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Local Optimal Polarization of Piezoelectric Material
1. 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 Piezoelectricity
30.09-02.10.2013
2. Introduction Local Optimal Polarization Numerical Examples Summary
Overview
General
linear continuum model
numerical approach based on ļ¬nite element method
PDE based optimization with high number of design variables
Optimization
optimization helps to understand systems better
manufacturability in mind
no real prototypes
3. Introduction Local Optimal Polarization Numerical Examples Summary
Structural Optimization = Topology Optimization + Material Design
Topology optimization
āwhere to put holesā/ material distribution
design of (piezoelectric) devices
macroscopic view
Material design
āassume you could have arbitrary material, what do you want?ā
realization might be another process
realizations might be metamaterials
4. Introduction Local Optimal Polarization Numerical Examples Summary
Motivation
stochastic orientation
Jayachandran, Guedes,
Rodrigues; 2011
Material Design
common homogeneous material seems to
be not optimal
Free Material Optimization ā why it does
not work
local optimal material ā new approach
5. Introduction Local Optimal Polarization Numerical Examples Summary
Standard Topology Optimization
distributes uniform polarized material/holes
āmacroscopic viewā
established in 2 1/2 dimensions (single layer)
scalar variable Ļe for each design element (= ļ¬nite element cell)
SIMP (solid isotropic material with penalization)
piezoelectric topology optimization KĀØogel, Silva; 2005
[cE
e ] = Ļe [cE
] [ee] = Ļe [e] [ĪµS
e ] = Ļe[ĪµS
], Ļe ā [Ļmin,1]
6. Introduction Local Optimal Polarization Numerical Examples Summary
Piezoelectric Free Material Optimization (FMO)
all tensor coeļ¬cients of every ļ¬nite element cell are design variable
[c] =
ļ£«
ļ£
c11 c12 c13
ā c22 c23
ā ā c33
ļ£¶
ļ£ø, [e] =
e11 e13 e15
e31 e33 e35
, [Īµ] =
Īµ11 Īµ12
ā Īµ22
properties
[c] and [Īµ] need to be symmetric positive deļ¬nite
[Īµ] only for sensor case (mechanical excitation) relevant
questions to be answered
[c] orthotropic?
[e] with only standard coeļ¬cients?
orientation of [c] and [e] coincides?
something like an optimal oriented polarization?
7. Introduction Local Optimal Polarization Numerical Examples Summary
FMO Problem Formulation (Actor)
min l u maximize compression
s.th. ĖK u = f, coupled state equation
Tr([c]e) ā¤ Ī½c, 1 ā¤ e ā¤ N, bound stiļ¬ness
Tr([c]e) ā„ Ī½c, 1 ā¤ e ā¤ N, enforce material
( [e]e 2)2
ā¤ Ī½e, 1 ā¤ e ā¤ N, bound coupling
[c]e āĪ½I 0, 1 ā¤ e ā¤ N. positive deļ¬niteness
realize positive deļ¬niteness by feasibility constraints
c11e āĪ½ ā¤ Īµ, 1 ā¤ e ā¤ N,
det2([c]e āĪ½I) ā¤ Īµ, 1 ā¤ e ā¤ N,
det3([c]e āĪ½I) ā¤ Īµ, 1 ā¤ e ā¤ N.
12. Introduction Local Optimal Polarization Numerical Examples Summary
Discussion of the FMO Results
objective
maximize vertical displacement of top electrode
observations
less vertical stiļ¬ness 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 (stiļ¬ness minimization)
inhomogeneity due to boundary conditions
boundary conditions
deformation
elasticity
coupling
13. Introduction Local Optimal Polarization Numerical Examples Summary
Electrode Design vs. Optimal Polarization
Electrode Design
pseudo polarization KĀØogel, Silva; 2005
[cE
e ] = [cE
], [ee] = [e], [ĪµS
e ] = Ļp[ĪµS
] Ļp ā [ā1,1]
(continuous) ļ¬ipping 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 āoptimal polarizationā
14. Introduction Local Optimal Polarization Numerical Examples Summary
Optimal Orientation
parametrization by design angle Īø
[cE
] = Q(Īø) [c]Q(Īø) [e] = R(Īø) [e]Q(Īø) [ĪµS
] = R(Īø) [Īµ]R(Īø)
R =
cosĪø sinĪø
āsinĪø cosĪø
Q =
ļ£«
ļ£
R2
11 R2
12 2R11 R12
R2
21 R2
22 2R21 R22
R11 R21 R12 R22 R11 R22 +R12 R21
ļ£¶
ļ£ø
concurrent orientation of all tensors
corresponds to local polarization
15. Introduction Local Optimal Polarization Numerical Examples Summary
Numerical System
linear FEM system (static)
Kuu KuĻ
KuĻ āKĻĻ
u
Ļ
=
f
ĀÆq
, short ĖKu = f
Kā assembled by local ļ¬nite element matrices Kāe
Kāe constructed by [cE
e ](Īø), [ee](Īø) and [ĪµS
e ](Īø)
f is discrete force vector, corresponding to mesh nodes.
ĀÆq from applied electric potential (inhomogeneous Dirichlet B.C.)
f = 0 for sensor, ĀÆq = 0 for actuator
16. Introduction Local Optimal Polarization Numerical Examples Summary
Function
discrete solution vector u = u1x u1y u2x u2y ...Ļ1 Ļ2 ...
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 nodes
horizontal displacement of a corner
diagonal displacement of a given region
selection of electric potential at electrode
. . .
17. Introduction Local Optimal Polarization Numerical Examples Summary
Sensitivity Analysis
the gradient vector āf
āĪø determines for every Īøe the impact on f
sensitivity analysis based on adjoint approach
f = uT
l,
āf
āĪøe
= Ī»e
āKe
āĻe
ue with Ī» solving ĖKĪ» = āl
one adjoint system ĖKĪ» = āl to be solved for every function f
āKe
āĻe
easily found by product rule
numerically very eļ¬cient, independent of number of design variables
iteratively problem solution by ļ¬rst order optimizer (SNOPT, MMA)
18. Introduction Local Optimal Polarization Numerical Examples Summary
Problem Formulation
generic problem formulation
min
Īø
l u objective function
s.th. ĖK u = f, coupled state equation
lk u ā¤ ck, 0 ā¤ k ā¤ M, arbitrary constraints
Īøe ā [ā
Ļ
2
,
Ļ
2
], 1 ā¤ Īøe ā¤ N, box constraints
for sensor and actuator problem
full material everywhere
individual polarization angle in every cell
19. Introduction Local Optimal Polarization Numerical Examples Summary
Regularization
orientational optimization in elasticity known to have local optimima
restricts local change of angle
ļ¬ltering Bruns, Tortorelli; 2001
Īøe =
ā
Ne
i=1 w(xi )Īøi
ā
Ne
i=1 w(xi )
w(xi ) = max(0,R ā|xe āxi |)
local slope constraints Petersson, Sigmund; 1998
gslope(Īø) = |< ei ,āĪø(x) >| ā¤ cs i ā {1,...,DIM}
gslope(Īøe,i) = |Īøe āĪøi | ā¤ c,
20. Introduction Local Optimal Polarization Numerical Examples Summary
Example Problems
A BC
actuator problems
maximize compression C ā
maximize compression C ā and limit A ā and B ā
twist A ā and B ā
sensor problem
maximize electric potential at C
21. Introduction Local Optimal Polarization Numerical Examples Summary
maximize compression C ā
initial |u| optimized |u|
gain: 6.1% of integrated y-displacement of C nodes
C is ļ¬attened
probably no global optimum reached
22. Introduction Local Optimal Polarization Numerical Examples Summary
maximize compression C ā and limit A ā and B ā
loss: 4.9% of integrated y-displacement of C nodes
but A and C bounded to 50 % of initial x-displacement
23. Introduction Local Optimal Polarization Numerical Examples Summary
twist A ā and B ā
note Īø ā [āĻ
2 , Ļ
2 ]
electrode design might be more eļ¬ective for this case
24. Introduction Local Optimal Polarization Numerical Examples Summary
maximize electric potential at C
gain: 0.6 % in diļ¬erence of potential
possibly due to poor local optima
25. Introduction Local Optimal Polarization Numerical Examples Summary
Coupling Tensor vs. Stiļ¬ness Tensor
what is the impact of the transversal isotropic stiļ¬ness tensor?
assume isotropic stiļ¬ness tensor
gain: 4.7 % vs. 6.1 % with PZT-5A tensors
26. Introduction Local Optimal Polarization Numerical Examples Summary
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)
27. Introduction Local Optimal Polarization Numerical Examples Summary
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 ļ¬eld
piezoelectric building blocks
. . . any suggestions?