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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
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
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
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
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]
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?
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.
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 ...
Introduction Local Optimal Polarization Numerical Examples Summary
Actuator Model Problem
Introduction Local Optimal Polarization Numerical Examples Summary
FMO Results - Elasticity Tensor [c]
orientational stiffness
orientational orthotropy norm
Introduction Local Optimal Polarization Numerical Examples Summary
FMO Results - Piezoelectric Coupling Tensor [e]
orientational stress coupling
orientational “zero norm”
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
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”
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
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
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
. . .
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)
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
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,
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
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
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
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
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
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
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)
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?
Introduction Local Optimal Polarization Numerical Examples Summary
End
thanks for your patience :)

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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 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 :)