Model Concurrency Topology Optimization Numerical Results Conclusions
Acoustic near field topology optimization of apiezoelectric loudspeaker
F. Wein, M. Kaltenbacher, E. Bansch, G. Leugering, F. Schury
ECCM-201020th May 2010
Fabian Wein (Uni-Erlangen, Germany) Acoustic near field topology optimization
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
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
Model Concurrency Topology Optimization Numerical Results Conclusions
Coupled Piezoelectric-Mechanical-Acoustic PDEs
PDEs: ρmu− BT(
[cE ]Bu + [e]T∇φ)
= 0 in Ωpiezo
BT(
[e]Bu− [εS ]∇φ)
= 0 in Ωpiezo
ρmu− BT [c]Bu = 0 in Ωplate
1
c2ψ −∆ψ = 0 in Ωair
1
c2ψ −A2 ψ = 0 in ΩPML
Interface conditions: n · u = −∂ψ∂n
on Γiface × (0,T )
σn = −n ρf ψ on Γiface × (0,T )
Full 3D FEM formulationFabian Wein (Uni-Erlangen, Germany) Acoustic near field topology optimization
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
Model Concurrency Topology Optimization Numerical Results Conclusions
Strain Cancellation
Linear Piezoelectricity: [σ] = [cE0 ][S]− [e0]T E
D = [e0][S] + [εS0 ]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
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
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
[cEe ] = ρe [cE ], ρm
e = ρeρm, [ee ] = ρe [e], [εS
e ] = ρe [εS ]
• Harmonic excitation: S(ω) = K + jω(αKK + αMM)− ω2M
• Piezoelectric-mechanical-acoustic couplingSψ ψ Cψ um 0 0
CTψ um
Sumum Sumup(ρ) 0
0 ST
umup(ρ) Supup(ρ) Kupφ(ρ)
0 0 KT
upφ(ρ) −Kφφ(ρ)
ψ(ρ)um(ρ)up(ρ)φ(ρ)
=
000
qφ
• Short form: S u = f
Fabian Wein (Uni-Erlangen, Germany) Acoustic near field topology optimization
Model Concurrency Topology Optimization Numerical Results Conclusions
Sound Power
Sound Power Pac =1
2
∫Γopt
<p v∗n dΓ
• Sound pressure p = ρf ψ
• Particle velocity v = −∇ψ = u; vn = −∇nψ = 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
Model Concurrency Topology Optimization Numerical Results Conclusions
Objective Functions for Pac = 12
∫Γopt<p v ∗n dΓ
Comparison: Wein et al.; 2009; WCSMO-08Structural approximation
• Assume Z constant on Γiface: vn = j ωun and p = Z vn
• Jst = ω2umT 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 ρfψ
• Jff = ω2ψT Lψ∗
• ≈ Duhring, Jensen, Sigmund; 2008
• Uncertainty on accuracy
Fabian Wein (Uni-Erlangen, Germany) Acoustic near field topology optimization
Model Concurrency Topology Optimization Numerical Results Conclusions
Acoustic Near Field Optimization
Continuous Problem: Pac = 12
∫Γopt<p v∗n dΓ
• Reformulate: vn = −∇nψ and p = j ω ρfψ
• Jnf = <j ωψT L∇nψ∗
• Interpret ∇n operator as constant matrix combined with L
• Jnf = <j ωψT Qψ∗
• Sensitivity: ∂Jnf∂ρ = 2<λT ∂bS
∂ρ u
• Adjoint problem: Sλ = −j ω (QT −Q)T u
• ≈ Jensen, Sigmund; 2005 and Jensen; 2007
Fabian Wein (Uni-Erlangen, Germany) Acoustic near field topology optimization
Model Concurrency Topology Optimization Numerical Results Conclusions
Full Plate Evaluation: |Ωair| = 20 cm
10-310-210-1100101102103104
0 500 1000 1500 2000
Obj
ectiv
e
Target Frequency (Hz)
Jnfc Jff
• 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
Model Concurrency Topology Optimization Numerical Results Conclusions
Full Plate Evaluation: |Ωair| = 6 cm
10-310-210-1100101102103104
0 500 1000 1500 2000
Obj
ectiv
e
Target Frequency (Hz)
Jnfc Jff
• 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
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
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0 500 1000 1500 2000
Obj
ectiv
e
Target Frequency (Hz)
c Pac(Jff)Jnf
full plate sweep
• Similar results for Jnf and Jff
• No reliable generation of resonating structures
Fabian Wein (Uni-Erlangen, Germany) Acoustic near field topology optimization
Model Concurrency Topology Optimization Numerical Results Conclusions
Selected Results
(a) 550Hz (b) 560 Hz (c) 980 Hz (d) 1510 Hz
10-510-410-310-210-1100101102103104
0 500 1000 1500 2000
Obj
ectiv
e
Target Frequency (Hz)
c Pac(Jff)Jnf
full plate sweep
• 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
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)
10-510-410-310-210-1100101102103104
0 500 1000 1500 2000
Obj
ectiv
e
Target Frequency (Hz)
Jnf(from left)Jnf(from right)
full plate sweep
• Blocked by resonances → Duhring, Jensen, Sigmund; 2008
Fabian Wein (Uni-Erlangen, Germany) Acoustic near field topology optimization
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
?
10-510-410-310-210-1100101102103104
0 500 1000 1500 2000
Obj
ectiv
e
Target Frequency (Hz)
Jnffull plate sweep
Fabian Wein (Uni-Erlangen, Germany) Acoustic near field topology optimization
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 eigenfrequencyanalysis
• Self-penalization: no regularization, constraints, (meshdepenency) . . .
• 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
Model Concurrency Topology Optimization Numerical Results Conclusions
Self-Penalization
• Piezoelectric setup often shows self-penalization
0
0.2
0.4
0.6
0.8
1
0 500 1000 1500 2000 0
0.2
0.4
0.6
0.8
1
Vol
ume
Gre
ynes
s
Target Frequency (Hz)
VolumeGreyness
• 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
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 0.41000 Hz 34 cm 3.4 cm 0.18
330 Hz 1 m 10.4 cm 0.058100 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
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