D.Ruiz J.C.Bellido & A.Donoso · D.Ruiz, J.C.Bellido & A.Donoso Optimal Design of Piezoelectric Transducers. Introduction Problem formulation Numerical approach and examples Conclusions

Optimal Design of Piezoelectric Transducers

D.Ruiz,J.C.Bellido & A.Donoso

Department of Mathematics, University of Castilla-la Mancha, Spain

11th WCSMO

Sydney, June 9th, 2015

IntroductionProblem formulation

Numerical approach and examplesConclusions and future work

1 Introduction

2 Problem formulation

3 Numerical approach and examples

4 Conclusions and future work

D.Ruiz, J.C.Bellido & A.Donoso Optimal Design of Piezoelectric Transducers

1 Introduction

Beginnings

Electronics + Mathematics = Great results!

Modal sensors/actuators

DescriptionMSA are piezoelectric-based devices that excite/measure a specific mode.

C.K.Lee and F.C.Moon. Modal Sensors/Actuators. Journal of AppliedMechanics, volume(57), 434-441, 1990.

Analytical expressions for one-dimensional plates.

Figure: Mode shape and top view of optimized sensors for 1st and 2nd eigenmodes

Modal sensors/actuators

DescriptionMSA are piezoelectric-based devices that excite/measure a specific mode.

C.K.Lee and F.C.Moon. Modal Sensors/Actuators. Journal of AppliedMechanics, volume(57), 434-441, 1990.

Analytical expressions for one-dimensional plates.

Figure: Mode shape and top view of optimized sensors for 1st and 2nd eigenmodes

Two-dimensional modal sensors/actuators

A. Donoso and J.C. Bellido. Systematic design of distributed piezoelectric modalsensor/actuators for rectangular plates by optimizing the polarization profile.Struct. Multidisc. Optim. volume(38), 347-356, 2009

J.L. Sánchez-Rojas, J. Hernando, A. Donoso, J.C. Bellido, T. Manzaneque, A.Ababneh, H. Seidel, U. Schmid. Modal optimization and filtering inpiezoelectric microplate resonators. J Micromech Microeng 20:055027,2010

The problem

Aim

Optimal design of two-dimensional MSA by optimizing simultaneously the topology and

the electrode polarization.

Why MSA?

Powerful applications: measurement of density and viscosity in fluids and substance

detection. Simplify electronic circuits.

The problem

Aim

Optimal design of two-dimensional MSA by optimizing simultaneously the topology and

the electrode polarization.

Why MSA?

Powerful applications: measurement of density and viscosity in fluids and substance

detection. Simplify electronic circuits.

1 Introduction

Continuous formulation

Output charge up to a scalling factor can be expressed as follows (C.K. Lee and F.C.Moon 1990):

q(t) =

∫Ωχp(x , y)

[(∂u

∂x+∂v

∂y

)−

(hs + hp)

2

(∂2w

∂x2+∂2w

∂y2

)]dΩ

?χs= 1

TOP VIEW

SIDE VIEW

Finχs=0

χp=

χp= -1

ELECTRODEPROFILE

VOID

STRUCTUREANDPIEZO

STRUCTUREPIEZO

PIEZO1

q

χp= 0

out

Ω

Continuous formulation (cont.)

Using modal expansion:

u(x , y , t) =∞∑

j=1

φj (x , y)ηj (t)

v(x , y , t) =∞∑

j=1

ψj (x , y)ηj (t)

w(x , y , t) =∞∑

j=1

ϕj (x , y)ηj (t)

Replacing:

q(t) =∞∑

j=1

Fjηj (t), Fj =

∫Ωχp(x , y)

[(∂φ

∂x+∂ψ

∂y

)−

(hs + hp)

2

(∂2ϕ

∂x2+∂2ϕ

∂y2

)]dΩ

Using modal expansion:

u(x , y , t) =∞∑

j=1

φj (x , y)ηj (t)

v(x , y , t) =∞∑

j=1

ψj (x , y)ηj (t)

w(x , y , t) =∞∑

j=1

ϕj (x , y)ηj (t)

Replacing:

q(t) =∞∑

j=1

Fjηj (t), Fj =

∫Ωχp(x , y)

[(∂φ

∂x+∂ψ

∂y

)−

(hs + hp)

2

(∂2ϕ

∂x2+∂2ϕ

∂y2

)]dΩ

maxχs ,χp

: Fk (χp ,Φk (χs ))

s.t. : A(χs ,Φj (χs )) = 0|Fj (χp ,Φj (χs ))| ≤ �Fk (χp ,Φk (χs )), j = 1, . . . , J, j 6= k

χs ∈ {0, 1}χp ∈ {−1, 0, 1}

Bound formulation:maxχs ,χp

: Fk (χp ,Φk (χs ))− α

s.t. : A(χs ,Φj (χs )) = 0|Fj (χp ,Φj (χs ))| ≤ α, j = 1, . . . , J, j 6= k

χs ∈ {0, 1}χp ∈ {−1, 0, 1}α ≥ 0

maxχs ,χp

: Fk (χp ,Φk (χs ))

s.t. : A(χs ,Φj (χs )) = 0|Fj (χp ,Φj (χs ))| ≤ �Fk (χp ,Φk (χs )), j = 1, . . . , J, j 6= k

χs ∈ {0, 1}χp ∈ {−1, 0, 1}

Bound formulation:maxχs ,χp

: Fk (χp ,Φk (χs ))− α

s.t. : A(χs ,Φj (χs )) = 0|Fj (χp ,Φj (χs ))| ≤ α, j = 1, . . . , J, j 6= k

χs ∈ {0, 1}χp ∈ {−1, 0, 1}α ≥ 0

Discrete approach

FEM

Integer variables (χs , χp) → Density variables (ρs , ρp) (RAMP)

Dicretized coefficients Fi can be expressed as follows: Fi = R(ρs )(2ρp − 1)BΦi

R(ρs ) is an interpolation scheme. R(ρ(e)s ) = e

−γ(1−ρ(e)s ) − (1− ρ(e)s )e−γ(2ρp − 1) is the polarization density.B is the strain displacement matrix.Φi is the i-th mode shape.

R

se

0 1

1=0=1=3=5

Discrete approach

FEM

R

se

0 1

1=0=1=3=5

Discrete approach

FEM

R

se

0 1

1=0=1=3=5

Discrete approach (cont.)

Discrete problemmax

ρs ,ρp ,α: Fk − α

s.t. : (K− µj M)Φj = 0, j = 1, . . . , JΦTj MΦl = 0, j , l = 1, . . . , J, j 6= l

ΦTj Φj = 1, j = 1 . . . , J|Fj | ≤ α, j = 1, . . . , J, j 6= kρs ∈ [0, 1]ρp ∈ [0, 1]α ≥ 0

Problem with eigenvectors in both, cost and constraints.

Introduction of an interpolation scheme in the expression of the collectedcharge.

Discrete approach (cont.)

Discrete problemmax

ρs ,ρp ,α: Fk − α

s.t. : (K− µj M)Φj = 0, j = 1, . . . , JΦTj MΦl = 0, j , l = 1, . . . , J, j 6= l

ΦTj Φj = 1, j = 1 . . . , J|Fj | ≤ α, j = 1, . . . , J, j 6= kρs ∈ [0, 1]ρp ∈ [0, 1]α ≥ 0

Problem with eigenvectors in both, cost and constraints.

Introduction of an interpolation scheme in the expression of the collectedcharge.

Difficulties

Classical issues in topology optimization problems

Local optima

Mode switching

Spurious modes

Normalization of eigenvectors

Large grey areas

Repeated eigenfrequencies

Figure: 2nd and 3rd mode shapes for a simply supported square plate

Difficulties

Classical issues in topology optimization problems

Local optima

Mode switching

Spurious modes

Normalization of eigenvectors

Large grey areas

Repeated eigenfrequencies

Figure: 2nd and 3rd mode shapes for a simply supported square plate

Computation of sensitivities

Optimizer used: Method of moving asymptotes (MMA).

Gradient-based method.Commonly used in structural optimization problems.

The sensitivity with respect to the polarization density is straighforward.

Fi = R(ρs )(2ρp − 1)BΦi (ρs )

Single eigenfrequencies → Adjoint + Nelson’s method.

Repeated eigenfrequencies → Dailey’s method.Computation of eigenvectors Φ.Computation of adjacent eigenvectors Z = ΦΓ.Computation of derivatives of Z, Z′.Computation of derivatives of Φ, Φ′ = Z′ΓT

1 Introduction

Numerical approach

Algorithm

1 Choose J mode shapes of the homogeneous plate. These are the reference forthe 1st iteration. J = 4, {Φ1,Φ2,Φ3,Φ4}

2 Initialize design variables ρs and ρp .

3 Compute L mode shapes for the plate to be optimized, with L > J, largeenough. L = 8, {Φ1,Φ2, . . . ,Φ8}

4 Identify the J closest modes to the ones of reference, among the set ofpreviously computed L modes by using MAC {Φ1,Φ5,Φ4,Φ8}. Relabel thesequence from 1 to J. This is the reference for the next iteration.

5 Check the multiplicity of eigenvalues by using a tolerance.

Simple: Compute Fi and its derivatives with Nelson’s method.

Repeated:

Find a new basis of eigenvectors using MAC.Compute Fi and its derivatives using Dailey’s method.

6 Update design variables by using MMA.

7 Until convergence go to step 3.

Numerical approach

Algorithm

Repeated:

Numerical approach

Algorithm

Repeated:

Numerical approach

Algorithm

Repeated:

Numerical approach

Algorithm

Repeated:

Numerical approach

Algorithm

Repeated:

Numerical approach

Algorithm

Repeated:

First example

Square plate clamped on its left edge. In-plane regime.

J = 4, L = 8.

Optimized mode shape: 1st.

Canceled mode shapes: 2nd, 3rd and 4th.

First example cont.

The gain obtained is 106% with respect to the optimization of only the polarityvariable ρp .

1 2 3 40

0.5

1

1.5

2

2.5

Simultaneous

optimization

Single

optimization

Mode shape

Norm

alized F

i

0 50 100 150 200 250 300500

1000

1500

2000

2500

3000

3500

4000

Iteration number

Eig

enfr

equencie

s

Square plate clamped on its left edge

In-plane

(a) 1st mode (b) 2nd mode (c) 3rd mode (d) 4th mode

Out-of-plane

(e) 1st mode (f) 2nd mode (g) 3rd mode (h) 4th mode

Square plate clamped on its left edge

In-plane

Out-of-plane

Square plate clamped on its left and right edges

In-plane

Out-of-plane

Square plate clamped on its left and right edges

In-plane

Out-of-plane

Square plate clamped on its four edges

In-plane

Out-of-plane

Square plate clamped on its four edges

In-plane

Out-of-plane

1 Introduction

Advantages of simultaneous optimization

Simultaneous optimization vs “Single” optimization

Isolated modes1st 2nd 3rd 4th

In-plane 106% 7% 2% 25%Out-of-plane 0% 36% 73% 56%

Troubleshooting

Local optima: Symmetry absence is the unique evidence. Solution:continuation methods.

Mode switching: Changes in topology modifies eigenmodes order. Solution:mode tracking (MAC).

0 50 100 150 200 250 3000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Iteration number

Mode s

wit

ch

ing

ω

Troubleshooting (cont.)

Spurious modes: Due to bad modelling. Spoils frequency spectrum. Solution:suitable relationship between stiffness and mass of the material.

Normalization of eigenvectors: M-orthonormal eigenvectors cause convergenceproblem. Solution: normalization of eigenvectors with respect to the identitymatrix.

Appearance of grey areas: Nature of the problem favours the appearance oflarge grey areas. Solution: introduction of a new interpolation scheme in theexpression of the collected charge.

Repeated eigenvalues: Can appear at any time. Solution: the use of referenceeigenmodes to fix the basis.

Future work

Introduction of a gap-phase in the electrode

Gap

Introduction of supports as a new variable

Fabrication and testing

Transducers with only one electrode

Thank you for your attention!

IntroductionProblem formulationNumerical approach and examplesConclusions and future work