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This is the extended version of a talk I held at EUROMECH 522 in 10/2011 in Erlangen/Germany

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- 1. Introduction PiezoelectricityStatic ElasticityDynamic Elasticity Conclusions Self-Penalization in Topology OptimizationFabian Wein extended version of the talk held atEUROMECH522October 10th-12th, 2011Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization

2. Introduction PiezoelectricityStatic ElasticityDynamic Elasticity Conclusions Introduction Anecdote starting piezoelectric topology optimization during my PhD thesis maximizing mean transduction w/o min compliance term w/o volume constraint vanishing optimal design vanishing design as validity test for problem formulation 10 maximization problems with linear interpolation w/o volume constraintFabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization 3. IntroductionPiezoelectricityStatic ElasticityDynamic Elasticity Conclusions Compliance Minimization solid material is the trivial optimal solution volume constraint solution is known to be gray (VTS) implicit penalization by power law and volume constraint Bendse; 1989 regularization Sigmund, Petersson; 1998 . . . . ill-posedness checkerboards mesh dependency most regularization approaches enforce some grayness Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization 4. Introduction PiezoelectricityStatic ElasticityDynamic Elasticity Conclusions Denition ersatz material topology optimization problems Denition of self-penalization linear continuous design variable only box constraints on the design variable suciently distinct black and white solution term self-penalization suggested by Ole Sigmund in communication at WCSMO-08 !!! self-penalization is an eect/ phenomenon and not a method !!!Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization 5. IntroductionPiezoelectricityStatic ElasticityDynamic Elasticity Conclusions References observable since mechanism design Sigmund; 1997 mentioned for wave guiding Sigmund, Jensen; 2003 proof extremal piezoelectric polarization Donoso, Bellido; 2008/2009 short discussion in piezoelectricity Rupp; 2009 remarks at multiphysics talks at ECCM-2010 piezoelectric self-penalization W. et al.; 2011 ...? Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization 6. Introduction PiezoelectricityStatic ElasticityDynamic Elasticity Conclusions Motivation standard penalization based on volume constraint fails when volume constraint is not active cost for volume constraint, regularization, penalization? actual weight for some applications of secondary interest self-penalization occurs why? early and basic considerations only!Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization 7. IntroductionPiezoelectricityStatic ElasticityDynamic Elasticity Conclusions Examples Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization 8. IntroductionPiezoelectricityStatic ElasticityDynamic Elasticity Conclusions Piezoelectric Topology Optimization Silva et al.; 1997 24+ Pb2+ O3Zr mechanical-electrical coupling elec. energy displ. (actuator)+ straining electric eld (sensor) ersatz material: [ cE ], [ e ], [ S ]a) T > Tc b) T < Tc Constitutive equations and FEM = [ cE ] S [ e ]T ESuu ()Ku ()u f =S D = [ e ] S + [ ]EKu ()T K () q Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization 9. IntroductionPiezoelectricity Static ElasticityDynamic Elasticity Conclusions Gedankenexperiment (Actuator) single design variable apply separately on [ cE ], [ e ] and [ S ] contradicting eects of optimal = [ cE ] S [ e ]T E grayness only for min < < maxD = [ e ] S + [ S ]E0.30 displacement in mm mech0.25elec0.20 mech+coupl+eleccoupling0.150.100.050.00 0 0.2 0.40.6 0.8 1 1.2 1.4 pseudo density > max = 1 no grayness but solid as optimal designFabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization 10. Introduction Piezoelectricity Static ElasticityDynamic Elasticity Conclusions Gedankenexperiment (Sensor) three nonlinear eects on[ cE ], [ e ] and [ S ] = [ cE ] S [ e ]T E D = [ e ] S + [ S ]E 0.030 elec. potential in V mech+coupl+elec 0.025elec 0.020 mech 0.015coupling 0.010 0.005 0.000 0 0.20.40.6 0.8 11.2 1.4pseudo density min : no grayness but void as optimal designFabian Wein (Uni-Erlangen, Germany)Self-Penalization in Topology Optimization 11. IntroductionPiezoelectricityStatic ElasticityDynamic Elasticity Conclusions Boundness model is arbitrary choice of geometry, thickness and material actuator: > 1 but ? displacement in mm100 10-1 elec. potential in V10-110-2-2-3101010-410-310-510-410-6actuator, displacement10-5 sensor, elec. potential10-710-610-810-4 10-2 100 102 104 pseudo density actuator: bounded; grayness may occur sensor: min ; no grayness expected only static problems max uz and max Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization 12. Introduction PiezoelectricityStatic ElasticityDynamic Elasticity Conclusions General Static Elastic Problem Static problem J Ke Jmin J(u());= T eue ; KT = e e uOptimal grayness {min ; max }Conditions Keue = 0 (1)e , ue =0 e e= 0 (2) B e , [ c ]B ue =0Sue= 0 (3) Se , [ c ]Sue = 0Se= 0 (4) Se , ue = 0Se ue (5) Se ue meant for vectors Se > 0 and ue > 0Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization 13. Introduction PiezoelectricityStatic ElasticityDynamic Elasticity Conclusions Again because it is easy but important! Static problem J Ke Jmin J(u());= T eue ; KT = e e uOptimal grayness {min ; max }Conditions Keue = 0 (6)e , ue =0 e e= 0 (7) B e , [ c ]B ue =0Sue= 0 (8) Se , [ c ]Sue = 0Se= 0 (9) Se , ue = 0Se ue (10) Se ue meant for vectors Se > 0 and ue > 0Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization 14. IntroductionPiezoelectricityStatic ElasticityDynamic ElasticityConclusions Force Inverter mechanism design example Sigmund; 1997 finuout min = 0.001 chosen carefullykin?kout some free gray regions (a) density (b) forward(c) adjoint Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization 15. Introduction Piezoelectricity Static ElasticityDynamic Elasticity Conclusions Analysing the Gradient Jmd Jmd (a) e < 0.00001 (b) e > 0.00001(c) | J | 0.00001 mde (a) desire for > max at support, loads, bars (b) desire for < min to improve mechanism (c) large region | J | 0.00001 out of [0.0073; 0.0734] mde mostly solid and void Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization 16. Introduction PiezoelectricityStatic ElasticityDynamic Elasticity Conclusions Identifying Actual Grayness Conditions(a) Se ue(b) strain Sue (c) strain Se only Sue = 0 and Se = 0 rigid displacementFabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization 17. IntroductionPiezoelectricity Static Elasticity Dynamic Elasticity Conclusions Varying Penalization for Static Elasticity Force Inverter with linear interpolation min is crucial void material at load points is a local optimum 2400 combinations of min and p for power law p SNOPT and SCPIP (MMA) have similar similar results grayness measure: g () = N N 4 e (1 e )1 e strong self-penalization for proper (p, min )0.02 0.9 3.5 0.8 3.50 0.7 -0.02 3.0 0.6 3.0 penalty p penalty p -0.04 0.5 2.5 -0.06 2.5 0.4 -0.08 0.3 2.0 2.0 0.2 -0.10.1 1.5 1.5 -0.12 0 1.0 -0.14 1.0-0.110-6 10-5 10-4 10-3 10-210-6 10-5 10-4 10-3 10-2lower bound minp lower bound minp (a) solution (b) objective (c) grayness Fabian Wein (Uni-Erlangen, Germany)Self-Penalization in Topology Optimization 18. IntroductionPiezoelectricityStatic ElasticityDynamic Elasticity Conclusions Dynamic Elasticity Wave guiding Sigmund, Jensen; 2003 J SJ(u()) = uT L u with= 2 Re Tu e e time-harmonic S = K + j C 2 M with C = K K + M M also self-penalization reported in Sigmund, Jensen; 2003 = 2 T SR uR T I uI T SR uI T I uR JR e R S I e ISe ee arbitrary combinations for = 0 e {min ; max }J e not much self penalization observed for wave guiding Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization 19. Introduction PiezoelectricityStatic ElasticityDynamic Elasticity Conclusions Pamping PampingCart (e , q) = q e (1 e ) M0 e penalized damping Jensen, Sigmund; 2005 vanishes at solid-void solutionS =K0 + j (K K + (M + q (1 )) M0 ) 2 M J= B R , [ c ] B uR 2 R , M0 uR K B R , [ c ] BuI e M R , M0 uI q (1 2 ) R , M0 uI + B I , [ c ] B uI 2 I , M0 uI K B I , [ c ] BuR M I , M0 uR q (1 2 ) I , M0 uRFabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization 20. IntroductionPiezoelectricityStatic ElasticityDynamic Elasticity Conclusions ?(a) wave guiding(b) no pamping(c) pamping q = 7 pamping qpamping q-20-15-10 -5 0 5 10 15 20 -20-15-10 -5 0 5 10 15 20 0.42.0q 0.4 5q objective in 101.8 Mgrayness1.6M0.31.40.31.20.21.00.20.80.20.60.124 26 28 30 32 24 26 28 30 32 damping M damping M example for xed K : varying K vs. varying q Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization 21. IntroductionPiezoelectricityStatic ElasticityDynamic Elasticity Conclusions Pamping: Discussion pamping is not self-penalization pamping might reduce objective value physical interpretation: adds dissipative material optimal design depends on q q : (q) {min ; max } Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization 22. IntroductionPiezoelectricityStatic ElasticityDynamic Elasticity Conclusions Self-Penalization Occurs! piezoelectricity counteracting eects possibly {min ; max } additional eects for dynamic case? better results (by appropriate inital designs) show stronger self-penalization static elasticity conditions for grayness unlikely to occur if, then likely no or rigid displacement mechanisms are intuitively solid-void based dynamic elasticity wave guiding: no explanation, not observed more likely to occur with mechanisms Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization 23. IntroductionPiezoelectricityStatic ElasticityDynamic Elasticity Conclusions Categories of Graynessnegligible relevance initial value due to rigid displacement insensitive to solid-void mapping remedies (post optimization) grayness constraint min vol penalty term min vol problem with argmin J constraint benecial damping corresponds to stiness, mass, damping pamping experiment: optimal damping {min ; max }benecial springs physical interpretation of grayness?... Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization 24. Introduction PiezoelectricityStatic ElasticityDynamic Elasticity Conclusions Summarizing Words self-penalization occurs for many problems this is a fact, independent if you believe it or not this presentation attempts to explain reasons self-penalization is self-penalization a phenomenon or an eect? scientic curiosity shall be reason enough to think about the problem we want to motivate you to consider your own topology optimization problem w/o volume constraints and penalization what is the dierence to the constrained solution?Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization 25. Introduction PiezoelectricityStatic ElasticityDynamic Elasticity Conclusions Where Shall it Lead To? a better understanding of self-penalization question optimality and methods of conventional solid-void solutions a collection of methods to solve problems w/o volume constraintsprojection methods (Heaviside type density lters)rigorous feature size control (MOLE constraints)pamping for dynamic systems (with care)... consider categories of grayness develop methods to avoid negligible grayness are there drawbacks in exploiting self-penalization?Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization 26. IntroductionPiezoelectricityStatic ElasticityDynamic Elasticity Conclusions The End Thank you for listening and for questions! Fabian Wein (Uni-Erlangen, Germany) Self-Penalization in Topology Optimization

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