this when
only [A] is stored ? We have already separate the unknowns into the basis variables
being part of the two M × 1 vectors {X +
I }, {X− I } and the out of basis unknowns in
}, {X−}.
A basis solution is obtained by setting
the two (N − M) × 1 vectors {X +
O
} = {X−}
= {0} , and we have
{X +
O
O
O
{ ˜ X +
I }−{˜ X −
I } =[AI] −1 {B} = { ˜ B} (19.9)
In fact we do not evaluate [AI] −1 , so here we just assume that it exists, i.e. that the
M × M matrix [AI] is regular.
In terms of the individual components the interpretation of the solution (19.9) is
then
˜Bi ≥ 0 ⇒ X +
i = ˜ Bi, X −
i =0
˜Bi < 0 ⇒ X +
i =0,X− i = − ˜ Bi (19.10)
i.e. the signs of the right-hand-side elements after elimination directly focus on half
of the unknowns in { ˜ X +
}, and we have as usual only M basis unknowns.
I }, { ˜ X −
I

Simplex and a modified simplex procedure 277
With the linear combinations [Z] :=[AI] −1 [AO], as in the classic simplex procedure,
the constraint (19.8) is written as
{ ˜ X +
I }−{˜ X −
I } = { ˜ B}−[Z]{X +
O
} +[Z]{X− O } (19.11)
and with only one out-of-basis variable Xo different from zero, equation (19.8) gives
{ ˜ X +
I }−{˜ X −
I } = { ˜ B}−{Zo}X + o + {Zo}X − o (19.12)
The limiting numerical value of X + o or of X− o is obtained by the conditions that
the basis variables do not change sign (in classic simplex do not become negative).
Two cases, each with two sub-cases, are considered here.
Case 1: Assume X + o to be introduced, i.e. X− o = 0, then the row i of (19.12) is
˜X +
i − ˜ X −
i = ˜ Bi − (Zi)oX + o (19.13)
According to (19.10) we have two sub-cases:
Case 1a:
˜Bi ≥ 0 ⇒ X +
i = ˜ Bi − (Zi)oX + o
and X +
i only keeps its sign for ˜ Bi − (Zi)oX + o ≥ 0 (19.14)
Case 1b: Value of
˜Bi < 0 ⇒ −X −
i = ˜ Bi − (Zi)oX + o
and X −
i only keeps its sign for ˜ Bi − (Zi)oX + o ≤ 0 (19.15)
Case 2: Assume X− o to be introduced, i.e. X + o = 0, then the row i of (19.12) is
˜X +
i − ˜ X −
i = ˜ Bi +(Zi)oX − o (19.16)
with the two sub-cases:
Case 2a:
˜Bi ≥ 0 ⇒ X +
i = ˜ Bi +(Zi)oX − o
and X +
i only keeps its sign for ˜ Bi +(Zi)oX − o ≥ 0 (19.17)
Case 2b: Two
˜Bi < 0 ⇒ −X −
i = ˜ Bi +(Zi)oX − o
and X −
i only keeps its sign for ˜ Bi +(Zi)oX − o ≤ 0 (19.18)
The condition (19.14) on X + o is only active for (Zi)o > 0, and in (19.15) only for
(Zi)o < 0. Together these two conditions are written ˜ Bi(Zi)o > 0 for case 1.
variable
introduced
sub-cases

Collected
results
278 Copyright c○Pauli Pedersen: Optimal designs ...
In (19.17) and (19.18) we have opposite signs, i.e. totally ˜ Bi(Zi)o < 0 for case 2.
With these conditions programmed we can determine the value of X + o or of X − o and
the corresponding new values of the old basis unknowns. This information is enough
to test the optimality criterion and if not optimal then decide the simplex iteration,
i.e. change of basis.
19.5 Summing up
In this chapter the important results to focus on are:
• A very compact description of standard simplex procedure is presented, (some
basis knowledge of linear programming is assumed).
• Three possibilities for deciding the change of basis are described.
• Still having a convex feasible solution space, we locate the best vertex solution,
also with a non-linear objective.
• From a procedure point of view, the doubling of the number of unknowns is
not necessary to obtain a formulation with non-negative unknowns.

Chapter 20
Linear programming
reformulations with SLP
20.1 Introduction
In a standard formulation, the linear programming problem is in matrix notation
stated Standard
LP
Determine non − negative variables {X} ≥{0} that
Minimize the scalar product {C} T {X}
subjected to the constraints that {X} satisfies the equation
[A]{X} = {B} (20.1)
where the vector {C} of cost coefficients and the matrix [A] as well as the vector {B}
are also assumed to be given.
Most formulations from initial an setup are not presented directly in this standard
form. The unknowns may not be non-negative, the objective and the constraints
may not be linear, the constraint equation may be a combination of equalities and
inequalities, maximizing may be the objective and not minimizing.
By proper reformulations we can obtain the standard formulation (20.1), for nonlinear
problems at least for an iterative step. Using linear programming iteratively
we name it sequential linear programming (SLP). Sequential
The goal of the present chapter is to show these reformulations. However, the
chapter also gives a more general, but very short, introduction to the aspects of linear
programming (LP), presented in matrix notation. The linear programming literature
279
LP, i.e. SLP

IMSL routine
DDLPRS
Maximize
by minimize
280 Copyright c○Pauli Pedersen: Optimal designs ...
is very rich and the present author is most familiar with the earlier books of (Gass
1964) and (Hadley 1964).
Using available software like the IMSL routine DDLPRS, some of the reformulations
are taken care of by the routine. The final part of the chapter is written to show
how input to this routine can be reformulated directly into a starting basis solution,
using notation related to this IMSL routine.
20.2 Actual optimization problem after sign reformulations
To get some uniformity we treat a problem where an objective Φ should be minimized
subjected to less than or equal to inequalities. Thus maximization and greater than
inequalities must be reformulated, which is simply done by proper change of sign
Maximize Φ ⇒ Minimize − Φ and
gj > ¯gj ⇒ −gj < −¯gj (20.2)
Greater than
by less than Without loss of generality we can therefore state the optimal design problem as
Taylor
expansion
Determine design variables hi for i =1, 2,...,I that
Minimize Φ as objective subjected to the constraints
gj ≤ ¯gj for j =1, 2,...,J orinvector notation
{g} ≤{¯g} (20.3)
20.3 From non-linear to sequential linear problem
Most practical optimal design problems are not linear and in general both the objective
Φ and the constraint vector {g} may be non-linear dependent on the vector of
design variables {h}.
A Taylor expansion is assumed to be valid, at least close to an actual design
specified at design iteration n by {h}n. The expansions in terms of sensitivity vector
{dΦ/dh}n for the objective and sensitivity matrix [d{g}/d{h}]n for the constraints
are
Φn+1 = Φn + {dΦ/dh} T n {∆h}n
{g}n+1 = {g}n +[d{g}/d{h}]n{∆h}n ≤{¯g} (20.4)

Linear programming reformulations with SLP 281
Then with Φn a given objective value corresponding to the design {h}n the optimal
redesign problem is stated
Determine redesign variables {∆h}n that
Minimize {dΦ/dh} T n {∆h}n as objective subjected to the constraints
[d{g}/d{h}]n{∆h}n ≤{¯g}−{g}n
(20.5)
After solving this problem the design is updated to {h}n+1 = {h}n + {∆h}n and the Redesign
errors of the Taylor expansion are taken care of in following iterations, assuming that
smaller and smaller move-limits are put on the changes {∆h}.
20.4 From inequalities to equalities
A standard technique to transform inequalities to equalities is to introduce additional
variables, named slack variables. With less than or equal to constraints these slack Slack
variables are by definition non-negative variables. We indicate them by a super-posed
˜, and show them here by the vector {˜g}
[d{g}/d{h}]n{∆h}n + {˜g}n = {¯g}−{g}n
(20.6)
The slack variables have no influence on the objective, normally specified by the
fact that their cost coefficients are zero. If the constraint without slack variables
must be equality, we obtain that by forcing the corresponding slack variable to be
zero. This is done by a relative large cost coefficient, as explained in more detail in
section 20.9.
20.5 Obtaining non-negative variables
formulation
variables
When a minimum allowable redesign vector {∆h}min is specified ({∆h}n ≥{∆h}min),
as is natural with Taylor expansions, then the transformation to non-negative redesign
variables {∆h} + can be obtained by the change of variables Shift of
unknowns
{∆h} + = {∆h}n −{∆h}min ≥{0} or
{∆h}n = {∆h} + + {∆h}min (20.7)
and this transformation must then be introduced into the objective as well as into the
constraints. Having solved the redesign problem in terms of {∆h} + we back-substitute
to obtain the ”physical” redesign {∆h}. Detail is given in section 20.8.
In addition to the minimum redesign vector {∆h}min we also specify a maximum
redesign vector {∆h}max, and therefore have the following bounds on the redesign
variables Bounds on
redesign

Non-negative by
move-limits
Equality by
slack variables
Constraint
types
282 Copyright c○Pauli Pedersen: Optimal designs ...
{∆h}min ≤{∆h} ≤{∆h}max or
{0} ≤{∆h} + ≤{∆h}max −{∆h}min
(20.8)
In order to get more compact notation and without direct reference to our design
variables, we in the following sections shift to the nomenclature used in the description
of the IMSL routine, see (Press et al. 1992). The description begins with the movelimits
as exemplified in (20.9).
20.6 Move-limits and non-negative variables
This section more or less repeats section 20.5 with alternative notation, the notation
of the mentioned IMSL routine. The solution vector is {X} and the move-limits {XL}
and {XU } give the constraints
{XL} ≤{X} ≤{XU } (20.9)
The number of variables is N. To convert to non-negative variables {X + } we define
this vector as
{X + } = {X}−{XL} ⇒ {X} = {X + } + {XL} (20.10)
and the move-limits are then written as the constraints
{0} ≤{X + }≤{XU }−{XL} (20.11)
To convert the right most inequality to equality we introduce the non-negative slack
variables { ˜ X}≥{0} and finally write the move-limit constraints
20.7 Problem constraints
{ ˜ X} + {X + } = {XU }−{XL} ≥{0} (20.12)
The right hand side of a specific constraint Ri is defined by
Ri = {Ai} T {X} =
A(i, j)X(j) (20.13)
For Ri a constraint type is specified by Ii =0, 1, 2, or3, and the constraints are for
the individual types:
j

Linear programming reformulations with SLP 283
Ii = 0 ⇒ Ri = BiL = BiU (20.14)
Ii = 1 ⇒ Ri ≤ BiU (20.15)
Ii = 2 ⇒ Ri ≥ BiL ⇒ −Ri≤−BiL (20.16)
Ii = 3 ⇒ BiL ≤ Ri ≤ BiU, i.e. (20.17)
Ii = 3 ⇒ Ri ≤ BiU and − Ri ≤−BiL (20.18)
where the BiL and the BiU are user specified limits. Double
The constraints of type Ii = 3 we treat as two separated constraints (type Ii =
1 and type Ii = 2). For this we define an extended matrix [ Ā] where the rows
corresponding to the actual Ii = 3 are repeated and an extended vector { ¯ B} with
elements BiL or BiU in agreement with the type Ii
dimension of the matrix [
as specified in (20.18). The
Ā] and the vector { ¯ B} is K
K = M + number of type Ii = 3 (20.19)
where M is the number of specified problem constraints.
The problem constraints can now all together be written as
{ ˜ Y } +[ Ā]{X} = { ¯ B} (20.20)
where we have introduced K further non-negative slack variables by the vector { ˜ Y }≥
{0}. Note that the slack variables corresponding to problem constraints of type Ii =0
constraints
separated
in the final solution must be zero for the solution to be feasible. Further
Changing the constraints (20.20) by introducing the non-negative variables {X + }
from (20.10) we get
{ ˜ Y } +[ Ā]{X+ } = { ¯ B}−[ Ā]{XL} (20.21)
If the right hand side of (20.21) is non-negative, i.e. { ¯ B}−[ Ā]{XL} ≥{0} then we
have a starting solution. However, for some elements in this vector this may not be
the case. Let there be L such element and we then introduce L artificial variables
collected in the vector {Z}. For the corresponding L equations in (20.20) we change
sign and write these L constraints
{Z}−{ ˜ Y − }−[ Ā− ]{X + } = −{ ¯ B − } +[ Ā− ]{XL} (20.22)
slack variables
now with non-negative right hand side. The remaining K − L constraints of (20.21)
is kept unchanged (except for the fewer number of constraints) Artificial
variables
{ ˜ Y + } +[ Ā+ ]{X + } = { ¯ B + }−[ Ā+ ]{XL} (20.23)

Collected
constraints
First basis
solution
284 Copyright c○Pauli Pedersen: Optimal designs ...
20.8 Total LP coefficient matrix for the constraints
Collecting all the unknown vectors { ˜ X}, {Z}, { ˜ Y + }, { ˜ Y − }, {X + } of dimensions N, L, K−
L, L, N, respectively, we write the 3 constraint matrix equations (20.12), (20.22),
(20.23) in one total matrix equation
⎡
⎣
[I] [0] [0] [0] [I]
[0] [I] [0] −[I] −[ Ā− ]
[0] [0] [I] [0] [ Ā+ ]
⎧
⎤
⎪⎨
⎦
⎪⎩
{ ˜ X}
{Z}
{ ˜ Y + }
{ ˜ Y − }
{X + }
⎫
⎧
⎪⎬ ⎨
=
⎩
⎪⎭
{XU }−{XL}
−{ ¯ B − } +[ Ā− ]{XL}
{ ¯ B + }−[ Ā+ ]{XL}
⎫
⎬
⎭ (20.24)
The three rows of equations have the dimensions N, L, K − L, so totally the order of
the total coefficient matrix is (N + K) × (2N + K + L). Note that in a programming
procedure, the original sequence of the slack variables { ˜ Y } need not be resequenced
as shown here by { ˜ Y + }, { ˜ Y − }; the change of signs are just done appropriately.
20.9 The vector of cost coefficients
We see directly from (20.24) that a possible solution to this system of equations is
⎧ ⎫ ⎧
⎫
⎪⎨
⎪⎩
{ ˜ X}
{Z}
{ ˜ Y + }
{ ˜ Y − }
{X + }
⎪⎬ ⎪⎨
=
⎪⎭
⎪⎩
{XU }−{XL}
−{ ¯ B − } +[ Ā− ]{XL}
{ ¯ B + }−[ Ā+ ]{XL}
{0}
{0}
⎪⎬
⎪⎭
(20.25)
Although all elements of the right hand side are non-negative, this is not necessary a
feasible solution (only a basic solution), because the artificial variables in {Z} are not
necessarily all zero and also the slack variables in { ˜ Y + } and/or { ˜ Y − } corresponding
to type Ii = 0 must be zero for the solution to be feasible.
To obtain this as part of the iterative improvement of the solution, the cost factors
for these variables are set to a large value, here indicated by ∞. The objective Φ to
be minimized is then given by
Φ={0} T { ˜ X}+{∞} T {Z}+{∞ or 0} T { ˜ Y + }+{∞ or 0} T { ˜ Y − }+{C} T {X + } (20.26)
where the vector {C} contains the coefficient of the original objective
Φ={C} T {X} = {C} T {X + } + {C} T {XL} (20.27)

Linear programming reformulations with SLP 285
with the last term {C} T {XL} without influence on the optimization, because {XL}
is a user given vector.
20.10 Sequential linear programming
In the presentation of {XL} and {XU } as move-limits it is assumed that the linear LP in SLP
programming solution is used in a sequence of LP to solve a non-linear optimization
problem. The total procedure is then named sequential linear programming (SLP).
For a non-linear problem absolute limits on the resulting final values of the Absolute
unknowns may also be given. Adjusting the move-limits in each linear programming
setup, we can account also for these constraints, without introducing further variables
and constraints.
20.11 Summing up
limits
In this chapter the important results to focus on are: Collected
results
• The initial sections directly focus on optimal design as the actual optimization
problem. Especially the formulation of sequential linear programming to solve
non-linear design problems is important, see also the illustrative figure 2.1 in
chapter 2.
• The chapter assumes some basic knowledge on linear programming from the
reader, and in the description from section 20.6 it is directly related to the often
used IMSL routine DDLPRS.
• The use of move-limits convert to non-negative variables is essential. Furthermore,
move-limits are very useful for user control of more difficult design
optimization.
• The use of slack variables is the tool to convert inequalities to equalities, and
this as a result gives information about non-used bounds.
• The use of artificial variables is the tool to obtain a first basis solution. Furthermore,
if the artificial variables are not zero after the final iteration, then
no solution exists. Thus the artificial-variables are also the tool that give information
about existence of solution.
• The sub-matrix presentation (20.24) is a valuable tool to get an overview of the
problem formulation.

286 Copyright c○Pauli Pedersen: Optimal designs ...

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Index
1D-design description, 125
2D-bulk modulus, 216
2D-design description, 126
2nd order polynomial, 257
3D, 246
3D analysis, 186
3D-bulk modulus, 216
3D-isotropic modulus relations, 217
A thesis, 187
Absolute limits, 285
Absolute side limits, 21
Active constraints or inactive ?, 22
active set strategy, 100
Active volume, 85
Adaptation, 187
Additional eigenvalue problem, 271
Adjoint problem, 264
Advanced FEM-analysis, 25
Aligned larger stress/strain, 179
Aligned orientations, 187
Aligned stresses and strains, 171
alignment
principal material direction, 254
principal strain direction, 254
principal stress direction, 254
Alignments, 254
Also best strength, 232
Also non-linear, 24
Alternative design variables, 83
Alternative materials, 206
Always sensitivities, 27
Analog in 3D, 214
analytical optimal design
explicit, 81
implicit, 81
analytical sensitivity analysis, 125
Approximately optimal hole, 72
artificial
297
cost, 284
variable, 283
Artificial variables, 283
Assumption for cost of supports, 252
Assumptions, 246
Available functions, 238
Axial load or bending, 57
axial stress flow, 57
Axisymmetric modelling, 138
bar
force, 245
long, medium and short, 59
short column, 250
slender column, 250
tensile, 249
zero string, 250
Bars and joints, 31
Base cell homogenization, 209
Basic assumption, 225, 234
basis
change, 275
Basis separation, 247
Basis solution, 247, 274
beam
active size constraint, 85
approximate gradient, 90
Bernoulli-Euler theory, 90
eigenfrequency, 85
exact gradient, 90
fixed-fixed, 93
long, medium and short, 65
minimum compliance, 82
second eigenfrequency, 90
simple beam theory, 86
simply supported, 93
slenderness, 90
statically indetermined, 85
stress constraint, 93

298 Index
stress energy, 82
Timoshenko theory, 89
beam-bar
long, medium and short, 62
beam/bridge truss
24 joints, 41
cantilever part, 42
hanging part, 42
parameter studies, 42
possible height, 41
possible support, 41
Beams and frames, 15
Beck’s column, 270
bending stiffness
contracted vector, 207
Bending stiffnesses, 207
Bernoulli-Euler beam theory, 98
Best vertex solution, 275
Better force flow, 46
black and white
design, 162
penalize, 162
bone
non-isotropic, 187
a thesis, 187
adaptation, 187
aligned orientations, 187
analysis, 186
constitutive constraints, 194
constitutive increments, 194
constitutive vector, 189
energy density, 186, 189
formulation in time, 186
frame model, 190
homogenization, 190
identification, 186
implants, 187
incremental formulation, 192
inverse homogenization, 191
load estimation, 189
material, 185
memory function, 188
micro-structure optimization, 190
note, 196
optimal re-modelling, 192
optimality criterion, 192
optimization formulation, 191
phenomenological re-modelling, 186
proximal femur, 188
rate function example, 194
rate of constitutive norm, 193
sensitivity of energy, 191
shape and size, 186
special case, 192
structure, 185
bound formulation, 130
boundary
displacement, 159
load, 159
post-process smoothing, 169
boundary condition, 261
boundary shapes, 129
Bounds on redesign, 281
bridge
nine problems, 65
bridge truss
16 joints, 39, 46
positions for joints, 46
Buckling load, 118
bulk modulus
Hashin-Shtrikman bound, 175
macroscopic, 175
cantilever
beam, 83
elementary load cases, 83
non-rectangular, 75
statically determined, 83
Cantilever + hanging part, 42
cantilever truss
17 joints, 38
7 to 12 joints, 46
9 joints, 33
Cases of no influence, 178
cavity
3D inclusion, 151
ellipsoid, 151
Chain rule of differentiation, 93
change basis, 275
Change in equilibrium, 275
Changed designs with forced displacements, 180
Changed with shape design, 182
Changing load direction, 35
Changing type of cross-section, 36
checkerboard, 162
circular cross-section, 31
Circular hole, 145
circulatory
force, 263
Citation from Frederiksen, 204

Index 299
Classes of optimization procedures, 27
Classical problem, 87
classification
behaviour, 263
dynamic system, 262
material, 112
names and concepts, 17
coefficient
cost, 279
Collected constraints, 284
Collected results, 218, 224, 228, 240, 244, 252,
260, 271, 278, 285
Columns at beam boundaries, 93
Combined bending and axial forces, 62
Combined shape design, 44
complementary energy, 225
complex
eigenvalue, 264
functional, 261
Complex functionals, 265
Compliance, 21
compliance
matrix, 219
isotropic, 221
non-dimensional, 220
non-orthotropic, 220
orthotropic, 220
measured by umean, 169
Compliance power law description, 220
Complicated laboratory tests, 198
composite, 107
compression bar, 31
Compressive - stability, 31
Concave cost, 249
concave function, 273
Concave transition, 251
Concentration factor of 2, 146
condition
optimality, 230, 249
proportional gradient, 172
stationarity, 255
Condition for proportionality, 222
Conservative systems, 262
Constant energy density, 131, 234
constitutive
coupling component, 215
eigenvalues, 213
individual parameter, 241
matrix, 211, 212, 223
non-dimensional, 242
normal component, 215
parameter, 257
alternative description, 211
positive semi-definite matrix, 241
rotational transformation, 212
shear component, 215
tensor, 211
vector, 212
Constitutive constraints, 194
Constitutive eigenvalues, 213
Constitutive Frobenius (length), 212
Constitutive increments, 194
Constitutive matrix, 109, 171
constitutive matrix
practical parameters, 110
constitutive parameter
transformation, 110
Constitutive trace, 212
Constitutive vector, 109, 189
constraint
displacement, 98
eigenfrequency, 87
equation, 274
geometrical, 159
manufacturing, 127
multi-physic, 160, 184
parametrization, 159
several eigenfrequencies, 87
side, 200
strength, 222
type, 282
vector, 280
Constraint on definiteness, 200
Constraint types, 282
Constraints, 20
Continuous - discrete, 19
Continuous problem, 269
contracted
constitutive matrix, 214
second order strains, 214
strain matrix, 214
stress matrix, 214
control of design process, 22
convergence
design variable, 259
objective, 259
Convergence tests, 20
Converted to non-constrained, 230
Convex, 23
Convex feasible space, 249

300 Index
corresponding displacement, 225
cost
change in total, 248
coefficient, 279
coefficient vector, 274
large coefficient, 281
support, 252
total, 247
Cost functions, 247
Cost non-increasing, 248
crack tip
allowable domain, 157
external load, 154
hole size, 154
material non-isotropy, 155
power law non-linearity, 155
stress release, 151
cross-section
constant, 82
hollow circular, 36
moment of inertia, 82
rectangular, 83
solid, 36
cross-sectional
parameter, 250
curvature
discontinuous, 135
Cylindrical pipe data, 33
damage
evolution, 194
non-bone materials, 194
damping
destabilizing, 261
DDLPRS
IMSL routine, 280
Deciding neighbour solution, 275
Definition of √ 2 contraction, 212
Deformation mode parameter, 117
degrees of freedom, 57
Derivation from elastic potential, 223
Derivative of cost, 248
design
advanced material, 159
code, 251
global parametrization, 123
iteration, 280
local parameter, 226
optimal strength, 124
thickness and orientation, 123
thickness distribution only, 122
design constraint
implicit, 21
life time, 21
list, 20
satisfied, 22
stability, 21
stiffness, 21
strength, 21
vibration, 21
violated, 22
design cycling, 78
design domain
graphical illustration, 115
lamination parameters, 113
Design for stiffness and strength, 235
Design functions, 127
Design independence, 122
Design independent loads, 233
design influence
finite element modelling, 163
initial design, 163
load modelling, 163
lower bound, 163
move-limit, 163
neglected modelling, 163
optimization procedure, 163
penalization approach, 163
smoothing (filtering), 163
total density, 163
design limit
absolute side, 21
move-limit, 22
design objective
convergence, 20
cost, merit, goal, 19
existence of solution, 20
local and global, 20
minimum and maximum, 19
optimized not optimal, 20
Design optimization, 229
Design problem, 33, 237, 242
design space
boundary, 23
convex, 23
direction, 23
exterior, 23
geometrical interpretation, 23
interior, 23
list of concepts, 23

Index 301
step size, 23
design variable
continuous, 19
convergence, 20
discrete, 19
distributed, 19
hierarchical, 19
linking, 29
local, global, 28
parametrization, 19
shape, 18
size, 18
topology, 18
vector, 280
design-independent
external load, 226
Designs independent of Poisson’s ratio, 174
Detail of proof, 234
Deviatoric stresses, 221
Different assumptions, 33
Different load cases, 175
Different objectives, 129, 254
Different objectives, but the same design, 159
Different sensitivities, 264
Direct conclusions, 242
direction
cosine, 246
principal strain/stress, 243
Discontinuous curvature, 135
Discussion of results, 169
displacement constraints, 46
displacement pattern, 117
Displacements and stiffness, 21
dissipative
force, 263
mutual energy, 266
Distinct eigenvalues, 270
Distributed parameter, 19
Divergence - quasi-static, 263
domain
non-shape, 235
shape, 235
dome frame, 100
Double constraints separated, 283
Down to 15 % concentration, 135
Down to 8 % concentration, 138
dyadic product, 214
dynamic behaviour, 261
Early references, 131, 245
Eccentricity, 92
effective
strain, 219, 242
stress, 219
Eigenfrequencies, 21, 118
eigenfrequency
calculated, 197, 199
measured, 197
thick plate formulation, 198
eigenfunction, 269
Eigenmodes identification, 270
eigenvalue
Sturm sequence check, 86
bimodal case
complex, 264
constitutive, 213
continuous problem, 269
double with one eigenvector, 271
inverse iteration, 86
multiple, 270
only one non-zero, 161
shift, 86
sub-space iteration, 86
Eigenvalue sensitivities, 203
eigenvector
left, adjoint, 264
normalization, 203
right, physical, 264
variation, 265
Eigenvector description, 214
eigenvectors
mutual orthogonal, 270
elastic energy
sensitivity analysis, 115
total, 55
elastic mutual energy, 266
elasticity
non-isotropic, 160, 219
non-linear, 219
powers n or p, 219
orthotropic, 219
power law non-linear, 160, 233, 241
Element energies, 254
Element level FEM, 267
Elementary strain cases, 209, 214
elementary strain state, 214
Eliminated variable from basis, 275
ellipsoidal shape, 151
Elliptical are not optimal, 146
Elongated better designs, 157

302 Index
Email, 11
energy concentration factor, 57
Energy densities, 223
Energy density, 186, 189
energy density
elastic, 55
initial mean, 57
maximum, 57
mean, 57
strain, 179, 182, 213, 223
stress, 179, 213, 223
total, 213
uniform, 55
Energy density RELATION, 223
Energy equilibrium, 225
Energy principles, 15
energy theorems of mechanics, 24
engineering parameters, 112
enlarged
flexibility, 80
stiffness, 80
Equal cases for small holes, 175
Equality by slack variables, 282
equilibrium
energy, 225
Equilibrium at joints, 246
Essential assumptions, 146
Estimated multiplier, 239
estimation, 197
applied load, 197
deep drawing, 197
hardness test, 197
material parameter, 197
orthotropic material, 197
stiffness model, 197
estimator
weighted least-square, 199
Exact gradients, 89
Example, 257
example
12.5% total relative density, 164
25% total relative density, 164
50% total relative density, 164
80% total relative density, 164
analytical solution, 170
Beck’s column, 269
boundary displacement, 175
boundary stress, 175
cantilever beam - Galerkin, 268
Hauger’s column, 269
identification - estimation
carbon-epoxy, 204
glass-epoxy, 203
orthotropic aluminum, 203
temperature dependence, 204
Leipholz’s column, 269
material with a single hole, 172
mixed problem, 175
optimal constitutive matrix, 160
quarter cell, 163
shape influence from boundary condition,
174
shape influence from material non-linearity,
180
shape influence from Poisson’s ratio, 178
shape influence from volume constraint, 173
two parameter cell model, 170
ultimate optimal field, 161
with penalizing, 163
without penalizing, 163
Example of rate function, 194
Example values, 259
Examples, 18
Examples in chapter 4, 161
Excitation source, 201
Existence, 20
Existence of solution, 259
expansion
adjoint eigenfunction, 268
physical eigenfunction, 268
expansion function
global, 267
local, 267
Expansion functions, 267
Experimental - numerical, 197
Experimental errors, 206
experimental setup
excitation source, 200
frequency analyzer, 200
identification program, 200
power amplifier, 200
response detector
accelerometer, 200
laser vibrometer, 200
microphone, 200
test specimen, 200
transient recorder, 200
Explicit optimal beam design, 84
Extended problems, 87
Extensions: stability, supports, 245

Index 303
external damping, 268
external load
design-independent, 226
work, 55
external mutual energy , 266
external potential, 225
extremum
local, 253
multiplicity, 255
Extremum relations, 226
fatigue damage repair, 151
FE model, 132
Feasibility, 23
Feasible basis solution, 276
Feasible solution, 274
feasible space
convex, 249
fillet
2D in tension, 131
3D bending, 138
3D tension, 138, 141
3D torsion, 138, 141
axisymmetric, 138
bending stress fields, 141
circular connection, 132
geometrical constraint, 131
length, 135
multi-parameter, 135
super-circular connection, 135
tension stress fields, 141
Finite cost increment, 249
Finite degrees of freedom, 261
Finite design regions, 81
finite element, 146, 238, 266
finite element modelling, 55
Finite element models, 57
First basis solution, 284
first ply failure, 124
fixed strain, 226, 254
fixed stress field, 235
flutter
frequency, 266
load, 263
Flutter - dynamic, 263
Flutter frequency sensitivity, 266
Flutter load sensitivity, 266
Focus on stress energy, 82
follower force, 268
For model by the FEM, 238
force
bar, 245
circulatory, 263
compressive, 245
dissipative, 263
equilibrium, 246
follower, 268
gyroscopic, 262
non-negative, 245
non-potential, 263
tensile, 245
Formulation in time, 186
Fortran subroutine, 273
foundation problem, 75
Fracture and creep, 21
frame
3D, 98
dome, 100
linking, 98
mobile crane, 103
optimal joint position, 98
portal, 81, 92
self-weight, 98
shape optimization, 82
side constraint, 99
total mass, 99
frame from beams, 81
Frame model with cubic symmetry, 190
Free material, 241
free material design, 160
frequency measure, 264
Frobenius norm, 207, 241
full pivoting, 273
Fully stressed, 22, 34
fully stressed, 31, 245
fully stressed design, 22
functional
complex, 261
mutual energy, 261
Further designs, 98
Further slack variables, 283
Further tools, 202
gain factor, 57
Galerkin
method, 267
Galerkin discretization, 268
gamma-function, 237
General data for examples, 32
General incompressible, 221

304 Index
General knowledge, 14, 129, 232
General non-isotropic, 220
General optimization, 229
General results, 219
geometrical constraint, 235
Given volume, 56
global
maximum, 253
minimum, 253
global design parameters, 125
curve, 125
orthogonal functions, 126
surface, 125
Global optimal, 274
Glossary, 14, 17
Good experience, 235
Gradient function, 268
gradient function, 264
Gradient information non-sufficient, 253
gray design
intermediate density, 169
Greater than by less than, 280
gyroscopic
force, 262
Half a model by symmetry, 42
Hashin-Shtrikman bound
bulk modulus, 175
Hauger’s column, 270
heuristic approach
successive iteration, 239
Heuristic methods, 259
heuristic procedure, 161
Higher order plate theory, 202
hole
approximately optimal, 72
biaxial load, 144
biaxial load , 72
classic solution, 144
limited domain, 144
orthotropic material, 148
relative hole size, 146
strain localization, 149
stress releasing, 149
Hole designs with resulting bulk modulus, 174
hollow circular cross-section, 36
homogeneous
energy relation, 231
mass relation, 231
system, 262
Homogeneous mass (volume) dependence, 231
Homogeneous non-homogeneous, 75
homogenization
constitutive parameter, 209
inverse, 209
material, 209
prescribed material parameter, 209
Homogenization analysis, 190
hydrostatic pressure
deviatoric stress, 221
incompressibility, 221
Identification, 13, 186
identification, 197
formulation, 199
micro-mechanical structure, 210
identification - estimation
bone material, 206
ceramic, 206
environmental influence, 206
errors
experimental, 204
numerical modelling, 204
physical modelling, 204
plate defects, 204
sandwich combination, 206
identification approach
eigenvalue analysis, 198
error estimation, 198
experiment, 198
optimization, 198
sensitivity analysis, 198
Illustrative example, 153
imaginary part, 265
Implants, 187
Implicit behaviour, 21
Importance of supports, 42
Important references, 81
Important RESULT, 226
IMSL
move-limit, 282
routine, 282
IMSL routine DDLPRS, 280
In-plane and out-of-plane, 113
incompressibility
hydrostatic pressure, 221
orthotropic, 221
Incremental formulation, 192
Individual bar design, 245
inequality

Index 305
greater than, 280
less than, 280
inequality to equality, 281
influence
material non-linearity, 159
material Poisson’s ratio, 159
maximum thickness, 78
non-isotropic, 79
non-linear, 79
Poisson’s ratio, 78
Influence from finite size, 146
Influence of gradients, 90
Influence of number of joints, 46
Influence of stiffness, 132
Influence on designs, 178
Influence studies, 172
Initial designs, 94
instability
divergent, 263
dynamic, 263
flutter, 263
initiation, 265
static, 263
intermediate density
gray design, 169
internal damping
Kelvin-Voigt, 268
invariant
determinant, 212
matrix, 241
matrix length, 241
trace, 212
vector length, 212
inverse
homogenization, 209
Inverse homogenization, 13, 191
inverse iteration, 86
Inverse material problems, 209
inverse problem, 197
bending stiffness, 207
laminate, 197, 207
material, 197
Inverse problems, 13, 24
inverse transformation
orthonormal, 213
Isotropic, 221
Isotropic 2 parameters, 113
iteration
monotonic convergence, 78
recursive formula, 75
relaxation, 78
Iteration strategy, 208
iterative improvement, 284
joints in trusses, 31
joints positions
optimization, 44
Kelvin-Voigt internal damping, 268
kinematically admissible, 226
kinetic mutual energy, 266
Kinetic, dissipative, elastic and
external potentials, 266
knee problem, 72
Known moment distribution, 83
Kronecker delta, 214
laboratory tests
eigenfrequency, 198
strain gauge, 198
Lagrange
function, 230
multiplier, 89, 230, 239
lamina, 107
laminate
analysis, 107
balanced, 116
bending stiffness, 199
identification, 124
material stiffness, 199
membrane stiffness, 199
ply layer lamina, 107
rotational transformation, 108
specially orthotropic, 116
stiffness, 113
Laminate design, 207
Laminate inverse problems, 197
Laminate parametrization for design, 123
Lamination parameters, 113
Large improvements, 120
layer, 107
Leipholz’s column, 270
level of non-isotropy, 112
limit
lower bound, 56
upper bound, 57
Linear dependence, 250, 251
Linear functions, 113
linear programming, 245
sequential, i.e. SLP, 279

306 Index
standard form, 274, 279
linear programming problem, 32
Linear systems, 262
Linking and no-linking, 104
linking design variables, 98
linking of design variables, 29
List of practical problems, 162
load
condition, 56
critical, 265
design dependent, 21
design independent, 21
distribution, 264
flutter, 265
level, 264
one case, 55
Load cases, 21
Load direction, 144
Load estimation, 189
Loads and boundary conditions, 21
local
design parameter, 226
determination of sensitivity, 226
Local - Global, 20
Local design parameter, 227
Local/global parametrization, 28
Localized determined sensitivities, 241
Localized energy change, 233
Localized volume change, 232
LP in SLP, 285
LP modifications, 273
Main contents of chapter, 81
Major importance, 135
Manufacturing constraints, 127
Many local extremum, 253
Many possibilities, 211
mass density, 82
material
base cell, 209
constitutive component, 160
effective property, 209
homogenization, 209
identification, 209
micro-structure, 209
non-isotropic, 253
orientational result, 171
orthotropic, 253
Material and structure, 185
Material as a structure, 159
material design, 107
free material, 241
negative Poisson’s ratio, 210
solution priority, 210
thermoelastic, 210
ultimate design, 241
Material models, 13
Material parameter, 178
Material, yes or no ?, 162
mathematical programming, 81, 124, 274
Matrices, 211
matrix
coefficient, 274
compliance, 219
inverse, 223
constitutive, 211
non-dimensional, 223
damping, 262
direction cosine’s, 246
element level, 267
Frobenius norm, 212
Hill strength, 222
invariant, 241
inverse, 274
length, 212
mass, 262
orthotropic constitutive, 170
positive definite, 212
projection, 221
regular, 274
sensitivity, 280
skew-symmetric, 262
stiffness, 262
symmetric and positive definite, 262
system, 264
system level, 267
total coefficient, 276
total equation, 284
total stiffness, 180
trace norm, 212
von Mises strength, 221
Matrix of Hill, 222
Matrix of von Mises, 222
Maximize by minimize, 280
maximum
stiffness, 253
strain energy density, 241
Maximum displacement, 118
mechanism in truss, 251
Medium slender columns, 96

Index 307
Member classes, 250
Memory function, 188
method
Galerkin, 267
heuristic, 259
Ritz, 267
subspace iteration, 202
weighted residual, 267
micro-structure
finite element analysis, 209
homogeneous, 209
inhomogeneous, 209
Micro-structure optimization, 190
minimize the total mass, 33
minimum
compliance, 234, 253
cost, 245
energy concentration, 239
gradient, 275
maximum strain energy density, 234
size, 245
total strain energy, 241
Minimum - Maximum, 19
Minimum of maximum, 130
Minimum stress concentration, 146
minimum-maximum formulation, 130
mode parameter, 117, 118
model/plane strain, 179
model/plane stress, 179
Modelling accuracy, 78
Modelling errors, 206
Modelling issues, 130
Moderately non-isotropic, 204
Modification functions, 239
Modified optimality criterion, 162
Modulus power law description, 223
monotonous behaviour, 248
Mostly numerical, 24
move-limit, 98
side constraint, 99
smaller and smaller, 281
Move-limits, 22, 27, 200
Moving in design space, 23
moving load, 48
multi-purpose plane truss, 46
multi-purpose space truss, 50
Multi-purpose trusses, 31
multiple
loads, 124
plies, 124
strength constraints, 124
multiple eigenvalue
sensitivity analysis, 270
Multiple eigenvalues, 24
Multiple load case, 101
Multiple load cases, 91, 124
multiple load cases, 46
Multiple plies, 124
Multiple solutions, 210
Multiple strength constraints, 124
mutual energy
dissipative, 266
elastic, 266
external, 266
kinetic, 266
specific, 264
variation, 265
Mutual potentials, 261
necessary condition, 229
a single constraint, 230
non-constrained, 229
positive definite, 242
proportional gradient, 243
proportionality, 230
necessary, sufficient, 24
Neighbouring redesign, 130
New design problem, 243
Nine problems, 65
No cost assumed, 251
No energy concentration, 120
nodal
displacement, 180
load, 180
Non-bone materials, 194
Non-changed loads, 226
non-conservative
load, 261
Non-conservative systems, 263
Non-dimensional parameters, 200
non-increasing function, 248
Non-intuitive behaviour, 261
Non-isotropic, 187
non-isotropic
material, 253
Non-isotropic parameters, 255
non-isotropy
degree, 253
level, 200
relative shear stiffness, 200

308 Index
Non-isotropy influences, 155
Non-linear - less influence, 157
Non-linear elasticity, 122
Non-linear, non-isotropic, 219
Non-negative by move-limits, 282
non-negative unknown
in basis, 274
out of basis, 274
non-potential
force, 263
Non-rectangular cantilever, 75
non-selfadjoint
problem, 261
Non-slender columns, 97
Non-trivial optimal orientation, 118
norm
Frobenius, 160, 242
trace, 160
Normalized mass matrix, 262
Not fully stressed, 48
Not optimal design, 197
Not proper references, 14
Not statically determined, 46
Notation, 219
Note, 196
Numerical errors, 206
numerical procedure, 75
Objective, 125, 153, 199, 207
objective, 161, 273, 280
concave, 275
concave function, 249
maximum bulk modulus, 174
non-linear, 275
stiffest design, 159
strongest design, 159
Objective Φmn, 117
Objective derivative, 275
Only axial forces, 59
Only bending forces, 65
Only deterministic, 24
Only four stationary solutions, 116
Only linear elasticity, 215
Only minimum, 130
Only single load case, 55
optimal
orientation, 253
orientation in plate, 118
Optimal 2D-modulus matrix, 161
Optimal 3D-modulus matrix, 160
optimal design
books, 13
Optimal designs, 13
Optimal energy ratio, 84
optimal material
aligned, 242
degenerate, 244
no shear stiffness, 242
non-zero eigenvalue, 243
orthotropic, 242
Optimal modulus matrix, 243
Optimal re-modelling, 192
Optimal redesign, 25
optimal redesign
analysis, 25
sensitivity analysis, 25
Optimal shape independent of load and size, 155
Optimal stress release, 151
optimal topology, 33
optimal truss
statically determinate, 245
Optimality condition, 238, 243
optimality condition, 230, 249
Optimality criterion, 82, 192
optimality criterion, 254
density distribution, 161
uniform energy density, 161
optimality criterion test, 89
optimization
material orientation, 230
non-linear by LP, 285
Optimization formulation, 191
optimization methods
list of formulations, 24
list of procedures, 27
Optimization problem, 208
optimize
stiffness, 229
strength, 229
Optimize bulk modulus, 172
Optimized design, 20
orientation
in strain parameter, 256
in stress parameter, 256
optimal, 253
orientational field, 254
Orthogonal [R] matrix, 109
Orthogonal [T ] matrix, 109
orthogonal matrix
fourth order tensor, 109

Index 309
second order tensor, 109
Orthonormal rotational transformations, 212
Orthotropic, 220
orthotropic
condition, 111
directions, 111
material, 253
Orthotropic analytical solutions, 116
Orthotropic engineering 4 parameters, 112
Orthotropic material, 170
Orthotropic materials, 111
Other classifications, 18
Overall parameters, 132
Parametrization, 19
parametrization
boundary shape, 235
super-ellipse, 235
penalize
stationarity condition, 162
Penalized and not-penalized, 164
Phenomenological re-modelling, 186
plate
buckling load, 118
eigenfrequencies, 117
eigenfrequency analysis, 202
higher order shear deformation, 202
mode parameter, 117
Rayleigh-Ritz method, 202
sensitivity analysis, 202
simply supported, 116
Plate errors, 206
Plies - layers - laminas - prepregs, 107
ply, 107
Points or domains, 13
Poisson’s ratio
global redistribution, 178
independence, 174
local change, 178
major, 200
scaling factor, 178
zero, 161, 244
portal frame, 92
Positive definite, 242, 257
positive definite
matrix, 212
post-process
boundary smoothing, 169
potential
derivative, 226
elastic complementary energy, 225
elastic strain energy, 225
elastic stress energy, 225
external, 225
stationary total, 226
total, 226
Potential definitions, 225
Potential relations, 226
Practical conclusions, 118
Practical definitions, 110
practical problem
checkerboard, 162
local optimal, 162
mesh dependence, 162
non-unique solution, 162
slow convergence, 162
stopping criterion dependence, 162
violate model, 162
prepreg, 107
Principal directions, 257
Principal strains only, 242
Problem of analysis, 85
procedure
simplex, 273
Procedure and examples in chapter 4, 161
Projection matrix, 221
proportional change, 182
proportional gradient, 162
Proportional gradients, 230
proportionality relation
elastic energy, 225
energy density, 223
Proximal femur, 188
pseudo-load, 93
psi-function, 238
Quarter cell example, 163
Rate of constitutive norm, 193
Rayleigh-Ritz or FEM, 202
reaction
support, 246
truss member, 247
recursive formula, 75
recursive iteration
optimality criterion, 197
Recursive procedures, 253
Redesign, 27
redesign
procedure, 208

310 Index
limit, 208
strategy, 208
Redesign formulation, 281
Redesign iterations, 75
redistribution
displacements, strains and stresses, 178
Reference coordinates, 211
Reference stress, 92
Reference values, 57
References to results, 210
reformulation
LP, 279
Regular sub-matrix, 274
relative angle, 108
Relative hole area or density, 237
Relative low or high shear stiffness, 256
relative shear stiffness, 112
Relative strains, 257
Repeated sub-structures, 39
Research groups, 198
response analysis
continuous, 261
discrete, 261
Response detector, 201
result
high energy density, 173
insensitive compliance, 173
load dependence, 173
non-linearity dependence, 173
Poisson’s ratio dependence, 173
sensitive concentration, 173
stationary compliance, 173
Results and conclusions, 173
Results in strains, 256
Results in stresses, 256
Results with different assumptions, 48
Robustness, convergence, 259
rotational transformation
constitutive relation, 109, 212
orthonormal, 212
strain, 108, 212
stress, 108, 212
Same optimal topology, 41
Satisfied or violated ?, 22
secant formulation, 219, 242
self-weight, 33
Self-weight included, 34
semi-analytical, 93
Semi-analytical sensitivities, 93
sensitivity
constant constitutive matrix, 227
constant volume, 227
distinct eigenfrequency, 202
flutter load, 266
local determination, 226
matrix, 280
multiple eigenfrequency, 203
to constitutive parameter, 227
to material orientation, 227
vector, 280
with respect to design, 265
with respect to load level, 265
Sensitivity analysis, 23, 178, 180, 182
sensitivity analysis, 93, 261, 264
pseudo-load, 93
effectiveness, 98
localized, 241
multiple eigenvalue, 270
robustness, 98
semi-analytical, 93
Sensitivity for simple beams, 86
Sensitivity functionals, 266
Sensitivity of energy, 191
Sensitivity result, 86
Separated constraints, 274
Separated objective, 274
separation
spatial problem, 264
sequential linear programming, 91
sequential linear programming (SLP), 285
Sequential LP, i.e. SLP, 279
sequential quadratic programming, 98
Severe concentration, 132
shaft
shoulder fillet, 138
Shape, 18
shape
boundary curve, 18
boundary surface, 18
crack tip, 151
curvature, 18
effective stress, 130
energy density, 130
heuristic approach, 129
influence from boundary condition, 172
influence from material non-linearity, 172
influence from Poisson’s ratio, 172
influence from volume constraint, 172
joint positions, 18

Index 311
length, 18
material boundary, 159
maximum bulk modulus, 151
minimum compliance, 129
minimum stress concentration, 129
neighbouring shape, 130
redesign, 130
single hole boundary, 172
smooth modelling, 130
stiffest design, 129
strongest design, 129
Shape and size optimization, 186
shape optimization
stiffness, 232
shear modulus, 200
shear stiffness
high, 112
low, 112
shift of eigenvalues, 86
Shift of unknowns, 281
short column, 251
side constraint
move-limit, 99
Side constraints, 56, 99
Simple example, 120
Simple iterations, 85
Simple laboratory tests, 198
Simple parametrization, 153
Simple unidirectional, 31
simplex
modified procedure, 249, 276
procedure, 249, 273
Single load case, 31, 100
Single value, 19
Size, 18
size
area of bar, 18
mass density, 18
material density, 159
material orientation, 159
orientation, 18
thickness of beam, plate, shell , 18
volume density, 18
Size and shape, 229
Size and shape with linking, 98
size optimization
stiffness, 231
strength, 232
Size, shape and topology, 32, 159
Slack variables, 281
slender column, 250
Slender columns, 95
Slender or short, 57, 250
solid cross-section, 36
solution
basis feasible, 274
existence, 259
neighbour feasible, 274
optimal, 274
starting basis, 280
vertex, 275
Solution equation, 255
space truss, 44
Special case, 192
Specific incompressible, 221
Specified linking, 100
Square root dependence, 250
stability
bar in a truss, 31
Stability (buckling), 21
stability measure, 264
Stabilizing or destabilizing, 265
Standard LP, 279
statically determinate
optimal truss, 245
Statically determinate solution, 246
Statically determined, 32
statically determined, 31
stationarity
condition, 255
energy functional, 261
Stationarity - extremum, 24
Stationary energy density, 254
Stationary objective, 230
Stationary systems, 262
statistics
uncertainties, 204
steepest decent
orientation, 253
Stiffest design, 231
stiffest design
shape, 234
size, 231
Stiffness and strength, 55
Stiffness constraints, 100
stiffness global measure, 55
Stiffnesses: membrane coupling bending, 113
Stopping condition, 29
strain
direction of larger principal, 253

312 Index
effective, 219, 242
elementary state, 214
mean energy density, 254
principal, 242
ratio, 171
related parameter, 255
rotational transformation, 212
tensor, 211
vector, 212
Strain concentration, 149
strain energy, 225
per plate area, 115
uniform density, 231
strain energy density, 213
θ function, 255
integrated, 223
Strain energy per area, 115
Strain or stress directions, 260
Strain ordering from stresses, 259
strategy
optimization, 259
strength, 31
constraint, 222
Hill, 222
von Mises, 222, 232
Strength constraints, 99
strength local measure, 55
Strength or stiffness, 238
stress
allowable in compression, 245
allowable in tension, 245
bound for optimal ratio, 257
characteristic, 253, 259
deviatoric, 221
direction of larger principal, 253
effective, 219
energy-consistent, 219
Hill, 222
von Mises, 219
limit of proportionality, 250
normal term, 220
principal ratio, 253
ratio, 171
rotational transformation, 212
shear term, 220
tension, bending, shear, torsion, 98
tensor, 211
unidirectional, 179
vector, 212
von Mises, 92
stress concentration
analytical, 145
factor, 169
measured by umax, 169
stress energy, 225
complementary formulation, 234
stress energy density, 213
θ function, 256
integrated, 223
Stress ordering from strains, 258
stress ratio
value, 259
Stress release, 149
Stress/strain parameters, 179
Stress/strain vectors, 108
Stresses and strength, 21
Strong influence from orthotropy, 148
strongest design
shape, 235
size, 232
Strongly non-isotropic, 204
Structural models, 13
Structure or material, 107
Structures of 1D-elements, 81
Structures or materials, 129
Sturm sequence check, 86
Sub-problems, 25
sub-space iteration, 86
Sub-vectors, sub-matrices, 220
Subspace iterations, 202
successive iteration
heuristic approach, 239
sufficient, necessary, 24
super-ellipse
power, 172
super-elliptic shape, 236
Supports as members, 247
surface traction, 225
Symmetry or non-symmetry, 91
system
conservative, gyroscopic, 262
conservative, non-gyroscopic, 262
homogeneous, 262
inhomogeneous, 262
instationary, 262
non-conservative, circulatory, 263
non-conservative, dissipative, 263
System matrix, 264
Table comments, 123

Index 313
tangential stress, 144
constant, 145
Taylor expansion, 280
tensile bar, 31, 250
tensor
constitutive, 211
strain, 211
stress, 211
Tensors, 211
Test of solutions, 89
Test specimens, 201
The θ function for strain energy density, 255
The θ function for stress energy density, 256
The match problem, 254
theorem
stiffest shape design, 235
strongest shape design, 235
truss, 246
Theoretical reference, 59
Theory and tools, 13
Theory in chapter 12, 161
Theory in chapter 15, 160
thermal expansion
negative, 210
Thicknesses or densities, 56
Three figures, 65
Timoshenko beam theory, 93, 98
Topology, 18
topology
a bar or no bar, 18
a beam or no beam, 18
a hole or no hole, 18
material, 159
truss, 245
topology optimization, 162
Total mutual potential, 265
Total potential, 226
tower space truss, 44
trace
invariant, 212
transition
slender - short, 251
transmission of external forces, 54
truss
positions for joints, 44
topology, 245
truss from bars, 81
Trusses, 14
Two level iterative optimization, 44
Two or only one parameter(s), 236
Two sub-cases, 277
ultimately optimal continua, 78
uncertainties
statistics, 204
Unchanged with size design, 182
Unchanged with stress loads, 179
Uniaxial stress state, 145
unidirectional state, 31
Uniform energy density, 14
uniform energy density, 55
uniform strain energy density, 231
Unknown support reactions, 246
Using virtual work principle, 226
Value of multiplier, 89
Value of variable introduced, 277
variable
artificial, 283
change, 281
non-negative, 273, 281
positive and negative, 276
slack, 281
variational analysis
general, 264
vector
acceleration, 262
constraint, 280
cost coefficient, 274, 284
design variable, 280
displacement, 262
element level, 267
force, 262
invariant length, 212
maximum redesign, 281
minimum redesign, 281
non-negative unknown, 274
redesign bound, 281
redesign variable, 281
sensitivity, 280
strain, 212
stress, 212
system level, 267
velocity, 262
Vectors, 211
vertex
solution, 275
vibration
frequency, 261
simple, free, undamped, 262

314 Index
Vibrations, 15
virtual work principle, 226
volume
relative density, 56
volume force, 225
von Mises stress, 92
Weakly non-isotropic, 203
weighted trigonometric integral, 113
Well defined instability criterion, 265
well-ordered strains, 257
With and without self-weights, 101
work, 55
zero string, 251