Acoustic near field topology optimization of a piezoelectric loudspeaker

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


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



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



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


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



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



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



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



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



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



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



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



Full Plate Evaluation: |Ωair| = 6 cm

10-310-210-1100101102103104

0 500 1000 1500 2000

Obj

ectiv

e


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



Topology Optimization: |Ωair| = 6 cm

• Several hundred mono-frequent optimizations!

• Max iterations: 250, SCPIP/MMA, generally no KKT reached

• Starting from full plate

10-510-410-310-210-1100101102103104

0 500 1000 1500 2000

Obj

ectiv

e


c Pac(Jff)Jnf

full plate sweep

• Similar results for Jnf and Jff

• No reliable generation of resonating structures



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


c Pac(Jff)Jnf

full plate sweep

• Strain cancellation and acoustic short circuits handled

• Self-penalization for ρ1, no regularization, no constraints, . . .



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


Jnf(from left)Jnf(from right)

full plate sweep

• Blocked by resonances → Duhring, Jensen, Sigmund; 2008



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


Jnffull plate sweep



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!



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


VolumeGreyness

• For most frequencies sufficient self-penalization

• Not as distinct as in structural optimization

• Stronger self-penalization for “global optima”



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



Experimental Prototype (200 µm Piezoceramic)

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

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


Acoustic near field topology optimization of a piezoelectric loudspeaker

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