Talk on the WCSMO-8 conference

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Motivation Model SIMP Objective Functions Numeriacal Results Conclusions
The End
Thank you for listening and for questions!
Fabian Wein Acoustic Piezoelectric Loudspeaker Optimization