My talk held at WCSMO-10, Mai 2013 in Orlando, Florida

1. Piezoelectricity Free Material Optimization Results Summary
Free Material Optimization of Piezoelectric Material
Fabian Wein1, M. Stingl1
WCSMO-10
Mai 19-24, 2013
1 Applied Mathematics, University Erlangen-Nuremberg, Germany
Fabian Wein Free Material Optimization of Piezoelectric Material

2. Piezoelectricity Free Material Optimization Results Summary
Piezoelectric Material: Overview
main property: convert electric energy ↔ mechanic energy
sinter sputter electrodes polarize
P P
electrodes
2D model electric excitation
standard assumption: homogeneous material with uniform polarization
Fabian Wein Free Material Optimization of Piezoelectric Material

3. Piezoelectricity Free Material Optimization Results Summary
Piezoelectric Polarization
base cell
electric neutral above temperature TC
dipole moment and deformation below TC
explains mechanical ↔ electric coupling
macroscopic view
randomly orientated domains (clusters)
electric neutral isotropic material
polarization
uniform alignment of domains
electric dipole moment
transversal isotropic (= orthotropic in 2D)
a) T > T
Pb ZrO
−
+
2+ 4+2−
3
c b) T < Tc
PZT cell
domains; wikipedia
Fabian Wein Free Material Optimization of Piezoelectric Material

4. Piezoelectricity Free Material Optimization Results Summary
Static Linear Piezoelectric Phenomenologic Continuum Model
constitutive equations and coupled FEM
σ = [c]S−[e] E
D = [e]S+[ε]E
→
Kuu Kuφ
Kuφ −Kφφ
u
φ
=
f
0
mechanic stress σ, strain S, electric displacement D, electric field E
elastic modulus [c], permittivity [ε], piezoelectric coupling [e]
mechanical displacement u, electric potential φ
stiffness matrices K∗∗, mechanical force f
Fabian Wein Free Material Optimization of Piezoelectric Material

5. Piezoelectricity Free Material Optimization Results Summary
Free Material Optimization (FMO)
general
FMO up to now applied to elasticity only
material tensors in every element are design variables
motivation
much larger design space than standard SIMP
results are generally not directly realizable
optimal solution as lower bound for realizable optimizations
. . .
technical
semi-definite optimization problem
strict feasibility not easy to maintain
Fabian Wein Free Material Optimization of Piezoelectric Material

6. Piezoelectricity Free Material Optimization Results Summary
Piezoelectric Free Material Optimization (FMO)
all tensor coefficients are design variable
[c] =


c11 c12 c13
− c22 c23
− − c33

, [e] =
e11 e13 e15
e31 e33 e35
, [ε] =
ε11 ε12
− ε22
properties
[c] and [ε] need to be symmetric positive definite
[ε] only for sensor case (mechanical excitation) relevant
questions 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 Summary
FMO Problem Formulation
min lT
u maximize compression
s.th. S u = f, coupled state equation
Tr([c]e) ≤ νc, 1 ≤ e ≤ N, bound stiffness
Tr([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 definiteness
realize positive definiteness by feasibility constraints
c11e −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 Summary
Tensor Visualization similar to [Marmier et al.; 2010]
[c] =


12.6 8.41 0
8.41 11.7 0
0 0 4.6

,[e] =


0 −6.5
0 23.3
17 0

,[ε] =
1.51 0
0 1.27
[c] [e] [ε] [c] “ortho” [e] “zeros” [ε] “ε12”
orientational stiffness
σ
[c]
x (θ) =


1
0
0

 [c](θ)


1
0
0

, σ
[e]
x (θ) =


1
0
0

 [e](θ)
1
0
, D
[ε]
x ...
Fabian Wein Free Material Optimization of Piezoelectric Material

9. Piezoelectricity Free Material Optimization Results Summary
Actuator Model Problem
Fabian Wein Free Material Optimization of Piezoelectric Material

10. Piezoelectricity Free Material Optimization Results Summary
FMO Results - Elasticity Tensor [c]
orientational stiffness
orientational orthotropy norm
Fabian Wein Free Material Optimization of Piezoelectric Material

11. Piezoelectricity Free Material Optimization Results Summary
FMO Results - Piezoelectric Coupling Tensor [e]
orientational stress coupling
orientational “zero norm”
Fabian Wein Free Material Optimization of Piezoelectric Material

12. Piezoelectricity Free Material Optimization Results Summary
Discussion of the Results
objective
optimize vertical displacement of top electrode
observations
less vertical stiffness to support compression
in coupling tensor e33 is dominant
characteristic orientational polarization
standard material classes (orthotropic)
coinciding orientation for [c] and [e]
ill-posed problem (stiffness minimization)
inhomogeneity due to boundary conditions
boundary conditions
initial deformation
elasticity
coupling
Fabian Wein Free Material Optimization of Piezoelectric Material

13. Piezoelectricity Free Material Optimization Results Summary
Lessons Learned and Motivation
lessons learned
results are plausible and to be expected
considered FMO problem is ill-posed
motivation
optimize piezoelectric devices
prescribed displacement, e.g. auxetic
use FMO to bound realizable approaches
possible realization of inhomogeneity
local optimal polarization
stiffness adaptation by doping
stochastic orientation
Jayachandran, Guedes,
Rodrigues; 2011
Fabian Wein Free Material Optimization of Piezoelectric Material

14. Piezoelectricity Free Material Optimization Results Summary
End
note: very early steps
thank you for your attention!
I hope you found it interesting and I was in time
Fabian Wein Free Material Optimization of Piezoelectric Material