Topology Optimization of a Piezoelectric Loudspeaker Coupled with the Acoustic Domain

Motivation Model SIMP Objective Functions Numeriacal Results Conclusions

Topology Optimization of a PiezoelectricLoudspeaker Coupled with the Acoustic Domain

F. Wein, M. Kaltenbacher, F. Schury, E. Bansch, G. Leugering

WCSMO-803. June 2009

Fabian Wein Acoustic Piezoelectric Loudspeaker Optimization


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)



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



Coupling to Acoustic Domain

• Discretization of Ωair via cair = 343 m/s = λacf

• 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



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 formulation (quadratic / linear with softening)



Solid Isotropic Material with Penalization

• Sample reference: Kogl, Silva; 2005

• 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:

[cEe ] = ρe [cE ], ρm

e = ρeρm, [ee ] = ρe [e], [εS

e ] = ρe [εS ]

• Intermediate values are non-physical!• Small ρ → low stiffness• Large ρ → high piezoelectric coupling



Linear Systems

• 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)

=

000qφ

• Piezoelectric-mechanical laminate Sumum Sumup (ρ) 0

STumup

(ρ) Supup (ρ) Kupφ(ρ)

0 KTupφ(ρ) −Kφφ(ρ)

um(ρ)

up(ρ)φ(ρ)

=

00qφ

• Harmonic excitation: S(ω) = K + jω(αKK + αMM)− ω2M

• Short form: S u = f



Sound Power

• What one wants to optimize

Pac =1

2

∫Γopt

Rep v∗n dΓ

Sound power Pac, pressure p, normal particle velocity vn

• Acoustic impedance Z

Z (x) =p(x)

vn(x)

• Z can bee assumed to be constant on Γopt

• For plane waves• In the acoustic far field (some wavelengths)



Optimizing for the Sound Power

Use∫

ΓoptRep v∗

n dΓ with Z = p/vn and u = (ψ um up φ)T

Optimization framework

• J = uTL u∗ with L selecting from u ≈ Sigmund, Jensen; 2003

∂J

∂ρe= 2 ReλT ∂S

∂ρeu where λ solves ST λ = −Lu∗

Acoustic optimization assume Z constant on Γopt

• Jac = ω2ψTLψ∗ with vn = p/Z and p = ρf ψ

• ≈ Duhring, Jensen, Sigmund; 2008

Structural optimization assume Z constant on Γiface

• Jst = ω2umTLu∗m with p = vn Z and v = u

• ≈ Du, Olhoff; 2007 (approximation is discussed!)



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)



Structural Optimization

• Objective function Jst = ω2umTLu∗m

• Several hundred mono-frequent optimizations

10-1100101102103104105

0 500 1000 1500 2000Obj

ectiv

e F

unct

ion

(m2 /s

2 )

Target Frequency (Hz)

Jst (max Jst)Jst (max Jac)

Jst (Sweep full plate)

• Robust shifting/ creation of resonances

• KKT-condition often not reached



Validate Acoustic Approximation

• Validate Jac = ω2ψTLψ∗ against∫

ΓoptRep v∗

n dΓ

• Use one mesh (23 cm height)

10-1100101102103104105106107

0 500 1000 1500 200010-810-710-610-510-410-310-210-1100

Obj

ectiv

e F

unct

ion

(Pa2 )

Aco

ustic

Pow

er (

W)


Jac (Sweep full plate)Pac (Sweep full plate)

• Good approximation above 1000 Hz (0.64 wave length)

• Two antiresonances (acoustic short circuits)

• Approximation detects only one antiresonance!



Acoustic Optimization

• Objective function Jac = ω2ψTLψ∗

10-1100101102103104105106107

0 500 1000 1500 200010-710-610-510-410-310-210-1100

Obj

ectiv

e F

unct

ion

(Pa2 )

Aco

ustic

Pow

er (

W)


Jac (max Jac)Pac (max Jst)

Jac (Sweep full plate)

• Jac avoids acoustic short circuits

• Structural approximation unfeasible (above first mode)

• Note Jac approximation unfeasible below 1000 Hz



Grayness and Volumes

• Remember No penalization, no volume constraint

• Normalized grayness: g(ρ) = 4∫

Ω(1− ρ) ρ dx ,

0

0.05

0.1

0.15

0.2

0.25

0.3

0 500 1000 1500 2000

Grayness over Target Frequenzy (Hz)

g(Jst)g(Jac)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 500 1000 1500 2000

Volume over Target Frequenzy (Hz)

Vol(Jst)Vol(Jac)

• Good avoidance of grayness

• Optimal volumes 25% . . . 100%



Selected Results

(a) 565Hz (b) 575 Hz (c) 875 Hz (d) 975 Hz

(e) 985 Hz (f) 1405Hz (g) 1585 Hz (h) 2160 Hz

Legend: black = full material; white = void material



Experimental Prototype (200 µm Piezoceramic)

(a) Original (b) Sputter (c) Lasing

(d) Temper (e) Polarize (f) Prototype



Conclusions and Future Work

Conclusions

• Structural approximation does not handle acoustic shortcircuits

• 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



The End

Thank you for listening and for questions!


Topology Optimization of a Piezoelectric Loudspeaker Coupled with the Acoustic Domain

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