The Finite Element Method for the Analysis of Non-Linear and ...

The Finite Element Method for the Analysis of 

Non-Linear and Dynamic Systems 

Prof. Dr. Eleni Chatzi 

Lecture 14 - 20 December, 2010 

Institute of Structural Engineering Method of Finite Elements II 1

Special Topics 

The standard finite element method has been used with great success 

in many fields with both academic and industrial applications. It is 

however not without limitations. Due to mesh-based interpolation, 

distorted or low quality meshes lead to higher errors, necessitate 

remeshing, a time and human labor consuming task, which is not 

guaranteed to be feasible in finite time for complex three-dimensional 

geometries. Additionally, they are not well suited to treat problems 

with discontinuities that do not align with element edges. 

Alternative/Extended Approaches 

Meshless Methods 

Multigrid Methods 

Extended Finite Element methods 




Traditional simulation algorithms relied on a grid or a mesh, Meshless 

methods in contrast use the geometry of the simulated object directly for 

calculations. Meshless (or Meshfree) methods (MMs) were born with the 

objective of eliminating part of the difficulties associated with reliance on a 

mesh to construct the approximation. 




Meshless approximations for a scalar function u in terms of the 

material (Lagrangian) coordinates can be written as 

u(x, t) = ∑ i∈S 

N i (x)u i (t) 

where N i : Ω → R are the shape functions and the u i are the nodal 

values at particle i located at position x i and S is the set of nodes i 

for which N i (x) ≠ 0. In contrast to FEM, the shape functions are 

only approximants and not interpolants, since u i ≠ u(x i ). 

Therefore, special techniques are needed to treat displacement 

boundary conditions. 




Advantages 

problems with moving 

discontinuities such as crack 

propagation and phase 

transformation can be treated 

with ease 

large deformation can be 

handled more robustly 

higher-order continuous 

shape functions 

non-local interpolation 

character 

no mesh alignment sensitivity. 

Disadvantages 

The MMs shape functions 

are rational functions which 

requires high-order 

integration scheme to be 

correctly computed 

The treatment of essential 

boundary conditions is not as 

straightforward 

They do not satisfy the 

Kronecker delta property 

In general, the computational 

cost of MMs is higher than 

one of FEM. 




Multigrid (MG) methods in numerical analysis are a group of 

algorithms for solving differential equations using a hierarchy of 

discretizations. They are a class of techniques very useful in 

problems exhibiting multiple scales of behavior. They are fast linear 

iterative solvers based on the multilevel or multi-scale paradigm. 

The typical application for 

multigrid is in the 

numerical solution of 

elliptic partial differential 

equations in two or more 

dimensions. 







There are many variations of multigrid algorithms, but the common 

features are that a hierarchy of discretizations (grids) is considered. 

The important steps are: 

Smoothing Perform some steps of a basic method in order to 

smooth out the error. 

Restriction Restrict the current state of the problem to a 

subset of the grid points, the so-called “coarse grid”, and solve 

the resulting projected problem. 

Interpolation or Prolongation Interpolate the coarse grid 

solution back to the original grid, and perform a number of 

steps of the basic method again. 



Multigrid Methods - Adaptive Mesh 

Adaptive multigrid exhibits adaptive mesh refinement, that is, it 

adjusts the grid as the computation proceeds, in a manner dependent 

upon the computation itself. The idea is to increase resolution of the 

grid only in regions of the solution where it is needed. 




Advantages 

MG can be applied in combination with any of the common 

discretization techniques. 

Multigrid is among the fastest solution techniques known today. 

In contrast to other methods, multigrid is general in that it can 

treat arbitrary regions and boundary conditions. 

Multigrid does not depend on the separability of the equations 

or other special properties of the equation. 

MG is also directly applicable to more complicated, 

non-symmetric and nonlinear systems of equations. 

Can be very easily parallelized. 



The Extended Finite Element Method (XFEM) 

The Extended Finite Element Method is a numerical method that 

enables a local enrichment of approximation spaces. The enrichment 

is realized through the partition of unity concept. 

The method is useful for the approximation of solutions with 

pronounced non-smooth characteristics in small parts of the 

computational domain, for example near discontinuities and 

singularities. In these cases, standard numerical methods such as the 

FEM often exhibit poor accuracy. 

The XFEM offers significant advantages by enabling optimal 

convergence rates for these applications. 



XFEM - Motivation 

Modeling of discontinuities, ex. structural flaws (cracks, holes) 



XFEM - Motivation 

Modeling of complicated crack geometries (ex. non planar cracks) 

A T-Joint with Crack on a 

Given Curved Plane 



XFEM vs FEM 

- XFEM locally enrich the standard FE approximation with local 

partitions of unity enrichment functions 

- This allows for the separation of geometry from the mesh. 

Standard FEM 

X-FEM 



XFEM Scheme 

The conventional shape function space is enriched with a set of 

enrichment functions H(x) and F j (x) 

u h (x) = 

n∑ ∑n J 

N I (x)u I + N I (x)H(x)a I + 

I=1 

Tip Enrichment 

I=1 

n T ∑ 

I=1 

⎡ 

⎣N I (x) 

4∑ 

j=1 

F j (x)b jI 

⎤ 

⎦ 

F j (r(x), θ(x)) = { √ r sin θ 2 , √ r cos θ 2 , √ r sin θ 2 sin θ, √ r cos θ 2 sin θ} j=1,..,4 

Jump Enrichment 

H(X) = 

{ 

1 above Γ + c 

0 bellow Γ − c 



XFEM Scheme - Tip and Jump Enrichments 



XFEM Applications - Composite Materials 

G XFEM 

= 4.762 

G ABAQUS 

= 4.703 

Gexp eriment 

= 4.784 

[lb in/in^2] 

Hart D., NSWCCD [2006] 

5 

4.5 

4 

3.5 

3 

G 

2.5 

2 

1.5 

1 

0.5 

0 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 

coordinate along crack 



XFEM Applications - Crack Growth 

Load P 

6.25 

Unit: mm 

Simplification 

BC 1 

Fix U,V 

15.63 

7.813 

BC 2 

Fix U 

35.0 

6.25 

37.5 

100 



Crack Initiation Site: 

Experimental Observation 





Introduction to FEM Software 


*The section on Introduction to FEM Software is based on 

Prof. M. Faber’s notes of the FEM II course - Fall 2009 

Method of Finite Elements II 

1 



Overview 

FEM development 

Introduction to the use of Finite Elements 

Modeling the physical problem 

Finite elements as a tool for computer aided design and 

assessment 

An overview of FEM software (open source and commercial) 

Example: ABAQUS 



FE development 

FEM development 

FE is the confluence of three ingredients: matrix structural 

analysis, variational approach and computer 

1950s, M.J. Turner at Boeing (aerospace industry in general): 

Direct Stiffness Method 

Academia: J.H. Argyris, R.W. Clough (name “finite element), 

H.C. Martin and O.C. Zienkiewicz popularization 

1960s, Melosh and De Veubeke: Variational Approach 

Commercial finite element computer codes 



FE Basics 

Within the framework of continuum mechanics dependencies 

between geometrical and physical quantities are formulated on a 

differentially small element and then extended to the whole 

continuum 

As a result we obtain differential, partial differential or integral 

equations for which, generally, an analytical solution is not 

available they have to be solved using some numerical 

procedure 

MFE is based on the physical discretization of the observed 

domain, thus reducing the number of the degrees of freedom; 

moreover the governing equations are in general algebraic 



FE Steps 

Continuum is discretized in a mesh of finite elements 

Elements are connected at nodes located on element boundaries 

State of deformation, stresses, etc. in each element is described 

by interpolation (shape) functions and corresponding values in 

the nodes; these node values are basic unknowns of the method 

of finite elements 

The way in which these three steps are approached has a great 

influence on the results of the calculations 



Modeling the Physical Problem 

Ideal 

Mathematical 

model 

§1.3 THE FEM ANALYSIS PRO 

generally 

irrelevant 

CONTINUIFICATION 

SOLUTION 

Physical 

system 

FEM 

Discrete 

model 

Discrete 

solution 

IDEALIZATION & 

DISCRETIZATION 

VERIFICATION 

solution error 

simulation error= modeling + solution error 

VALIDATION 



Modeling the Physical Problem 

MFE is only a way of solving the mathematical model 

The solution of the physical problem depends on the quality of 

the mathematical model the choice of the mathematical model 

is crucial 

Thus, mathematical model must be reliable and effective 

The chosen mathematical model is reliable if the required 

response can be predicted within a given level of accuracy 

measured on the response of a very comprehensive 

mathematical model 

The most effective mathematical model for the analysis is the 

one that gives the required response with sufficient accuracy 

and at least costs 



Complex physical problem (steel bracket) modelled 

p p y p ( ) 

by a simple mathematical model 

Institute of Structural Engineering Method of Finite Elements II 28 

Example 

Complex physical problem (steel bracket) modeled by a simple 

mathematical model


Example 

Detailed reference model 2D plane stress model for FE analysis 



Example 

Choice of mathematical model must correspond to desired 

response 

The most effective mathematical model delivers reliable answers 

with the least amount of effort 

Any solution (including MFE) of a mathematical model is 

limited to information contained in or fed into the model: bad 

input bad output (garbage in garbage out) 

Assessment of accuracy is based on comparisons with the 

results from very comprehensive models but in practice it has 

to be based on experience (experiments) 



FE as a Tool for CAD 

Practical application requires that solutions obtained by MFE 

are reliable and efficient 

However, it is also necessary that the use of FE is robust this 

implies that minor changes in any input to an FE analysis 

should not change the response quantity significantly 

The engineer (user) should be able to judge the quality of the 

obtained results (i.e. for plausibility) 



Implementation of FE 

Calculation of system matrices K, M, C and R whichever 

applicable (nodal point and element information are read; 

element stiffness matrices, mass and damping matrices and 

equivalent loads are calculated; structure matrices are 

assembled) 

Solution of equilibrium equations 

Evaluation of element stresses 



Implementation of FE - Flowchart 



Open Source Packages 

CalculiX is an Open Source FEA project. The solver uses a 

partially compatible ABAQUS file format. 

OpenSees, the Open System for Earthquake Engineering 

Simulation, is an object-oriented, open source software 

framework created at the NSF-sponsored Pacific Earthquake 

Engineering (PEER) Center. 

FEAP is a general purpose finite element analysis program 

which is designed for research and educational use. 

http://www.ce.berkeley.edu/projects/feap/ 

ForcePAD: educational software forcepad.sourceforge.net 

Sundance: a software package developed at Sandia National 

Laboratories 



Commercial Packages 

ABAQUS: Franco - American software from SIMULIA 

ADINA R&D, Inc. See http://www.adina.com/ 

ANSYS: American software 

COMSOL Multiphysics 

COSMOSWorks: A SolidWorks module 

GTSTRUDL30 

LS-DYNA, LSTC - Livermore Software Technology Corporation 

Nastran: American software 

CUBUS Swiss Software Structurally Oriented 

SAP: Structurally Oriented 



ABAQUS Features 

Wide material modeling capability 

Ability to be customized 

A good collection of multiphysics capabilities, such as coupled 

acoustic - structural piezoelectric etc. 

Attractive for simulations where multiple fields need to be 

coupled 



ABAQUS Multiphysics Example 

Example: Shock Response and Acoustic Radiation Analysis for marine 

structures 

Most naval organizations require some form of shock survival assessment to 

be performed on marine structures before they are commissioned 

Besides shock survivability, another important consideration in submarine 

design is stealth. Since their inception, submarines have been deployed in 

strategic situations where susceptibility to detection is equated with 

reduced effectiveness of the vessel. Various acoustic treatments are used to 

reduce the target strength and acoustic signature of a submarine. 

ABAQUS provides fully coupled structural acoustic capabilities in the 

frequency domain that predict near and far-field sound pressure levels due 

to machinery-induced vibrations. 




The structure was meshed with quadrilateral shells and the fluid with 

acoustic tetrahedra. 




Figure 14 and Figure 15 show the Mises stress 

distribution on the external surface of the submarine 

and in the internal compartments, respectively. 

Stresses are notably lower in the areas where the 

hulls overlap, as is expected due to the enhanced 

thickness there. 

Figure 14. Mises stress distribution on the exterior 

surface of the submarine after 10 ms. 

Figure 12. Acoustic pressure distribution in the 

outer water after 10 ms. 

presence of the terminating fluid boundary. This 

indicates that the fluid zone is sufficiently large, so 

there are no significant spurious reflections from the 

boundary. Since the effect of hydrostatic pressure is 

Figure 15. Mises stress distribution in the internal 

structures of the submarine after 10 ms. 

The results of the acoustic radiation analysis are 

examined next. Figure 16, Figure 17, and Figure 18

The Finite Element Method for the Analysis of Non-Linear and ...

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