My presentation at the 9th Int. Workshop on Direct and Inverse Problems in Piezoelectricity in Weimar (30.09-02.10.2013).

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 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

3. Introduction Local Optimal Polarization Numerical Examples Summary
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

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 → why it does
not work
local optimal material → new approach

5. Introduction Local Optimal Polarization Numerical Examples Summary
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 K¨ogel, Silva; 2005
[cE
e ] = ρe [cE
] [ee] = ρe [e] [εS
e ] = ρe[εS
], ρe ∈ [ρmin,1]

6. Introduction Local Optimal Polarization Numerical Examples Summary
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 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?

7. Introduction Local Optimal Polarization Numerical Examples Summary
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 −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.

8. Introduction Local Optimal Polarization Numerical Examples 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 ...

9. Introduction Local Optimal Polarization Numerical Examples Summary
Actuator Model Problem

10. Introduction Local Optimal Polarization Numerical Examples Summary
FMO Results - Elasticity Tensor [c]
orientational stiffness
orientational orthotropy norm

11. Introduction Local Optimal Polarization Numerical Examples Summary
FMO Results - Piezoelectric Coupling Tensor [e]
orientational stress coupling
orientational “zero norm”

12. Introduction Local Optimal Polarization Numerical Examples Summary
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

13. Introduction Local Optimal Polarization Numerical Examples Summary
Electrode Design vs. Optimal Polarization
Electrode Design
pseudo polarization K¨ogel, Silva; 2005
[cE
e ] = [cE
], [ee] = [e], [εS
e ] = ρp[εS
] 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”

14. Introduction Local Optimal Polarization Numerical Examples Summary
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 =


R2
11 R2
12 2R11 R12
R2
21 R2
22 2R21 R22
R11 R21 R12 R22 R11 R22 +R12 R21


concurrent orientation of all tensors
corresponds to local polarization

15. Introduction Local Optimal Polarization Numerical Examples Summary
Numerical System
linear FEM system (static)
Kuu Kuφ
Kuφ −Kφφ
u
φ
=
f
¯q
, short ˜Ku = f
K∗ assembled by local finite element matrices K∗e
K∗e constructed by [cE
e ](θ), [ee](θ) and [εS
e ](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

16. Introduction Local Optimal Polarization Numerical Examples Summary
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 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 ∂f
∂θ determines for every θe the impact on f
sensitivity analysis based on adjoint approach
f = uT
l,
∂f
∂θe
= λe
∂Ke
∂ρe
ue 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)

18. Introduction Local Optimal Polarization Numerical Examples Summary
Problem Formulation
generic problem formulation
min
θ
l u objective function
s.th. ˜K u = f, coupled state equation
lk 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

19. Introduction Local Optimal Polarization Numerical Examples Summary
Regularization
orientational optimization in elasticity known to have local optimima
restricts local change of angle
filtering Bruns, Tortorelli; 2001
θe =
∑
Ne
i=1 w(xi )θi
∑
Ne
i=1 w(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,

20. Introduction Local Optimal Polarization Numerical Examples Summary
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

21. Introduction Local Optimal Polarization Numerical Examples Summary
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

22. Introduction Local Optimal Polarization Numerical Examples Summary
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

23. Introduction Local Optimal Polarization Numerical Examples Summary
twist A ↓ and B ↑
note θ ∈ [−π
2 , π
2 ]
electrode design might be more effective for this case

24. Introduction Local Optimal Polarization Numerical Examples Summary
maximize electric potential at C
gain: 0.6 % in difference of potential
possibly due to poor local optima

25. Introduction Local Optimal Polarization Numerical Examples Summary
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

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 field
piezoelectric building blocks
. . . any suggestions?

28. Introduction Local Optimal Polarization Numerical Examples Summary
End
thanks for your patience :)