1. Model Concurrency Topology Optimization Numerical Results Conclusions
Acoustic near field topology optimization of a
piezoelectric loudspeaker
F. Wein, M. Kaltenbacher, E. B¨nsch, G. Leugering, F. Schury
a
ECCM-2010
20th May 2010
Fabian Wein (Uni-Erlangen, Germany) Acoustic near field topology optimization

2. Model Concurrency Topology Optimization Numerical Results Conclusions
Piezoelectric-Mechanical Laminate
Bending due to inverse piezoelectric effect
Piezoelectric layer: PZT-5A, 5 cm×5 cm, 50 µm thick, ideal electrodes
Mechanical layer: Aluminum, 5 cm×5 cm, 100 µm thick, no glue layer
Fabian Wein (Uni-Erlangen, Germany) Acoustic near field topology optimization

3. Model Concurrency Topology Optimization Numerical Results Conclusions
Coupling to Acoustic Domain
• Discretization of Ωair determined by acoustic wave length λac
• Discretization of Ωpiezo / Ωplate determined by optimization
• Non-matching grids Ωplate → Ωair to solve scale problem
Fabian Wein (Uni-Erlangen, Germany) Acoustic near field topology optimization

4. Model Concurrency Topology Optimization Numerical 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
Fabian Wein (Uni-Erlangen, Germany) Acoustic near field topology optimization

5. Model Concurrency Topology Optimization Numerical Results Conclusions
Structural Resonance
• Resonance is relevant for any maximization
• Piezoelectric-mechanical eigenfrequency analysis
(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
Fabian Wein (Uni-Erlangen, Germany) Acoustic near field topology optimization

6. Model Concurrency Topology Optimization Numerical Results Conclusions
Strain Cancellation
Linear Piezoelectricity: [σ] = [cE ][S] − [e0 ]T E
0
S
D = [e0 ][S] + [ 0 ]E
(a) First mode w/o electrodes (b) First mode with electrodes
(c) Higher mode w/o electrodes (d) Higher mode with electrodes
• Most structural resonance modes have strain cancellation
• No piezoelectric excitation of these vibrational patterns
Fabian Wein (Uni-Erlangen, Germany) Acoustic near field topology optimization

7. Model Concurrency Topology Optimization Numerical Results Conclusions
Acoustic Short Circuit
• “Elimination of sound radiation by out of phase sources”
• Most structural resonance modes are out of phase
• Strain cancelling patterns are out of phase
Fabian Wein (Uni-Erlangen, Germany) Acoustic near field topology optimization

8. Model Concurrency Topology Optimization Numerical Results Conclusions
Solid Isotropic Material with Penalization
• Fully coupled piezoelectric-mechanical-acoustic FEM system
• Replace piezoelectric material constants: Silva, Kikuchi; 1999
[cE ] = ρe [cE ],
e ρm = ρe ρm ,
e [ee ] = ρe [e], [εS ] = ρe [εS ]
e
• Harmonic excitation: S(ω) = K + jω(αK K + αM M) − ω 2 M
• 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)
˜
• Short form: S u = f
Fabian Wein (Uni-Erlangen, Germany) Acoustic near field topology optimization

9. Model Concurrency Topology Optimization Numerical Results Conclusions
Sound Power
1 ∗
Sound Power Pac = {p vn } dΓ
2 Γopt
• Sound pressure p = ρf ψ˙
• Particle velocity v = − ψ = u; vn = − n ψ = un on Γopt
˙ ˙
• Acoustic potential ψ solves the acoustic wave equation
• Acoustic impedance Z (x) = p(x)/vn (x)
• Objective functions are proportional with negative sign
Fabian Wein (Uni-Erlangen, Germany) Acoustic near field topology optimization

10. Model Concurrency Topology Optimization Numerical Results Conclusions
1 ∗
Objective Functions for Pac = 2 Γopt {p vn } dΓ
Comparison: Wein et al.; 2009; WCSMO-08
Structural approximation
• Assume Z constant on Γiface : vn = j ωun and p = Z vn
• Jst = ω 2 um T L u∗
m
• ≈ Du, Olhoff; 2007, framework: Sigmund, Jensen; 2003
• Creation of resonance patterns: Wein et. al.; 2009
• Ignores acoustic short circuits
Acoustic far field optimization
• Assume Z constant on Γopt : vn = p/Z and p = j ω ρf ψ
• Jff = ω 2 ψ T L ψ ∗
• ≈ D¨hring, Jensen, Sigmund; 2008
u
• Uncertainty on accuracy
Fabian Wein (Uni-Erlangen, Germany) Acoustic near field topology optimization

11. Model Concurrency Topology Optimization Numerical Results Conclusions
Acoustic Near Field Optimization
Continuous Problem: Pac = 1 ∗
{p vn } dΓ
2 Γopt
• Reformulate: vn = − n ψ and p = j ω ρf ψ
• Jnf = {j ωψ T L n ψ ∗ }
• Interpret n operator as constant matrix combined with L
• Jnf = {j ωψ T Q ψ ∗ }
˜
• Sensitivity: ∂Jnf = 2 {λT ∂ S u}
b
∂ρ ∂ρ
˜
• Adjoint problem: S λ = −j ω (QT − Q)T u
• ≈ Jensen, Sigmund; 2005 and Jensen; 2007
Fabian Wein (Uni-Erlangen, Germany) Acoustic near field topology optimization

12. Model Concurrency Topology Optimization Numerical Results Conclusions
Full Plate Evaluation: |Ωair | = 20 cm
104
Jnf
103 c Jff
102
Objective
101
0
10
-1
10
-2
10
10-3
0 500 1000 1500 2000
Target Frequency (Hz)
• Frequency response for full plate with large acoustic domain
• Grey bars represent structural eigenfrequencies
• Most eigenmodes cannot be excited piezoelectrically
• Good far field approximation with 20 cm
Fabian Wein (Uni-Erlangen, Germany) Acoustic near field topology optimization

13. Model Concurrency Topology Optimization Numerical Results Conclusions
Full Plate Evaluation: |Ωair | = 6 cm
104
Jnf
103 c Jff
102
Objective
101
0
10
-1
10
10-2
10-3
0 500 1000 1500 2000
Target Frequency (Hz)
• Frequency response for full plate with small acoustic domain
• Jff resolves acoustic short circuit inexact
• Jff does not resolve negative Pac
• Negative Pac indicates too small acoustic domain
• Note: Γopt is top surface of Ωair
Fabian Wein (Uni-Erlangen, Germany) Acoustic near field topology optimization

14. Model Concurrency Topology Optimization Numerical Results Conclusions
Topology Optimization: |Ωair | = 6 cm
• Several hundred mono-frequent optimizations!
• Max iterations: 250, SCPIP/MMA, generally no KKT reached
• Starting from full plate
4
103
102
101
Objective
100
10
10-1
10-2 c Pac(Jff)
10-3 Jnf
10-4 full plate sweep
10-5
0 500 1000 1500 2000
Target Frequency (Hz)
• Similar results for Jnf and Jff
• No reliable generation of resonating structures
Fabian Wein (Uni-Erlangen, Germany) Acoustic near field topology optimization

15. Model Concurrency Topology Optimization Numerical Results Conclusions
Selected Results
(a) 550 Hz (b) 560 Hz (c) 980 Hz (d) 1510 Hz
4
103
102
101
Objective
100
10
10-1
10-2
-3
c Pac(Jff)
10-4 Jnf
10-5 full plate sweep
10
0 500 1000 1500 2000
Target Frequency (Hz)
• Strain cancellation and acoustic short circuits handled
• Self-penalization for ρ1 , no regularization, no constraints, . . .
Fabian Wein (Uni-Erlangen, Germany) Acoustic near field topology optimization

16. Model Concurrency Topology Optimization Numerical Results Conclusions
Topology Optimization Starting From Previous Result
• Start max Jnf (fi ) from left/right result arg max Jnf (fi k )
4
103
102
101
Objective
100
10
10-1
10-2 Jnf(from left)
10-3 Jnf(from right)
10-4 full plate sweep
10-5
0 500 1000 1500 2000
Target Frequency (Hz)
• Blocked by resonances → D¨hring, Jensen, Sigmund; 2008
u
Fabian Wein (Uni-Erlangen, Germany) Acoustic near field topology optimization

17. Model Concurrency Topology Optimization Numerical Results Conclusions
Interpolated Eigenmodes as Initial Designs
• Good optimal results reflect eigenmode vibrational patterns
• These patterns are hard to reach from full plate
• Interpolate ρ from positive real u of lower/ upper eigenmode
?
104
103
102
101
Objective
100
10-1
10-2
10-3 Jnf
10-4 full plate sweep
10-5
0 500 1000 1500 2000
Target Frequency (Hz)
Fabian Wein (Uni-Erlangen, Germany) Acoustic near field topology optimization

18. Model Concurrency Topology Optimization Numerical Results Conclusions
Conclusions
• We introduced acoustic near field optimization
• Surprisingly good results for “old” far field optimization
• Promising construction of start design from eigenfrequency
analysis
• Self-penalization: no regularization, constraints, (mesh
depenency) . . .
• Based on CFS++ (M. Kaltenbacher ) using SCPIP (Ch.
Zillober )
Thank you very much for your attention!
Fabian Wein (Uni-Erlangen, Germany) Acoustic near field topology optimization

19. Model Concurrency Topology Optimization Numerical Results Conclusions
Self-Penalization
• Piezoelectric setup often shows self-penalization
1 1
Volume
0.8 Greyness 0.8
Greyness
Volume
0.6 0.6
0.4 0.4
0.2 0.2
0 0
0 500 1000 1500 2000
Target Frequency (Hz)
• For most frequencies sufficient self-penalization
• Not as distinct as in structural optimization
• Stronger self-penalization for “global optima”
Fabian Wein (Uni-Erlangen, Germany) Acoustic near field topology optimization

20. Model Concurrency Topology Optimization Numerical Results Conclusions
Coupling to Acoustic Domain - cont.
• Acoustic wave length: λair = f /cair with cair = 343 m/s
• Discretization: hac ≤ λair /10 for 2nd order FEM elements
• Acoustic domain: 6 × 6 × 6 cm3 plus PML layer
Frequency wave length hac |Ωair |/λ
2300 Hz 15 cm 1.5 cm 0.4
1000 Hz 34 cm 3.4 cm 0.18
330 Hz 1m 10.4 cm 0.058
100 Hz 3.4 m 34 cm 0.018
• Plate surface: 5 × 5 cm2 by 30 × 30 elem. with hst = 1.7 mm
• Non-matching grids Ωplate → Ωair to solve scale problem
Fabian Wein (Uni-Erlangen, Germany) Acoustic near field topology optimization

21. Model Concurrency Topology Optimization Numerical Results Conclusions
Experimental Prototype (200 µm Piezoceramic)
(a) Original (b) Sputter (c) Lasing
(d) Temper (e) Polarize (f) Prototype
Fabian Wein (Uni-Erlangen, Germany) Acoustic near field topology optimization