free material optimization of piezoelectric material

Piezoelectricity Free Material Optimization Results Summary

Free Material Optimization of Piezoelectric Material

Fabian Wein1, M. Stingl1

WCSMO-10Mai 19-24, 2013

1 Applied Mathematics, University Erlangen-Nuremberg, Germany

Fabian Wein Free Material Optimization of Piezoelectric Material


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 polarizationFabian Wein Free Material Optimization of Piezoelectric Material


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

cb) T < T

c

PZT cell

domains; wikipedia



Static Linear Piezoelectric Phenomenologic Continuum Model

constitutive equations and coupled FEM

σ = [c ]S− [e ]>E

D = [e ]S+ [ε]E→

(Kuu Kuφ

K>uφ−Kφφ

)(uφ

)=

(f0

)

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



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



Piezoelectric 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

− 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?



FMO Problem Formulation

min lTu 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 −ν I � 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.



Tensor Visualization similar to [Marmier et al.; 2010]

[c ] =

12.6 8.41 08.41 11.7 0

0 0 4.6

, [e ] =

0 −6.50 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) =

100

> [c ](θ)

100

, σ[e ]x (θ) =

100

> [e ](θ)

(10

), D

[ε]x . . .



Actuator Model Problem



FMO Results - Elasticity Tensor [c ]

orientational stiffness

orientational orthotropy norm



FMO Results - Piezoelectric Coupling Tensor [e ]

orientational stress coupling

orientational “zero norm”



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

couplingFabian Wein Free Material Optimization of Piezoelectric Material


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



End

note: very early steps

thank you for your attention!

I hope you found it interesting and I was in time


free material optimization of piezoelectric material

Education

homogeneous material

material ee

c e c ortho e zeros

orientation of c

bound stinesstrce c

electric displacement

state equationtrce c

tensor e33