1. Institute for Structural Analysis A Multi-objective Optimization Approach with a View to Robustness Improvement www.tu-dresden.de/isd 5 th Conference on Reliable Engineering Computing A. Serafinska, W. Graf, M. Kaliske Institute for Structural Analysis, Technische Universität Dresden

2. 2 1Motivation 2Deterministic multi-objective optimization 3Uncertainty modelling -uncertainty model Fuzziness 4Aggregate objective function evaluating fuzzy variables -uncertainty measures -robustness measure 5 Nondominated Sorting Genetic Algorithm (NSGA-II) with fuzzy variables - domination of fuzzy objective vectors - Fuzzy-Pareto-Front 6Example - design of passenger car tire (195/60 R15) Institute for Structural Analysis Content Motivation

3. Institute for Structural Analysis noise handling aquaplaning Multi-objective optimization task Motivation 3 wear fatigue structural design as multi-objective optimization task design variables objective functions constraints uncertain quantities example: tire design optimization task defined through:

4. Institute for Structural Analysis Uncertain quantities material properties tensile test results of rubber compound considering different charges Motivation 4

5. Institute for Structural Analysis Uncertain quantities loads optimization task: design, which provides quite uniform wear for all possibly appearing inner pressures underinflated overinflated properly inflated 5 changing tire inner pressure → different wear images Motivation

6. Institute for Structural Analysis Deterministic optimization task Deterministic MOO 6 optimization task design variables result variables objective function equality constraints inequality constraints optimization approaches, single-objective optimization problem multi-objective optimization problem → e.g. aggregate objective function weighted sum method, -dimensional objective vector weight factors, classical multi-objective optimization problem comparison of –dimensional vectors with dominance concept

7. Institute for Structural Analysis Deterministic optimization task 7 dominance concept Pareto-optimal set Pareto-optimal set – solutions not dominated by any other member of set. Strength-Pareto Evolutionary Algorithm SPEA-2, Zitzler et al. [2001] Nondominated Sorting Genetic Algorithm NSGA-II, Deb et al. [2002], if no component of is greater than the corresponding com- ponent of and at least one component is smaller Pareto-front multi-objective optimization methods if, than – Pareto-optimal set Pareto-Archieved Evolution Strategy PAES, Knowles, Corne [1999] Deterministic MOO Pareto-front – visualization of Pareto-optimal set in objective space.

8. Institute for Structural Analysis Optimization with uncertain quantities Uncertainty modeling 8 Optimization methods with uncertainty quantification and evaluation: M1: aggregate function (weighted sum method) M2: Nondominated Sorting Genetic Algorithm NSGA-II noise handling aquaplaning wear fatigue

9. Institute for Structural Analysis uncertainty epistemic aleatoric stochastic methods randomness objective data non-stochastic methods evaluation subjective/ objective data 9 Uncertainty quantification Uncertainty modeling

10. 1 0,5 0 2 46 810 x 1 0 246 8 x 1 0 2468 x stochastic variable fuzzy variable fuzzy-stochastic variable limited data, small sample size, linguistic assessments, expert evaluations probabilistic models fuzziness f(x) μ (x) Institute for Structural Analysis Modeling uncertainty 10 convex modeling interval mathematics fuzzy-randomness imprecise probability … model chosen for further consideration: Uncertainty modeling

11. fuzzy set/fuzzy variable x xlxl xrxr membership function 1.0 0 0.5 0.75 0.25 α -level α -level-discretization: Institute for Structural Analysis 11 Uncertainty model fuzziness convexity: Uncertainty modeling

12. Institute for Structural Analysis MOO with fuzzy quantities (M1) design variables uncertain a-priori parameters result variables objective functions information reducing measures weighted sum method design A design B μ(z1)μ(z1) μ(z1)μ(z1) μ(z2)μ(z2) μ(z2)μ(z2) M1: Aggregate objective function evaluating fuzzy variables 12

13. Institute for Structural Analysis uncertainty measure SHANNON's entropy zeroth moment second central moment first moment Uncertainty measures for fuzzy variables 13 MOO with fuzzy quantities (M1)

14. Institute for Structural Analysis x d2 x d1 Modified Evolution Strategy Möller, Graf, Beer [2000] gradient method Monte Carlo simulation optimization method: (1+1) evolution strategy Optimization concept 14 MOO with fuzzy quantities (M1)

15. Institute for Structural Analysis ℓ – number of load cases np – number of inputs (a-priori param.) k – number of results w – weighting factor P – penalty function – uncertainty measures Definitions of robustness ratio of all uncertain input parameter according to uncertain responses uncertainty is given as a fuzzy variable  limit the variations of structural responses  redundancy  ratio direct risks / indirect risk robustness measure 15 MOO with fuzzy quantities (M1)

16. Institute for Structural Analysis Objective space with k-dimensional objective vectors k-dimensional fuzzy result vector 16 goal: finding the set of nondominated fuzzy objective vectors (not considered in the aggregate function approach) MOO with fuzzy quantities (M2)

17. Institute for Structural Analysis Fuzzy-Pareto-Front (M2 method) domination of fuzzy objective vectors 17 dominations: fuzzy objective vector is non-dominated, if in the set of all fuzzy objective vectors, there exists no other vector, so that every element dominates all elements. MOO with fuzzy quantities (M2)

18. Institute for Structural Analysis M2: NSGA-II coupled with fuzzy analysis NSGA-II 18 - elitism preservation - nondominated sorting - binary tournament selection - simulated binary crossover - mutation NSGA-II coupled with fuzzy analysis - nondominated sorting applied to fuzzy objective vectors - fuzzy analysis performed for each design vector - domination criteria for fuzzy objective vectors MOO with fuzzy quantities (M2)

19. Institute for Structural Analysis design variables belt angle thickness of tread layer number of capplies a-priori parameter inner pressure fibre spacing in bodyply stiffness of tread compound objectives minimal contact pressure ratio minimal energy density amplitude high robustness Example: optimization of passenger car tire (M1 method) Example 19 bodyply rim tread capply belt bead optimization task

20. Institute for Structural Analysis wear model rate of wear distance of sliding friction coefficient contact pressure strain energy density constant strain distribution in tire footprint stress objective function goal Uniform wear – first objective 20 Muhr, Roberts [1992] Example

21. Institute for Structural Analysis predictors for fatigue crack tearing energy (energy release rate) strain energy density crack length function of extension ratio objective function goal Resistance to fatigue – second objective 21 Gent et al. [1964] Example

22. Institute for Structural Analysis 1.230 0.190 0.0206 0.0162 1.972 1.196 p 0 1.230 1.268 μ(p) 0.321 0.172 ΔWΔW 0 0.190 0.222 μ(ΔW)μ(ΔW) result interval results for chosen design result p ΔW ΔW A1 A1 A2 A2 computational effort: 160 designs analysed for each design: parallel computing - 2 fuzzy analysis - ca. 1000 computations Institute for Structural Analysis Optimal design 22 Example

23. chosen best design initial design x d1 = 26.54 x d2 = -1.5 mm x d3 = 2 x d1 = 24.00 x d2 = 0 mm x d3 = 2 p = 1.51 p a1 = 0.27 p a2 = 1.31 p a3 = 0.875 p = 1.73 p a1 = 0.23 p a2 = 1.37 p a3 = 1.075 p = 1.23 p a1 = 0.27 p a2 = 1.17 p a3 = 0.875 p = 1.27 p a1 = 0.23 p a2 = 1.44 p a3 = 1.050 Institute for Structural Analysis Optimal contact pressure 23 N/mm 2 mm N/mm 2 mm p a1 inner pressure p a2 fibre spacing: body ply p a3 stiffness: tread compound x d1 belt angle x d2 thickness: tread layer x d3 number of capplies Example

24. Institute for Structural Analysis Optimal strain energy density 24 chosen best design initial design x d1 = 26.54 x d2 = -1.5 mm x d3 = 2 x d1 = 24.00 x d2 = 0 mm x d3 = 2 p a1 = 0.23 p a2 = 1.44 p a3 = 0.954 p a1 = 0.27 p a2 = 1.18 p a3 = 1.075 ΔW = 0.190 p a1 = 0.23 p a2 = 1.17 p a3 = 0.875 p a1 = 0.27 p a2 = 1.44 p a3 = 1.075 N/mm 2 mm N/mm 2 mm ΔW = 0.222N/mm 2 ΔW = 0.241 ΔW = 0.261 N/mm 2 Example p a1 inner pressure p a2 fibre spacing: bodyply p a3 stiffness: tread compound x d1 belt angle x d2 thickness: tread layer x d3 number of capplies N/mm 2

25. Institute for Structural Analysis 1.230 0.190 0.0206 0.0162 1.972 1.196 p 0 1.230 1.268 μ(p) 0.321 0.172 ΔWΔW 0 0.190 0.222 μ(ΔW)μ(ΔW) result interval results for chosen design result p ΔW ΔW A1 A1 A2 A2 computational effort: 160 designs analysed for each design: parallel computing - 2 fuzzy analysis - ca. 1000 computations Institute for Structural Analysis Optimal design 25 Example

26. Institute for Structural Analysis Summary Conclusions 26 multi-objective optimization approaches with consideration of uncertainties epistemic uncertainty described with model fuzziness robustness optimization implicitly included in optimization task coupled approaches of optimization and fuzzy analysis example: durability of tire design improved by consideration of two objectives: due to high computational cost of the approach parallel computing and neuronal networks applied - reduction of wear - resistance to fatigue - aggregate objective function - NSGA-II

27. Institute for Structural Analysis A Multi-objective Optimization Approach with a View to Robustness Improvement www.tu-dresden.de/isd 5 th Conference on Reliable Engineering Computing A. Serafinska, W. Graf, M. Kaliske Institute for Structural Analysis, Technische Universität Dresden

28. low middlehigh μ(x) 1.0 0.5 0.0 010 3050 x μ(x) 1.0 0.5 0.0 0 10 30 50 n 1 0 x fuzzy variable x μ(x) 1.0 0.0 7.5 μ(d) 1.0 0 d 2.05 2.20 limited data base linguistic quantity single uncertain measure expert knowledge Institute for Structural Analysis Uncertainty model fuzziness 28 Uncertainty modeling

29. Institute for Structural Analysis chosen best design worst design X 4 = 26.54 X 5 = -1.5 mm X 6 = 2 X 4 = 18.00 X 5 = 0.4 mm X 6 = 0 ΔW = 0.309 X 1 = 0.24 X 2 = 1.440 X 3 = 0.892 ΔW = 0.190 X 1 = 0.23 X 2 = 1.170 X 3 = 0.875 Institute for Structural Analysis Optimization results Optimal strain energy density 29 N/mm 2 mm N/mm 2 mm N/mm 2 X 1 inner pressure X 2 fibre spacing: bodyply X 3 stiffness: tread compound X 4 belt angle X 5 thickness: tread layer X 6 number of capplies

30. Institute for Structural Analysis ΔW = 0.321 X 1 = 0.27 X 2 = 1.170 X 3 = 1.075 ΔW = 0.222 X 1 = 0.27 X 2 = 1.440 X 3 = 1.075 Institute for Structural Analysis Optimization results Optimal strain energy density 30 N/mm 2 mm N/mm 2 mm N/mm 2 X 1 inner pressure X 2 fibre spacing: bodyply X 3 stiffness: tread compound X 4 belt angle X 5 thickness: tread layer X 6 number of capplies chosen best design worst design X 4 = 26.54 X 6 = 2 X 4 = 18.00 X 5 = 0.4 mm X 6 = 0 ΔW = 0.309 X 1 = 0.24 X 2 = 1.440 X 3 = 0.892 ΔW = 0.190 X 1 = 0.23 X 2 = 1.170 X 3 = 0.875 X 5 = -1.5 mm

31. Institute for Structural Analysis Optimization results Optimal strain energy density 31 X 1 inner pressure X 2 fibre spacing: bodyply X 3 stiffness: tread compound X 4 belt angle X 5 thickness: tread layer X 6 number of capplies chosen best design worst design X 4 = 26.54 X 6 = 2 X 4 = 18.00 X 5 = 0.4 mm X 6 = 0 ΔW = 0.309 X 1 = 0.24 X 2 = 1.440 X 3 = 0.892 ΔW = 0.190 X 1 = 0.23 X 2 = 1.170 X 3 = 0.875 ΔW = 0.321 X 1 = 0.27 X 2 = 1.170 X 3 = 1.075 ΔW = 0.222 X 1 = 0.27 X 2 = 1.440 X 3 = 1.075 N/mm 2 mm N/mm 2 mm N/mm 2 X 5 = -1.5 mm

32. chosen best design worst design X 4 = 26.54 X 5 = -1.5 mm X 6 = 2 X 4 = 18.00 X 5 = 1.0 mm X 6 = 0 p = 1.68 X 1 = 0.27 X 2 = 1.367 X 3 = 0.875 p = 1.97 X 1 = 0.23 X 2 = 1.316 X 3 = 0.962 p = 1.23 X 1 = 0.27 X 2 = 1.170 X 3 = 0.875 p = 1.27 X 1 = 0.23 X 2 = 1.440 X 3 = 1.050 p s = 0.510 p c = 0.415 p c = 0.367 p s = 0.427 p s = 0.720 p c = 0.429 p c = 0.374 p s = 0.738 Institute for Structural Analysis Optimization results Optimal contact pressure 32 N/mm 2 mm N/mm 2 mm X 1 inner pressure X 2 fibre spacing: bodyply X 3 stiffness: tread compound X 4 belt angle X 5 thickness: tread layer X 6 number of capplies

33. Institute for Structural Analysis Optimization results Optimal strain energy density 33 X 1 inner pressure X 2 fibre spacing: bodyply X 3 stiffness: tread compound X 4 belt angle X 5 thickness: tread layer X 6 number of capplies chosen best design worst design X 4 = 26.54 X 6 = 2 X 4 = 18.00 X 5 = 0.4 mm X 6 = 0 ΔW = 0.309 X 1 = 0.24 X 2 = 1.440 X 3 = 0.892 ΔW = 0.190 X 1 = 0.23 X 2 = 1.170 X 3 = 0.875 X 5 = -1.5 mm N/mm 2 mm N/mm 2 mm N/mm 2

34. x1x1 µ(x 1 ) 0.270.23 0.25 1 0 1.441.17 1.30 1 0 x2x2 µ(x 2 ) x1x1 x2x2 αiαi αiαi 1 0 z µ(z) αiαi fuzzy input variable fuzzy result variable deterministic solution d min zmax z d – FE solution d – response surface Institute for Structural Analysis Numerical realization Fuzzy analysis 34

35. Institute for Structural Analysis Optimization results Optimal strain energy density 35 X 1 inner pressure X 2 fibre spacing: bodyply X 3 stiffness: tread compound X 4 belt angle X 5 thickness: tread layer X 6 number of capplies chosen best design worst design X 4 = 26.54 X 6 = 2 X 4 = 18.00 X 5 = 0.4 mm X 6 = 0 ΔW = 0.309 X 1 = 0.24 X 2 = 1.440 X 3 = 0.892 ΔW = 0.190 X 1 = 0.23 X 2 = 1.170 X 3 = 0.875 ΔW = 0.321 X 1 = 0.27 X 2 = 1.170 X 3 = 1.075 ΔW = 0.222 X 1 = 0.27 X 2 = 1.440 X 3 = 1.075 N/mm 2 mm N/mm 2 mm N/mm 2 X 5 = -1.5 mm

36. Institute for Structural Analysis ℓ – number of load cases n – number of inputs m – number of results k – weighting factor P – penalty function – uncertainty measures Definitions of robustness the ratio of all uncertain input parameter according the uncertain response uncertainty is given as a fuzzy variable  limit the variations of structural responses  redundancy  ratio direct risks / indirect risk Robustness measure robustness measure Sickert et al. [2009] 36

37. response surface and neural network coupling Institute for Structural Analysis Numerical realization Response surface 37

38. 2 feed forward neural networks (NN) DOE: 200 sampling points (FEM) 160 points – training of NN 40 points – test of NN mean square error: MSE p = 0.029 MSE ΔW = 0.0028 maximal error: ME p = 0.092 ME ΔW = 0.0077 x1x1 x2x2 xjxj xmxm … … 1. hidden layer 2. hidden layer output layer input layer synapses input signal output signal y=z z original z prediction test points result z NN – contact pressure ratio result z NN – strain energy amplitude Institute for Structural Analysis Numerical realization Neural networks 38