local optimal polarization of piezoelectric material

Introduction Local Optimal Polarization Numerical Examples Summary

Local Optimal Polarization of Piezoelectric Material

Fabian Wein, M. Stingl

9th Int. Workshop on Direct and Inverse Problems in Piezoelectricity30.09-02.10.2013


Overview

General

linear continuum model

numerical approach based on finite element method

PDE based optimization with high number of design variables

Optimization

optimization helps to understand systems better

manufacturability in mind

no real prototypes


Structural Optimization = Topology Optimization + Material Design

Topology optimization

“where to put holes”/ material distribution

design of (piezoelectric) devices

macroscopic view

Material design

“assume you could have arbitrary material, what do you want?”

realization might be another process

realizations might be metamaterials


Motivation

stochastic orientationJayachandran, Guedes,Rodrigues; 2011

Material Design

common homogeneous material seems tobe not optimal

Free Material Optimization → why it doesnot work

local optimal material → new approach


Standard Topology Optimization

distributes uniform polarized material/holes

“macroscopic view”

established in 2 1/2 dimensions (single layer)

scalar variable ρe for each design element (= finite element cell)

SIMP (solid isotropic material with penalization)

piezoelectric topology optimization Kogel, Silva; 2005

[ cEe ] = ρe [cE ] [ ee ] = ρe [e ] [ εSe ] = ρe [ε

S ], ρe ∈ [ρmin,1]


Piezoelectric Free Material Optimization (FMO)

all tensor coefficients of every finite element cell 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 (Actor)

min l>u maximize compression

s.th. K 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 FMO Results

objective

maximize 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

deformation

elasticity

coupling


Electrode Design vs. Optimal Polarization

Electrode Design

pseudo polarization Kogel, Silva; 2005

[ cEe ] = [cE ], [ ee ] = [e ], [ εSe ] = ρp[ε

S ] ρp ∈ [−1,1]

(continuous) flipping of polarization (+ topology optimization)

applied on single layer piezoelectric plates

only scales polarization, does not change angle

known to result in -1 and 1 full polarization (static)

erroneously called “optimal polarization”


Optimal Orientation

parametrization by design angle θ

[ cE ] = Q(θ)>[c ]Q(θ) [ e ] = R(θ)>[e ]Q(θ) [ εS ] = R(θ)>[ε]R(θ)

R =

(cosθ sinθ

−sinθ cosθ

)Q =

R211 R2

12 2R11R12

R221 R2

22 2R21R22

R11R21 R12R22 R11R22 +R12R21

concurrent orientation of all tensors

corresponds to local polarization


Numerical System

linear FEM system (static)(Kuu Kuφ

K>uφ−Kφφ

)(uφ

)=

(fq

), short Ku = f

K∗ assembled by local finite element matrices K∗e

K∗e constructed by [ cEe ](θ), [ ee ](θ) and [ εSe ](θ)

f is discrete force vector, corresponding to mesh nodes.

q from applied electric potential (inhomogeneous Dirichlet B.C.)

f = 0 for sensor, q = 0 for actuator


Function

discrete solution vector u =(u1x u1y u2x u2y . . .φ1 φ2 . . .

)>displacement (each direction) and electric potential at mesh nodes

generic function f identifying solution

f = u>l

scalar product of solution with selection vector l = (0 . . . 1 . . .0)>

f can be maximized or used to specify a restriction

vertical displacement of all upper electrode nodeshorizontal displacement of a cornerdiagonal displacement of a given regionselection of electric potential at electrode. . .


Sensitivity Analysis

the gradient vector ∂ f∂θ

determines for every θe the impact on f

sensitivity analysis based on adjoint approach

f = uT l,∂ f

∂θe= λ>e

∂ Ke

∂ρeue with λ solving Kλ =−l

one adjoint system Kλ =−l to be solved for every function f

∂ Ke∂ρe

easily found by product rule

numerically very efficient, independent of number of design variables

iteratively problem solution by first order optimizer (SNOPT, MMA)


Problem Formulation

generic problem formulation

minθ

l>u objective function

s.th. K u = f, coupled state equation

l>k u ≤ ck , 0≤ k ≤M, arbitrary constraints

θe ∈ [−π

2,π

2], 1≤ θe ≤ N, box constraints

for sensor and actuator problem

full material everywhere

individual polarization angle in every cell


Regularization

orientational optimization in elasticity known to have local optimima

restricts local change of angle

filtering Bruns, Tortorelli; 2001

θe =∑Nei=1w(xi )θi

∑Nei=1w(xi )

w(xi ) = max(0,R−|xe −xi |)

local slope constraints Petersson, Sigmund; 1998

gslope(θ) = |< ei ,∇θ(x) >| ≤ cs i ∈ {1, . . . ,DIM}gslope(θe , i) = |θe −θi | ≤ c ,


Example Problems

A BC

actuator problems

maximize compression C ↓maximize compression C ↓ and limit A ← and B →twist A ↓ and B ↑

sensor problem

maximize electric potential at C


maximize compression C ↓

initial |u| optimized |u|

gain: 6.1% of integrated y -displacement of C nodes

C is flattened

probably no global optimum reached


maximize compression C ↓ and limit A ← and B →

loss: 4.9% of integrated y -displacement of C nodes

but A and C bounded to 50 % of initial x-displacement


twist A ↓ and B ↑

note θ ∈ [−π

2 ,π

2 ]

electrode design might be more effective for this case


maximize electric potential at C

gain: 0.6 % in difference of potential

possibly due to poor local optima


Coupling Tensor vs. Stiffness Tensor

what is the impact of the transversal isotropic stiffness tensor?

assume isotropic stiffness tensor

gain: 4.7 % vs. 6.1 % with PZT-5A tensors


Conclusion

General

local polarization works in principle

solutions might be far from global optimium

more feasible than piezoelectric Free Material Optimization

simple support would change everything

Applications

not to improve performance

exact tuning of devices

metamaterial not yet possible (e.g. auxetic material)


Future Work

Examples

dynamic problems, shift of resonance frequencies possible?

metamaterials (e.g. auxetic material)

Mathematical

novel tensor based solver

very promising for elasticity

Technical Realization

polarization by local electric field

piezoelectric building blocks

. . . any suggestions?


End

thanks for your patience :)

local optimal polarization of piezoelectric material

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