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USNCCM07 Structure Preserving Model Reduction for Damped Resonant MEMS David Bindel USNCCM 07, 23 Jul 2007

Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose second: really high-pitch guitars. USNCCM07 The Mechanical Cell Phone filter Mixer

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Page 1: Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose second: really high-pitch guitars. USNCCM07 The Mechanical Cell Phone filter Mixer

USNCCM07

Structure Preserving Model Reduction forDamped Resonant MEMS

David Bindel

USNCCM 07, 23 Jul 2007

Page 2: Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose second: really high-pitch guitars. USNCCM07 The Mechanical Cell Phone filter Mixer

USNCCM07

Collaborators

Tsuyoshi KoyamaSanjay GovindjeeSunil BhaveEmmanuel QuévyZhaojun Bai

Page 3: Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose second: really high-pitch guitars. USNCCM07 The Mechanical Cell Phone filter Mixer

USNCCM07

Resonant MEMS

Microguitars from Cornell University (1997 and 2003)

MHz-GHz mechanical resonatorsFavorite application: radio on chipClose second: really high-pitch guitars

Page 4: Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose second: really high-pitch guitars. USNCCM07 The Mechanical Cell Phone filter Mixer

USNCCM07

The Mechanical Cell Phone

filter

Mixer

Tuning

control

...RF amplifier /

preselector

Local

Oscillator

IF amplifier /

Your cell phone has many moving parts!What if we replace them with integrated MEMS?

Page 5: Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose second: really high-pitch guitars. USNCCM07 The Mechanical Cell Phone filter Mixer

USNCCM07

Ultimate Success

“Calling Dick Tracy!”

Page 6: Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose second: really high-pitch guitars. USNCCM07 The Mechanical Cell Phone filter Mixer

USNCCM07

Example Resonant System

DC

AC

Vin

Vout

Page 7: Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose second: really high-pitch guitars. USNCCM07 The Mechanical Cell Phone filter Mixer

USNCCM07

Example Resonant System

AC

C0

Vout

Vin

Lx

Cx

Rx

Page 8: Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose second: really high-pitch guitars. USNCCM07 The Mechanical Cell Phone filter Mixer

USNCCM07

Model Reduction: Basic set-up

Linear time-invariant system:

Mu′′ + Ku = bφ(t)y(t) = pT u

Frequency domain:

−ω2Mu + K u = bφ(ω)

y(ω) = pT u

Transfer function:

H(ω) = pT (−ω2M + K )−1by(ω) = H(ω)φ(ω)

Page 9: Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose second: really high-pitch guitars. USNCCM07 The Mechanical Cell Phone filter Mixer

USNCCM07

Model Reduction: Basic set-up

Have a rational transfer function relating input and output:

H(ω) = pT (−ω2M + K )−1b

Can approximate H by Galerkin projection:

H(ω) = (Vp)T (−ω2V T MV + V T KV )−1(Vb)

Could also try to approximate H directly (often equivalent).

Page 10: Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose second: really high-pitch guitars. USNCCM07 The Mechanical Cell Phone filter Mixer

USNCCM07

The Designer’s Dream

Ideally, would likeCompact models for behavioral simulationParameterized for design optimizationIncluding all relevant physicsWith reasonably fast and accurate set-up

We aren’t there yet.

Page 11: Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose second: really high-pitch guitars. USNCCM07 The Mechanical Cell Phone filter Mixer

USNCCM07

Standard Projection Model Reduction

Approximate H by Galerkin projection:

H(iω) = (Vp)T (−ω2V T MV + V T KV )−1(Vb)

1 Define Kσ := K − σ2M; build a Krylov subspace

span(V ) = Kn(K−1σ M, K−1

σ b) = span{(K−1σ M)jK−1

σ b}nj=0

Has the moment-matching property:

H(k)(iσ) = H(k)(iσ), k = 0, . . . , n

Get 2n moments for symmetric systems (or forseparate left and right subspaces).

2 Project onto one or more modal vectors.

Page 12: Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose second: really high-pitch guitars. USNCCM07 The Mechanical Cell Phone filter Mixer

USNCCM07

The Hero of the Hour

Major theme: use problem structure for better reducedmodels

ODE structureComplex symmetric structurePerturbative structureGeometric structure

Page 13: Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose second: really high-pitch guitars. USNCCM07 The Mechanical Cell Phone filter Mixer

USNCCM07

SOAR and ODE structure

Damped second-order system:

Mu′′ + Cu′ + Ku = Pφ

y = V T u.

Projection basis Qn with Second Order ARnoldi (SOAR):

Mnu′′n + Cnu′

n + Knun = Pnφ

y = V Tn u

where Pn = QTn P, Vn = QT

n V , Mn = QTn MQn, . . .

Page 14: Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose second: really high-pitch guitars. USNCCM07 The Mechanical Cell Phone filter Mixer

USNCCM07

Checkerboard Resonator

Page 15: Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose second: really high-pitch guitars. USNCCM07 The Mechanical Cell Phone filter Mixer

USNCCM07

Checkerboard Resonator

� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �

� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �

�����

�����

�����

�����

� �� �� �� �

� �� �

�����

� �� �

�����

�����

� �� �

D+

D−

D+

D−

S+ S+

S−

S−

Anchored at outside cornersExcited at northwest cornerSensed at southeast cornerSurfaces move only a few nanometers

Page 16: Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose second: really high-pitch guitars. USNCCM07 The Mechanical Cell Phone filter Mixer

USNCCM07

Performance of SOAR vs Arnoldi

N = 2154 → n = 80

0.08 0.085 0.09 0.095 0.1 0.105

100

105

Mag

nitu

de

Bode plot

0.08 0.085 0.09 0.095 0.1 0.105−200

−100

0

100

200

Pha

se(d

egre

e)

ExactSOARArnoldi

Page 17: Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose second: really high-pitch guitars. USNCCM07 The Mechanical Cell Phone filter Mixer

USNCCM07

Complex Symmetry

Model with radiation damping (PML) gives complex problem:

(K − ω2M)u = f , where K = K T , M = MT

Forced solution u is a stationary point of

I(u) =12

uT (K − ω2M)u − uT f .

Eigenvalues of (K , M) are stationary points of

ρ(u) =uT KuuT Mu

First-order accurate vectors =⇒second-order accurate eigenvalues.

Page 18: Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose second: really high-pitch guitars. USNCCM07 The Mechanical Cell Phone filter Mixer

USNCCM07

Disk Resonator Simulations

Page 19: Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose second: really high-pitch guitars. USNCCM07 The Mechanical Cell Phone filter Mixer

USNCCM07

Disk Resonator Mesh

PML region

Wafer (unmodeled)

Electrode

Resonating disk

0 1 2 3 4

x 10−5

−4

−2

0

2x 10

−6

Axisymmetric model with bicubic meshAbout 10K nodal points in converged calculation

Page 20: Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose second: really high-pitch guitars. USNCCM07 The Mechanical Cell Phone filter Mixer

USNCCM07

Symmetric ROM Accuracy

Frequency (MHz)

Tra

nsf

er(d

B)

Frequency (MHz)

Phase

(deg

rees

)

47.2 47.25 47.3

47.2 47.25 47.3

0

100

200

-80

-60

-40

-20

0

Results from ROM (solid and dotted lines) nearindistinguishable from full model (crosses)

Page 21: Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose second: really high-pitch guitars. USNCCM07 The Mechanical Cell Phone filter Mixer

USNCCM07

Symmetric ROM Accuracy

Frequency (MHz)

|H(ω

)−

Hreduced(ω

)|/H

(ω)|

Arnoldi ROM

Structure-preserving ROM

45 46 47 48 49 50

10−6

10−4

10−2

Preserve structure =⇒get twice the correct digits

Page 22: Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose second: really high-pitch guitars. USNCCM07 The Mechanical Cell Phone filter Mixer

USNCCM07

Perturbative Structure

Dimensionless continuum equations for thermoelasticdamping:

σ = Cε− ξθ1u = ∇ · σθ = η∇2θ − tr(ε)

Dimensionless coupling ξ and heat diffusivity η are 10−4

=⇒ perturubation method (about ξ = 0).

Large, non-self-adjoint, first-order coupled problem →Smaller, self-adjoint, mechanical eigenproblem + symmetriclinear solve.

Page 23: Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose second: really high-pitch guitars. USNCCM07 The Mechanical Cell Phone filter Mixer

USNCCM07

Thermoelastic Damping Example

Page 24: Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose second: really high-pitch guitars. USNCCM07 The Mechanical Cell Phone filter Mixer

USNCCM07

Performance for Beam Example

10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

6

7

8

9

10

Beam length (microns)

Tim

e (s

)

Perturbation methodFirst−order form

Page 25: Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose second: really high-pitch guitars. USNCCM07 The Mechanical Cell Phone filter Mixer

USNCCM07

Aside: Effect of Nondimensionalization

100 µm beam example, first-order form.

Before nondimensionalizationTime: 180 snnz(L) = 11M

After nondimensionalizationTime: 10 snnz(L) = 380K

Page 26: Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose second: really high-pitch guitars. USNCCM07 The Mechanical Cell Phone filter Mixer

USNCCM07

Semi-Analytical Model Reduction

We work with hand-build model reduction all the time!Circuit elements: Maxwell equation + field assumptionsBeam theory: Elasticity + kinematic assumptionsAxisymmetry: 3D problem + kinematic assumption

Idea: Provide global shapesUser defines shapes through a callbackMesh serves defines a quadrature ruleReduced equations fit known abstractions

Page 27: Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose second: really high-pitch guitars. USNCCM07 The Mechanical Cell Phone filter Mixer

USNCCM07

Global Shape Functions

Normally:u(X ) =

∑j

Nj(X )uj

Global shape functions:

u = ul + G(ug)

Then constrain values of some components of ul , ug .

Page 28: Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose second: really high-pitch guitars. USNCCM07 The Mechanical Cell Phone filter Mixer

USNCCM07

“Hello, World!”

Which mode shape comes from the reduced model (3 dof)?

Student Version of MATLABStudent Version of MATLAB

(Left: 28 MHz; Right: 31 MHz)

Page 29: Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose second: really high-pitch guitars. USNCCM07 The Mechanical Cell Phone filter Mixer

USNCCM07

Conclusions

Respecting problem structure is a Good Thing!ODE structureComplex symmetric structurePerturbative structureGeometric structure

Result:Better accuracy, faster set-up, better understanding.