Motivation         Model    SIMP         Objective Functions         Numeriacal Results           Conclusions




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Motivation         Model     SIMP         Objective Functions         Numeriacal Results           Conclusions



 Motivat...
Motivation     Model       SIMP         Objective Functions         Numeriacal Results           Conclusions



 Piezoelec...
Motivation         Model     SIMP         Objective Functions         Numeriacal Results           Conclusions



 Couplin...
Motivation    Model     SIMP         Objective Functions         Numeriacal Results           Conclusions



 Coupled Piez...
Motivation         Model       SIMP          Objective Functions         Numeriacal Results           Conclusions



 Soli...
Motivation         Model          SIMP         Objective Functions         Numeriacal Results           Conclusions



 Li...
Motivation         Model     SIMP          Objective Functions         Numeriacal Results           Conclusions



 Sound ...
Motivation            Model   SIMP         Objective Functions         Numeriacal Results           Conclusions



 Optimi...
Motivation         Model     SIMP         Objective Functions         Numeriacal Results           Conclusions



 Eigenfr...
Motivation                                 Model   SIMP         Objective Functions         Numeriacal Results           C...
Motivation                              Model   SIMP         Objective Functions         Numeriacal Results            Con...
Motivation                              Model   SIMP         Objective Functions         Numeriacal Results            Con...
Motivation          Model            SIMP               Objective Functions              Numeriacal Results          Concl...
Motivation        Model   SIMP         Objective Functions         Numeriacal Results           Conclusions



 Selected R...
Motivation   Model       SIMP         Objective Functions         Numeriacal Results           Conclusions



 Experimenta...
Motivation         Model     SIMP         Objective Functions         Numeriacal Results           Conclusions



 Conclus...
Motivation   Model         SIMP         Objective Functions         Numeriacal Results           Conclusions



 The End

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Topology Optimization of a Piezoelectric Loudspeaker Coupled with the Acoustic Domain

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Topology Optimization of a Piezoelectric Loudspeaker Coupled with the Acoustic Domain

  1. 1. Motivation Model SIMP Objective Functions Numeriacal Results Conclusions Topology Optimization of a Piezoelectric Loudspeaker Coupled with the Acoustic Domain F. Wein, M. Kaltenbacher, F. Schury, E. B¨nsch, G. Leugering a WCSMO-8 03. June 2009 Fabian Wein Acoustic Piezoelectric Loudspeaker Optimization
  2. 2. Motivation Model SIMP Objective Functions Numeriacal Results Conclusions Motivation Overview • Ersatz material/ “SIMP” optimization • Piezoelectric-mechanical laminate coupled with acoustics • Piezoelectric layer is design domain Topics • Acoustic optimization vs. structural approximation • Self-penalized piezoelectric optimization without . . . • penalization (e.g. RAMP) • volume constraint • regularization (filtering) Fabian Wein Acoustic Piezoelectric Loudspeaker Optimization
  3. 3. Motivation Model SIMP Objective Functions Numeriacal Results Conclusions Piezoelectric-Mechanical Laminate Bending due to inverse piezoelectric effect Piezoelectric layer: PZT-5A, 5 cm×5 cm, 50 µm thick, ideal electrodes Mechanicial layer: Aluminum, 5 cm×5 cm, 100 µm thick, no glue layer Fabian Wein Acoustic Piezoelectric Loudspeaker Optimization
  4. 4. Motivation Model SIMP Objective Functions Numeriacal Results Conclusions Coupling to Acoustic Domain • Discretization of Ωair via cair = 343 m/s = λac f • Dimension of Ωair determined by acoustic far field • Fine discretization of Ωpiezo and Ωplate • Non-matching grids Ωplate → Ωair-fine and Ωair-fine → Ωair-coarse Fabian Wein Acoustic Piezoelectric Loudspeaker Optimization
  5. 5. Motivation Model SIMP Objective Functions Numeriacal Results Conclusions Coupled Piezoelectric-Mechanical-Acoustic PDEs PDEs: ρm u − B T [cE ]Bu + [e]T φ ¨ = 0 in Ωpiezo B T [e]Bu − [ S ] φ = 0 in Ωpiezo ρm u − B T [c]Bu = 0 in Ωplate ¨ 1 ¨ ψ − ∆ψ = 0 in Ωair c2 1 ¨ ψ − A2 ψ = 0 in ΩPML c2 ∂ψ Interface conditions: n · u = − ˙ on Γiface × (0, T ) ∂n σn ˙ = −n ρf ψ on Γiface × (0, T ) Full 3D FEM formulation (quadratic / linear with softening) Fabian Wein Acoustic Piezoelectric Loudspeaker Optimization
  6. 6. Motivation Model SIMP Objective Functions Numeriacal Results Conclusions Solid Isotropic Material with Penalization • Sample reference: K¨gl, Silva; 2005 o • Design vector ρ (pseudo density ) • ρ = (ρ1 . . . ρne )T is piecewise constant within elements • ρe ∈ [ρmin : 1] from void (ρmin ) to full (1) material • Replace piezoelectric material constants: [cE ] = ρe [cE ], e ρm = ρe ρm , e [ee ] = ρe [e], [εS ] = ρe [εS ] e • Intermediate values are non-physical! • Small ρ → low stiffness • Large ρ → high piezoelectric coupling Fabian Wein Acoustic Piezoelectric Loudspeaker Optimization
  7. 7. Motivation Model SIMP Objective Functions Numeriacal Results Conclusions Linear Systems • Piezoelectric-mechanical-acoustic coupling ¯   Sψ ψ Cψ um 0  ¯ 0   0 CT Sum um Sum up (ρ)  ψ(ρ) 0  ψ um  um (ρ)  0   T  =   0  Sum up (ρ) Sup up (ρ) Kup φ (ρ)   up (ρ)   0   T φ(ρ) ¯ qφ 0 0 Kup φ (ρ) −Kφ φ (ρ) • Piezoelectric-mechanical laminate  Su m u m˜ Sum up (ρ) 0     um (ρ) 0 ˜T ˜ up up (ρ) Ku φ (ρ)   up (ρ)  =  0  ˜ Sum up (ρ) S p  0 ˜ T φ (ρ) −Kφφ (ρ) Kup ˜ φ(ρ) ¯ qφ • Harmonic excitation: S(ω) = K + jω(αK K + αM M) − ω 2 M ˜ • Short form: S u = f Fabian Wein Acoustic Piezoelectric Loudspeaker Optimization
  8. 8. Motivation Model SIMP Objective Functions Numeriacal Results Conclusions Sound Power • What one wants to optimize 1 ∗ Pac = Re{p vn } dΓ 2 Γopt Sound power Pac , pressure p, normal particle velocity vn • Acoustic impedance Z p(x) Z (x) = vn (x) • Z can bee assumed to be constant on Γopt • For plane waves • In the acoustic far field (some wavelengths) Fabian Wein Acoustic Piezoelectric Loudspeaker Optimization
  9. 9. Motivation Model SIMP Objective Functions Numeriacal Results Conclusions Optimizing for the Sound Power Use ∗ ¯ Re{p vn } dΓ with Z = p/vn and u = (ψ um up φ)T Γopt Optimization framework • J = uT L u∗ with L selecting from u ≈ Sigmund, Jensen; 2003 ∂J ∂S = 2 Re{λT u} where λ solves ST λ = −L u∗ ∂ρe ∂ρe Acoustic optimization assume Z constant on Γopt ˙ • Jac = ω 2 ψ T L ψ ∗ with vn = p/Z and p = ρf ψ • ≈ D¨hring, Jensen, Sigmund; 2008 u Structural optimization assume Z constant on Γiface • Jst = ω 2 um T L u∗ with p = vn Z and v = u m ˙ • ≈ Du, Olhoff; 2007 (approximation is discussed!) Fabian Wein Acoustic Piezoelectric Loudspeaker Optimization
  10. 10. Motivation Model SIMP Objective Functions Numeriacal Results Conclusions Eigenfrequency Analysis We want resonating structures, so consider eigenmodes (a) 1. mode (b) 2./3. m (c) 4. mode (d) 5. mode (e) 6. mode (f) 7./8. m (g) 9./10. m (h) 11. mode • Only (a) and (e) can be excited electrically • Acoustic short circuits for all patterns but (a) and (e) Fabian Wein Acoustic Piezoelectric Loudspeaker Optimization
  11. 11. Motivation Model SIMP Objective Functions Numeriacal Results Conclusions Structural Optimization • Objective function Jst = ω 2 um T L u∗ m • Several hundred mono-frequent optimizations Objective Function (m /s ) 2 2 5 10 4 10 103 2 10 1 10 Jst (max Jst) Jst (max Jac) 100 Jst (Sweep full plate) 10-1 0 500 1000 1500 2000 Target Frequency (Hz) • Robust shifting/ creation of resonances • KKT-condition often not reached Fabian Wein Acoustic Piezoelectric Loudspeaker Optimization
  12. 12. Motivation Model SIMP Objective Functions Numeriacal Results Conclusions Validate Acoustic Approximation • Validate Jac = ω 2 ψ T L ψ ∗ against Γ Re{p vn } dΓ ∗ opt • Use one mesh (23 cm height) Objective Function (Pa2) 7 0 Acoustic Power (W) 10 10 106 10-1 105 10-2 104 10-3 103 10-4 102 10-5 101 Jac (Sweep full plate) 10-6 100 Pac (Sweep full plate) 10-7 10-1 10-8 0 500 1000 1500 2000 Target Frequency (Hz) • Good approximation above 1000 Hz (0.64 wave length) • Two antiresonances (acoustic short circuits) • Approximation detects only one antiresonance! Fabian Wein Acoustic Piezoelectric Loudspeaker Optimization
  13. 13. Motivation Model SIMP Objective Functions Numeriacal Results Conclusions Acoustic Optimization • Objective function Jac = ω 2 ψ T L ψ ∗ Objective Function (Pa2) 7 0 Acoustic Power (W) 10 10 106 10-1 105 10 -2 104 10-3 3 10 -4 102 Jac (max Jac) 10 101 Pac (max Jst) 10-5 100 Jac (Sweep full plate) 10-6 10-1 10-7 0 500 1000 1500 2000 Target Frequency (Hz) • Jac avoids acoustic short circuits • Structural approximation unfeasible (above first mode) • Note Jac approximation unfeasible below 1000 Hz Fabian Wein Acoustic Piezoelectric Loudspeaker Optimization
  14. 14. Motivation Model SIMP Objective Functions Numeriacal Results Conclusions Grayness and Volumes • Remember No penalization, no volume constraint • Normalized grayness: g (ρ) = 4 Ω (1 − ρ) ρ dx, 0.3 1 g(Jst) Vol(Jst) g(Jac) 0.9 Vol(Jac) 0.25 0.8 0.2 0.7 0.15 0.6 0.5 0.1 0.4 0.05 0.3 0 0.2 0 500 1000 1500 2000 0 500 1000 1500 2000 Grayness over Target Frequenzy (Hz) Volume over Target Frequenzy (Hz) • Good avoidance of grayness • Optimal volumes 25% . . . 100% Fabian Wein Acoustic Piezoelectric Loudspeaker Optimization
  15. 15. Motivation Model SIMP Objective Functions Numeriacal Results Conclusions Selected Results (a) 565 Hz (b) 575 Hz (c) 875 Hz (d) 975 Hz (e) 985 Hz (f) 1405 Hz (g) 1585 Hz (h) 2160 Hz Legend: black = full material; white = void material Fabian Wein Acoustic Piezoelectric Loudspeaker Optimization
  16. 16. Motivation Model SIMP Objective Functions Numeriacal Results Conclusions Experimental Prototype (200 µm Piezoceramic) (a) Original (b) Sputter (c) Lasing (d) Temper (e) Polarize (f) Prototype Fabian Wein Acoustic Piezoelectric Loudspeaker Optimization
  17. 17. Motivation Model SIMP Objective Functions Numeriacal Results Conclusions Conclusions and Future Work Conclusions • Structural approximation does not handle acoustic short circuits • Piezoelectric-mechanical laminate allows optimization . . . • without penalization and regularization • without constraints (especially volume) Future Work • Use Taylor-Hood elements to compute p and v simultanuously • Find suitable structural approximation • Create more resonance patterns Fabian Wein Acoustic Piezoelectric Loudspeaker Optimization
  18. 18. Motivation Model SIMP Objective Functions Numeriacal Results Conclusions The End Thank you for listening and for questions! Fabian Wein Acoustic Piezoelectric Loudspeaker Optimization

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