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  • 1. Piezoelectricity Free Material Optimization Results SummaryFree Material Optimization of Piezoelectric MaterialFabian Wein1, M. Stingl1WCSMO-10Mai 19-24, 20131 Applied Mathematics, University Erlangen-Nuremberg, GermanyFabian Wein Free Material Optimization of Piezoelectric Material
  • 2. Piezoelectricity Free Material Optimization Results SummaryPiezoelectric Material: Overviewmain property: convert electric energy ↔ mechanic energysinter sputter electrodes polarizeP Pelectrodes2D model electric excitationstandard assumption: homogeneous material with uniform polarizationFabian Wein Free Material Optimization of Piezoelectric Material
  • 3. Piezoelectricity Free Material Optimization Results SummaryPiezoelectric Polarizationbase cellelectric neutral above temperature TCdipole moment and deformation below TCexplains mechanical ↔ electric couplingmacroscopic viewrandomly orientated domains (clusters)electric neutral isotropic materialpolarizationuniform alignment of domainselectric dipole momenttransversal isotropic (= orthotropic in 2D)a) T > TPb ZrO−+2+ 4+2−3c b) T < TcPZT celldomains; wikipediaFabian Wein Free Material Optimization of Piezoelectric Material
  • 4. Piezoelectricity Free Material Optimization Results SummaryStatic Linear Piezoelectric Phenomenologic Continuum Modelconstitutive equations and coupled FEMσ = [c]S−[e] ED = [e]S+[ε]E→Kuu KuφKuφ −Kφφuφ=f0mechanic stress σ, strain S, electric displacement D, electric field Eelastic modulus [c], permittivity [ε], piezoelectric coupling [e]mechanical displacement u, electric potential φstiffness matrices K∗∗, mechanical force fFabian Wein Free Material Optimization of Piezoelectric Material
  • 5. Piezoelectricity Free Material Optimization Results SummaryFree Material Optimization (FMO)generalFMO up to now applied to elasticity onlymaterial tensors in every element are design variablesmotivationmuch larger design space than standard SIMPresults are generally not directly realizableoptimal solution as lower bound for realizable optimizations. . .technicalsemi-definite optimization problemstrict feasibility not easy to maintainFabian Wein Free Material Optimization of Piezoelectric Material
  • 6. Piezoelectricity Free Material Optimization Results SummaryPiezoelectric Free Material Optimization (FMO)all tensor coefficients are design variable[c] =c11 c12 c13− c22 c23− − c33, [e] =e11 e13 e15e31 e33 e35, [ε] =ε11 ε12− ε22properties[c] and [ε] need to be symmetric positive definite[ε] only for sensor case (mechanical excitation) relevantquestions to be answered[c] orthotropic?[e] with only standard coefficients?orientation of [c] and [e] coincides?something like an optimal oriented polarization?Fabian Wein Free Material Optimization of Piezoelectric Material
  • 7. Piezoelectricity Free Material Optimization Results SummaryFMO Problem Formulationmin lTu maximize compressions.th. S u = f, coupled state equationTr([c]e) ≤ νc, 1 ≤ e ≤ N, bound stiffnessTr([c]e) ≥ νc, 1 ≤ e ≤ N, enforce material( [e]e 2)2≤ νe, 1 ≤ e ≤ N, bound coupling[c]e −νI 0, 1 ≤ e ≤ N. positive definitenessrealize positive definiteness by feasibility constraintsc11e −ν ≤ ε, 1 ≤ e ≤ N,det2([c]e −νI) ≤ ε, 1 ≤ e ≤ N,det3([c]e −νI) ≤ ε, 1 ≤ e ≤ N.Fabian Wein Free Material Optimization of Piezoelectric Material
  • 8. Piezoelectricity Free Material Optimization Results SummaryTensor Visualization similar to [Marmier et al.; 2010][c] =12.6 8.41 08.41 11.7 00 0 4.6,[e] =0 −6.50 23.317 0,[ε] =1.51 00 1.27[c] [e] [ε] [c] “ortho” [e] “zeros” [ε] “ε12”orientational stiffnessσ[c]x (θ) =100 [c](θ)100, σ[e]x (θ) =100 [e](θ)10, D[ε]x ...Fabian Wein Free Material Optimization of Piezoelectric Material
  • 9. Piezoelectricity Free Material Optimization Results SummaryActuator Model ProblemFabian Wein Free Material Optimization of Piezoelectric Material
  • 10. Piezoelectricity Free Material Optimization Results SummaryFMO Results - Elasticity Tensor [c]orientational stiffnessorientational orthotropy normFabian Wein Free Material Optimization of Piezoelectric Material
  • 11. Piezoelectricity Free Material Optimization Results SummaryFMO Results - Piezoelectric Coupling Tensor [e]orientational stress couplingorientational “zero norm”Fabian Wein Free Material Optimization of Piezoelectric Material
  • 12. Piezoelectricity Free Material Optimization Results SummaryDiscussion of the Resultsobjectiveoptimize vertical displacement of top electrodeobservationsless vertical stiffness to support compressionin coupling tensor e33 is dominantcharacteristic orientational polarizationstandard material classes (orthotropic)coinciding orientation for [c] and [e]ill-posed problem (stiffness minimization)inhomogeneity due to boundary conditionsboundary conditionsinitial deformationelasticitycouplingFabian Wein Free Material Optimization of Piezoelectric Material
  • 13. Piezoelectricity Free Material Optimization Results SummaryLessons Learned and Motivationlessons learnedresults are plausible and to be expectedconsidered FMO problem is ill-posedmotivationoptimize piezoelectric devicesprescribed displacement, e.g. auxeticuse FMO to bound realizable approachespossible realization of inhomogeneitylocal optimal polarizationstiffness adaptation by dopingstochastic orientationJayachandran, Guedes,Rodrigues; 2011Fabian Wein Free Material Optimization of Piezoelectric Material
  • 14. Piezoelectricity Free Material Optimization Results SummaryEndnote: very early stepsthank you for your attention!I hope you found it interesting and I was in timeFabian Wein Free Material Optimization of Piezoelectric Material