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# Local Optimal Polarization of Piezoelectric Material

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

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### Local Optimal Polarization of Piezoelectric Material

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