Acoustic near field topology optimization of a piezoelectric loudspeaker
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