Probability-based Robust Design Optimization of a ... - Dynardo GmbH

Introduction 

Probability-based robust design optimization 

Sequential robust design optimization 

Probability-based Robust Design 

Optimization of a Radial Compressor 

concerning Fluid-Structure Interaction 

Kevin Cremanns, Dirk Roos 

Institute of Modelling and High-Performance Computing, 

Niederrhein University of Applied Sciences 

Johannes Einzinger 

ANSYS Germany GmbH 

Hochschule Niederrhein 

University of Applied Sciences 

IMH 

Institut für Modellbildung 

und Hochleistungsrechnen 

Institute of Modelling 

and High-Performance Computing 

K. Cremanns, D. Roos & J. Einzinger Probability-based Robust Design Optimization WOST 2012 1 / 49

Introduction 



Motivation 

CAE process integration 

Radial compressor example 

Optimization concerning uncertainties 

Power plant 1000 

MW, η ca. 50% 

Increasing of 2% 

results in 

additional +20 

MW power 

Per person, P = 1/6 kW 

Electricity for 20 MW / 1/6 kW = 120.000 inhabitants 

Maximal efficiency vs. maximal life time 

Six Sigma Design P(F) ≤ 3.4 · 10−6 Hochschule Niederrhein 


IMH 






Introduction 



Workbench and optiSLang 

Motivation 



Integration of structural, modal and fluid analysis (FSI) 

CAD- and CAE-based parameterization, automatic meshing 

Process integration, stochastic analysis and optimization 


Introduction 




Motivation 







Introduction 




Motivation 







Introduction 




Motivation 







Introduction 




Motivation 







Introduction 




Motivation 







Introduction 




Motivation 







Introduction 




Motivation 







Introduction 



Parametric CAE model with 

Motivation 



n d = 38 design parameters (geometry), nr = 51 random 

variables (geometry, material and process), mu = 71 constraints 

(stress and eigenfrequency resonance), objective: efficiency η 


Introduction 




Sigma level vs. failure probability 

Iterative robust design optimization 


f (d1,d2,...dn ,σ d 2 X1 ,σ2 X2 ,...σ2 Xnr 

,P(F)) → min 

g k(d1,d2,...dn d )=0; k = 1,me 

h l(d1,d2,...dn d ) ≥ 0; l = 1,mu 

σ 2 X = i 1 

N − 1 

d i ∈ [d l,du] ⊂ R n d 

d l ≤ d i ≤ du 

d i = E[X i] 

1 − P(F) 

Pt ≥ 0; 

(F) 

N 

∑ 

k=1 

σ j 

L 

σt − 1 ≥ 0 

L 

fX j (x j) 

� 

x k i − μX 

� � 

2 

; P(F)= 

i 

� 

nr ... 

g j(x)≤0 

x j 

d j = E[X j] 

fX(x)dx 

g(x i)=0 

unfeasible 

domain 

f (xi)=const di = E[Xi] fX i (x i) 

K. Cremanns, D. Roos & J. Einzinger Probability-based Robust Design Optimization WOST 2012 5 / 49 

x i

Introduction 






Methods solving PBRDO problems 

Coupled PBRDO, e.g. (Choi, Tu, and Park 2001) 

Nesting of two distinct levels of optimization (design level and 

reliability level) 

Double-loop problem needs sensitivity analysis to analytically 

compute the design gradients 

Single-loop PBRDO, e.g. (Kharmanda, Mohamed, and Lemaire 

2002) 

Simultaneously optimization the objective function and searches 

for the most probable failure point. 

Satisfying the probabilistic constraint only at the optimal solution 

Sequential PBRDO, e.g. (Chen, Liu, Sheng, and Gea 2003) 

Iteration between optimization and uncertainty quantification 

Updating the optimization goals based on the most recent 

probabilistic assessment results 

Quantification of the constraints based on safety factors 


Introduction 






Sigma level vs. probability of (non-)exceedance 

Sigma level σL is defined by a 

upper and lower limit state value x u,l 

g 

depending on limit state condition 

x u,l 

g := {X|g(x)=0}: 

E[X] ± σL · σX 

symmetrical case 

! 

≶ x u,l 

g 

σL ≥ xg − E[X] 

σX 

fX(x) 

x l g 

σX 

σX 

E[X] 

x 

σL · σX 

P(E)/2 P(E)/2 

(Non-)exceedance probability as function of the sigma level 

P(E)=P({X|X≶x u,l 

g })=f (σL) 



x u g = xg 

IMH 






Introduction 






Sigma level vs. probability of failure 

Failure probability as function of the 

sigma level, e.g. 

P(F)=P({X|X > xg})=f (σL) 

probability density function f X(x) 

0.4 

0.35 

0.3 

0.25 

0.2 

0.15 

0.1 

0.05 

0 

Gaussian distribution 

μ X = 0,σ X = 1 

xg 

-2 -1 0 1 2 3 4.5 6 

value x of the random variable X 

P(X|X > xg) 

10 0 

10 −2 

10 −4 

10 −6 

10 −8 

10 −10 

P T (F)=3.4 · 10 −6 

0 1 2 3 4.5 6 


σ L 

σ T L

Introduction 







Initial design 

Modification of 

constraints g(X) and 

safety factors γ 

Sensitivity analysis 

Sensitive parameters 

Iteration 0 

Optimization 

Optimal design 

Iteration I, II, III, IV, ... 

Robustness evaluation 

σ L ≶ σ t L 

σ L ∼ = σ t L 

Reliability analysis 

P(F) ≶ P t (F) 

P(F) ∼ = P t (F) 

σI I 

L II 

σ II 

L 

σ III 

L 

σ t L 

gI (X) gt g (X) 

II (X) 

III 

Robust and safety 

optimal design 

= 4.5 

g III (X) 


Introduction 



σe [Pa] 

Sensitivity analysis (initial design 0) 

Optimization and stochastic analysis I, II, III, IV 



3e+08 

2.5e+08 

2e+08 

1.67e+08 

1.5e+08 

History of the von Mises stress σe 

5%-quantil of the yield strength nominal value σ y,5% 

Nominal design stress σ I d 

1e+08 

5e+07 

Sensitivity analysis 0 

N = 122 

0 20 40 60 80 100 120 140 

N number of design evaluations 


Introduction 







8 

INPUT parameter 

4 6 

2 

Coefficients of Prognosis (using MoP) 

full model: CoP = 69 % 

INPUT: ReturnVaneHubThk3 

3 % 

INPUT: ReturnVaneShdThk2 

4 % 

INPUT: InletWidth 

4 % 

INPUT: RImpeller 

6 % 

INPUT: RotorHubBeta1 

7 % 

INPUT: ReturnVaneShdBeta1 

9 % 

INPUT: RotorShdBeta1 

18 % 

INPUT: ReturnVaneHubBeta1 

18 % 

20 40 60 80 

CoP [%] of OUTPUT: myeta 

100 

Latin hypercube sampling 

and 

Metamodel and coefficients 

of optimal prognosis 

(MoP) and (CoP) for the 

efficiency η 


Introduction 







8 


4 6 

2 



INPUT: ExitWidth 

1 % 


1 % 


3 % 


6 % 


7 % 


17 % 


24 % 


31 % 

0 20 40 60 80 

CoP [%] of OUTPUT: Equivalent_Stress_Maximum 

Most important multivariate 

dependencies and design 

parameters for the maximal 

von Mises stress σe 


Introduction 







5 


2 3 4 

1 




0 % 

INPUT: RotorThkHubTE 

3 % 


6 % 


9 % 


77 % 

20 40 60 80 100 

CoP [%] of OUTPUT: Total_Deformation_Reported_Frequency 


dependencies 

and design parameters 

for the first eigenfrequency 


Introduction 



η 





0.9 

0.893 

0.889 

0.88 

0.87 

0.86 

0.85 

History of the efficiency η 

Evolutionary optimization on MoP N = 1 

Efficiency of the initial design 

0.84 

0.83 

Sensitivity analysis 0 

N = 122 

0 20 40 60 80 100 120 140 



Introduction 



η 

Design optimization I (γ = 1.5) 

0.91 

0.905 

0.893 

0.889 

Sensitivity 

analysis 0 

N = 122 





0.86 

0.84 


Optimization I 

(ARSM) 

N = 197 

0 123 320 350 



Introduction 



σe [Pa] 

Design optimization I (γ = 1.5) 

3e+08 

2.5e+08 

1.67e+08 

1e+08 







5e+07 

Sensitivity analysis 

N = 122 

Optimization (ARSM) 

N = 197 

0 123 320 350 



Introduction 



η 

Robustness evaluation (design I) 

0.91 

0.905 

0.9 

0.89 

0.88 

0.87 

Sensitivity 

analysis 0 

N = 122 





0.86 



(ARSM) 

N = 197 

Robustness 

evaluation I 

N = 93 

0.84 

0 123 320 413 500 



Introduction 



σe [Pa] 


3e+08 

2.5e+08 

1.67e+08 







1e+08 

5e+07 

Sensitivity 

analysis 0 

N = 122 


(ARSM) 

N = 197 


evaluation I 

N = 93 

0 123 320 413 500 



Introduction 



Stress limit state g(X)=σy − σe 

5 

4 

PDF [1e-8] 

2 3 

1 

0 

OUTPUT: Equivalent_Stress_Maximum 

Fitted PDF 

Histogram 

Limit line 

1.4 1.6 1.8 2 2.2 2.4 

OUTPUT: Equivalent_Stress_Maximum [1e8] 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

� 

�� 

2.5 

2 

PDF [1e-8] 

1 1.5 

0.5 

0 

INPUT: Tensile_Yield_Strength 

Defined PDF 

Histogram 

Limit line 

2.4 2.6 2.8 3 3.2 

INPUT: Tensile_Yield_Strength [1e8] 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 




2 

1.5 

PDF [1e-8] 

1 

0.5 

0 

0 

0.25 

Fitted PDF 

Histogram 

Limit line 

INEQUAL: lsc 

0.5 0.75 1 1.25 

INEQUAL: lsc [1e8] 

1.5 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 


1.75

Introduction 




0.06 

PDF 

0.04 

0.02 

0 

INPUT: myomega 

685 690 695 700 705 710 


Defined PDF 

Histogram 

Limit line 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

715 

OUTPUT: Total_Deformation_Reported_Frequency 

0.008 

0.006 

PDF 

0.004 

0.002 

0 

Fitted PDF 

Histogram 

Limit line 

800 1000 1200 1400 1600 1800 2000 

OUTPUT: Total_Deformation_Reported_Frequency 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 




PDF 

200 

150 

100 

50 

0 

0.86 

OUTPUT: myeta 

Fitted PDF 

Histogram 

Limit line 

0.87 0.88 0.89 

OUTPUT: myeta 

�� 

�� 

�� 

0.9 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 


σ L 

Introduction 




5.13 

4.5 

Sigma level vs. safety factor 

II 

1.32 1.5 

γ of the optimized designs I-II 

I 




Sigma level of the optimized 

design 

σ I L = EI [σe] 

σI = 

σe 

1.106 · 108 

= 5.13 

2.154 · 107 regarding safety factor used 

for the first optimization 

γ I = 1.5 

Target sigma level σt L = 4.5 to 

ensure probability of failure 

P(F)=3.14 · 10−6 In case of lack of prior 

knowledge “rule of proportion” 

γ II = γ I · σt L 

σI = 1.32 

L 


Introduction 



Design optimization II (γ = 1.32) 

0.908 

0.905 

0.893 

0.889 

Sensitivity 

analysis 0 

N = 122 





0.86 



(ARSM) 

N = 197 


evaluation I 

N = 93 

Evolutionary 

optimization II 

N = 208 

0.84 

0 123 320 413 621 700 



Introduction 



Design optimization II (γ = 1.32) 

3e+08 

2.5e+08 

1.89e+08 

1.67e+08 




History of the von Mises stress σe [Pa] 


Nominal design stress σ II 

d 

1e+08 

5e+07 

Sensitivity 

analysis 0 

N = 122 


(ARSM) 

N = 197 


evaluation I 

N = 93 

Evolutionary 


N = 208 

0 123 320 413 621 700 



Introduction 



Robustness evaluation (design II) 

0.908 

0.905 

0.893 

0.889 

Sensitivity 

analysis 0 

N = 122 





0.86 



(ARSM) 

N = 197 


evaluation I 

N = 93 

Evolutionary 


N = 208 

RE 

II 

N = 46 

0.84 

0 123 320 413 621 667 



Introduction 




3e+08 

2.5e+08 

1.89e+08 

1.67e+08 






Nominal design stress σ II 

d 

1e+08 

5e+07 

Sensitivity 

analysis 0 

N = 122 


(ARSM) 

N = 197 


evaluation I 

N = 93 

Evolutionary 


N = 208 

RE 

II 

N = 46 

0 123 320 413 621 667 



Introduction 




3 

2.5 

2 

PDF [1e-8] 

1.5 

1 

0.5 

0 


Fitted PDF 

Histogram 

Limit line 

1.6 1.8 2 2.2 2.4 


�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

1.75 

1.5 

1.25 

PDF [1e-8] 

0.75 1 

0.5 

0.25 

0 

0 

0.2 

Fitted PDF 

Histogram 

Limit line 

INEQUAL: lsc 

0.4 0.6 0.8 1 





1.2 

1.4 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

140 

120 

100 

PDF 

80 

60 

40 

20 

0 

0.86 

0.87 

OUTPUT: myeta 

Fitted PDF 

Histogram 

Limit line 

0.88 0.89 

OUTPUT: myeta 

0.9 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 


σ L 

Introduction 




5.13 

4.5 

3.6 

Sigma level vs. safety factor 

II 

II 

III 

1.32 1.426 1.5 

γ of the optimized designs I-III 

I 





design II is actual 

σ II 

L = EII [σe] 

σII = 

σe 

9.182 · 107 

= 3.6 < 4.5 

2.551 · 107 Linear interpolation of the new 

safety factor 

γ III = 1.426 

We obtain the nominal design 

stress of the third optimization 

step 

σ III 

d 

σy,0.05 2.5 · 108 

= = = 1.75 · 108 

γIII 1.426 


Introduction 



Design optimization III (γ = 1.43) 

0.909 

0.905 

0.893 

0.889 

SA 

N = 122 





0.86 


DO I 

ARSM 

N = 197 

RE 

I 

N = 93 

DO II 

EA 

N = 208 

RE 

II 

N = 46 

DO III 

EA 

N = 206 

0.84 

0 123 320 413 621 667 873 



Introduction 



Design optimization III (γ = 1.43) 

3e+08 

2.5e+08 

1.89e+08 

1.75e+08 

1.67e+08 






1e+08 

SA DO I RE DO II RE DO III 

5e+07 

0 

N = 122 

ARSM 

N = 197 

I 

N = 93 

EA 

N = 208 

II 

N = 46 

EA 

N = 206 

0 123 320 413 621667 873 



σ III 

d

Introduction 



Robustness evaluation (design III) 

0.909 

0.905 

0.893 

0.889 

SA 

N = 122 





0.86 


DO I 

ARSM 

N = 197 

RE 

I 

N = 93 

DO II 

EA 

N = 208 

RE 

II 

N = 46 

DO III 

EA 

N = 206 

RE 

III 

N = 72 

0.84 

0 123 320 413 621 667 873 945 



Introduction 




3e+08 

2.5e+08 

1.89e+08 

1.75e+08 

1.67e+08 






1e+08 

SA DO I RE DO II RE DO III RE 

5e+07 

0 

N = 122 

ARSM 

N = 197 

I 

N = 93 

EA 

N = 208 

II 

N = 46 

EA 

N = 206 

III 

N = 72 

0 123 320 413 621667 873 945 



σ III 

d

Introduction 




3 

2.5 

PDF [1e-8] 

1.5 2 

1 

0.5 

0 


Fitted PDF 

Histogram 

Limit line 

1.6 1.8 2 2.2 2.4 


�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

2.5 

2 

PDF [1e-8] 

1 1.5 

0.5 

0 

0 

Fitted PDF 

Histogram 

Limit line 

INEQUAL: lsc 

0.2 0.4 0.6 0.8 1 





1.2 

1.4 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

200 

150 

PDF 

100 

50 

0 

0.86 

0.87 

OUTPUT: myeta 

Fitted PDF 

Histogram 

Limit line 

0.88 0.89 

OUTPUT: myeta 

0.9 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 


0.91

Introduction 



6.5 

5.13 

4.5 

4.1 

3.6 


6 

3 

Sigma level σ L vs. safety factor γ 

II III IV I 

1.32 1.43 1.46 1.5 1.55 

γ of the optimized designs I-IV 





design III is actual 

σ III 

L = EIII [σe] 

σIII = 

σe 

9.989 · 107 

= 4.1 < 4.5 

2.415 · 107 Quadratic interpolation of the 

new safety factor 

γ IV = 1.46 

We obtain the nominal design 

stress of the fourth optimization 

step 

σ IV 

d 

σy,0.05 2.5 · 108 

= = = 1.71 · 108 

γIV 1.46 


Introduction 



Design optimization IV (γ = 1.46) 

0.91 

0.905 

0.893 

0.889 

SA 

N = 122 





0.86 


DO I 

ARSM 

197 

RE 

I 

93 

DO II 

EA 

208 

RE 

II 

46 

DO III 

EA 

206 

RE 

III 

72 

DO IV 

EA 

93 

0.84 

0 123 320 413 621667 873 945 1038 



Introduction 



Design optimization IV (γ = 1.46) 

3e+08 

2.5e+08 

1.89e+08 

1.71e+08 






1e+08 

SA DO I RE DO II RE DO III RE DO IV 

5e+07 

0 ARSM 

N = 122 197 

I 

93 

EA 

208 

II 

46 

EA 

206 

III 

72 

EA 

93 

0 123 320 413 621667 873 945 1038 



σ IV 

d

Introduction 



Robustness evaluation (design IV) 

0.91 

0.905 

0.893 

0.889 

SA 

N = 122 





0.86 


DO I 

ARSM 

197 

RE 

I 

93 

DO II 

EA 

208 

RE 

II 

46 

DO III 

EA 

206 

RE 

III 

72 

DO IV RE 

EA IV 

93 142 

0.84 

0 123 320 413 621667 873 945 1038 1180 



Introduction 



Robustness evaluation (design IV) 

3e+08 

2.5e+08 

1.71e+08 






1e+08 

SA DO I RE DO II RE DO III RE DO IV RE 

5e+07 

0 ARSM 

N = 122 197 

I 

93 

EA 

208 

II 

46 

EA 

206 

III 

72 

EA 

93 

IV 

142 

0 123 320 413 621667 873 945 1038 1180 



2 

1.5 

PDF [1e-8] 

1 

0.5 

0 

Introduction 



0 

Robustness evaluation IV 

Fitted PDF 

Histogram 

Limit line 

INEQUAL: lsc 

0.2 0.4 0.6 0.8 1 1.2 1.4 


�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

1.6 

�� 

Sigma level of the 

optimized design IV: 

σ IV 

L = EIV [σe] 

σIV σe 

= 1.004 · 108 

2.24 · 107 = 4.48 ≈ 4.5 

Small standard 

deviation of the 

efficiency and 

very small probability 

(0.02%) of violation 

of initial efficiency 




PDF 

200 

150 

100 

50 

0 

0.86 

0.87 

OUTPUT: myeta 

Fitted PDF 

Histogram 

Limit line 

0.88 0.89 

OUTPUT: myeta 

0.9 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

0.91 

�� 

�� 

�� 


Introduction 







6 


4 

2 




2 % 


2 % 

INPUT: myAirR 

2 % 


9 % 


9 % 

INPUT: Poissons_Ratio 

11 % 

INPUT: Density 

18 % 

0 20 40 60 80 100 


Metamodel and coefficients 

of optimal prognosis 

(MoP) and (CoP) for the 

maximal von Mises stress 

σe 


Introduction 







6 


4 

2 




2 % 


2 % 

INPUT: myAirR 

2 % 


9 % 


9 % 

INPUT: Poissons_Ratio 

11 % 

INPUT: Density 

18 % 

0 20 40 60 80 100 



dependencies and design 

parameters for the maximal 

von Mises stress σe 


Introduction 



Multi-domain reliability analysis 




g(x) ⇒ ˜g(x) Advanced moving 

least square 

approximation (Roos, 

Adam, and Bayer 2006) 

of 

highly nonlinear state 

and limit state functions 

Independency of the 

probability levels 

using directional 

sampling 

Cluster analysis is 

given to detect 

multi-domain failure 

states (Roos 2011) 


4 

INPUT: Tensile_Yield_Strength [1e8] 

2.5 

3 

3.5 

2 

Introduction 



Reliability analysis design IV 

660 

INPUT: myomega vs. INPUT: Tensile_Yield_Strength 

( 4. Approximation ) 

Design point 

Safe domain 

Supports 

Unsafe domain 

680 

700 


720 

740 




N = 56 design evaluations 

failure probability: P(F)=2.5 · 10 −6 


Introduction 




0.91 

0.905 

0.893 

0.889 

SA 

N = 122 





0.86 


DO I 

ARSM 

197 

RE 

I 

93 

DO II 

EA 

208 

RE 

II 

46 

DO III 

EA 

206 

RE 

III 

72 

DO IV 

EA 

93 

RE 

IV 

142 

Reliability 

analysis 

N = 56 

0.84 

0 123 320 413 621667 873 945 1038 1236 



Introduction 




3e+08 

2.5e+08 

1.71e+08 






1e+08 

SA DO I RE DO II RE DO III RE DO IV RE Reliability 

5e+07 

0 

122 

ARSM 

197 

I 

93 

EA 

208 

II 

46 

EA 

206 

III 

72 

EA 

93 

IV 

142 

analysis 

N = 56 

0 123 320 413 621667 873 945 1038 1236 



σ IV 

d

Introduction 



Summary 




Probability- and variance-based robust design optimization 

n d = 38 design parameters (geometry) 

nr = 51 random variables (geometry, material and process) 

Adjust the design parameter to maximize the efficiency η 

about 5% 

Electricity for 5 · 20 MW / 1/6 kW = 600.000 inhabitants 

(population of Stuttgart) 

Optimize the design such that it is insensitive to 

uncertainties 

Consideration of the failure probability (defects per million) 

More robust and safety designs 



IMH 






Introduction 



Summary 




Nonlinear multi-physics analysis (FSI) with N = 1236 

design evaluations 

ANSYS Workbench (CFX, Mechanical, TurboTools): 4 

Parallel Tasks 

Calculation time: 2 weeks (on 12 Cores: 2 Intel® Xeon® 

X5680 Six Core, 3.33 GHz, 12MB Cache) 

Acknowledgement: thanks to Ulrike Adams and Daniela 

Ochsenfahrt of the DYNARDO GmbH (software 

implementation into the optiSLang software package) 



IMH 






Introduction 



Are there any questions? 

Thank you for your attention! 






IMH 






Literature I 

Appendix 

References 

Chen, W., H. Liu, J. Sheng, and H. C. Gea (2003, September 2–6). 

Application of the sequential optimization and reliabilty assessment method 

to structural design problems. 

In Proceedings of DETC’03, ASME 2003 Design Engineering Technical 

Conferences and Computers and Information in Engineering 

Conference, Chicao, Illinois USA. 

Choi, K. K., J. Tu, and Y. H. Park (2001). 

Extensions of design potential concept for reliability-based design 

optimization to nonsmooth and extreme cases. 

Structural and Multidisciplinary Optimization 22, 335–350. 

Kharmanda, G., A. Mohamed, and M. Lemaire (2002). 

Efficient reliability-based design optimization using a hybrid space 

withapplication to finite element analysis. 

Structural and Multidisciplinary Optimization 24, 233 – 245. 



IMH 






Literature II 

Appendix 

References 

Roos, D. (2011, November 17-18). 

Multi-domain adaptive surrogate models for reliability analysis. 

In H. Budelmann, A. Holst, and D. Proske (Eds.), Proceedings of the 9th 

International Probabilistic Workshop, pp. 191 – 207. Braunschweig, 

Germany: Technical University Carolo-Wilhelmina zu Braunschweig. 

Roos, D., U. Adam, and V. Bayer (2006). 

Design reliability analysis. 

In 24th CAD-FEM USERS’ Meeting 2006. Stuttgart, Schwabenlandhalle, 

Germany, October 26-27, 2006: International Congress on FEM 

Technology with 2006 German ANSYS Conference. 

Wolpert, D. H. and W. G. Macready (1997). 

No free lunch theorems for optimization. 

IEEE Trans, Evolutionary Computation 1(1), 67 – 82. 



IMH

Probability-based Robust Design Optimization of a ... - Dynardo GmbH

Create successful ePaper yourself

Delete template?

Save as template?