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Structure-preserving model reduction for MEMS modeling David Bindel Department of Computer Science Cornell University SIAM Annual Meeting, 12 Jul 2010

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Page 1: Structure-preserving model reduction for MEMS modelingbindel/present/2010-07-siam.pdfStructure-preserving model reduction for MEMS modeling David Bindel Department of Computer Science

Structure-preserving model reduction forMEMS modeling

David Bindel

Department of Computer ScienceCornell University

SIAM Annual Meeting, 12 Jul 2010

Page 2: Structure-preserving model reduction for MEMS modelingbindel/present/2010-07-siam.pdfStructure-preserving model reduction for MEMS modeling David Bindel Department of Computer Science

Collaborators

I Tsuyoshi KoyamaI Sanjay GovindjeeI Sunil BhaveI Emmanuel QuévyI Zhaojun Bai

Page 3: Structure-preserving model reduction for MEMS modelingbindel/present/2010-07-siam.pdfStructure-preserving model reduction for MEMS modeling David Bindel Department of Computer Science

Resonant MEMS

Microguitars from Cornell University (1997 and 2003)

I MHz-GHz mechanical resonatorsI Favorite application: radio on chipI Close second: really high-pitch guitars

Page 4: Structure-preserving model reduction for MEMS modelingbindel/present/2010-07-siam.pdfStructure-preserving model reduction for MEMS modeling David Bindel Department of Computer Science

The Mechanical Cell Phone

filter

Mixer

Tuning

control

...RF amplifier /

preselector

Local

Oscillator

IF amplifier /

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

Page 5: Structure-preserving model reduction for MEMS modelingbindel/present/2010-07-siam.pdfStructure-preserving model reduction for MEMS modeling David Bindel Department of Computer Science

Ultimate Success

“Calling Dick Tracy!”

I Old dream: a Dick Trace watch phone!I New dream: long battery life for smart phones

Page 6: Structure-preserving model reduction for MEMS modelingbindel/present/2010-07-siam.pdfStructure-preserving model reduction for MEMS modeling David Bindel Department of Computer Science

Example Resonant System

DC

AC

Vin

Vout

Page 7: Structure-preserving model reduction for MEMS modelingbindel/present/2010-07-siam.pdfStructure-preserving model reduction for MEMS modeling David Bindel Department of Computer Science

Example Resonant System

AC

C0

Vout

Vin

Lx

Cx

Rx

Page 8: Structure-preserving model reduction for MEMS modelingbindel/present/2010-07-siam.pdfStructure-preserving model reduction for MEMS modeling David Bindel Department of Computer Science

The Designer’s Dream

Ideally, would likeI Simple models for behavioral simulationI Parameterized for design optimizationI Including all relevant physicsI With reasonably fast and accurate set-upI Backed by error analysis

We aren’t there yet.

Page 9: Structure-preserving model reduction for MEMS modelingbindel/present/2010-07-siam.pdfStructure-preserving model reduction for MEMS modeling David Bindel Department of Computer Science

The Hero of the Hour

Major theme: use problem structure for better modelsI Algebraic

I Structure of ODEs (e.g. second-order structure)I Structure of matrices (e.g. complex symmetry)

I AnalyticI Perturbations of physics (thermoelastic damping)I Perturbations of geometry (near axisymmetry)I Perturbations of boundary conditions (clamping)

I GeometricI Simplified models: planar motion, axisymmetry, ...I Substructures

Page 10: Structure-preserving model reduction for MEMS modelingbindel/present/2010-07-siam.pdfStructure-preserving model reduction for MEMS modeling David Bindel Department of Computer Science

SOAR and ODE structure

Damped second-order system:

Mu′′ + Cu′ + Ku = Pφy = 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 11: Structure-preserving model reduction for MEMS modelingbindel/present/2010-07-siam.pdfStructure-preserving model reduction for MEMS modeling David Bindel Department of Computer Science

Checkerboard Resonator

Page 12: Structure-preserving model reduction for MEMS modelingbindel/present/2010-07-siam.pdfStructure-preserving model reduction for MEMS modeling David Bindel Department of Computer Science

Checkerboard Resonator

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

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

�����

�����

�����

�����

� �� �� �� �

� �� �

�����

� �� �

�����

�����

� �� �

D+

D−

D+

D−

S+ S+

S−

S−

I Anchored at outside cornersI Excited at northwest cornerI Sensed at southeast cornerI Surfaces move only a few nanometers

Page 13: Structure-preserving model reduction for MEMS modelingbindel/present/2010-07-siam.pdfStructure-preserving model reduction for MEMS modeling David Bindel Department of Computer Science

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

deBode 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 14: Structure-preserving model reduction for MEMS modelingbindel/present/2010-07-siam.pdfStructure-preserving model reduction for MEMS modeling David Bindel Department of Computer Science

Aside: Next generation

Page 15: Structure-preserving model reduction for MEMS modelingbindel/present/2010-07-siam.pdfStructure-preserving model reduction for MEMS modeling David Bindel Department of Computer Science

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 16: Structure-preserving model reduction for MEMS modelingbindel/present/2010-07-siam.pdfStructure-preserving model reduction for MEMS modeling David Bindel Department of Computer Science

Disk Resonator Simulations

Page 17: Structure-preserving model reduction for MEMS modelingbindel/present/2010-07-siam.pdfStructure-preserving model reduction for MEMS modeling David Bindel Department of Computer Science

Disk Resonator Mesh

PML region

Wafer (unmodeled)

Electrode

Resonating disk

0 1 2 3 4

x 10−5

−4

−2

0

2x 10

−6

I Axisymmetric model with bicubic meshI About 10K nodal points in converged calculation

Page 18: Structure-preserving model reduction for MEMS modelingbindel/present/2010-07-siam.pdfStructure-preserving model reduction for MEMS modeling David Bindel Department of Computer Science

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 19: Structure-preserving model reduction for MEMS modelingbindel/present/2010-07-siam.pdfStructure-preserving model reduction for MEMS modeling David Bindel Department of Computer Science

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 20: Structure-preserving model reduction for MEMS modelingbindel/present/2010-07-siam.pdfStructure-preserving model reduction for MEMS modeling David Bindel Department of Computer Science

Perturbative Structure

Dimensionless continuum equations for thermoelastic damping:

σ = Cε− ξθ1u = ∇ · σθ = η∇2θ − tr(ε)

Dimensionless coupling ξ and heat diffusivity η are 10−4 =⇒perturbation method (about ξ = 0).

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

Page 21: Structure-preserving model reduction for MEMS modelingbindel/present/2010-07-siam.pdfStructure-preserving model reduction for MEMS modeling David Bindel Department of Computer Science

Thermoelastic Damping Example

Page 22: Structure-preserving model reduction for MEMS modelingbindel/present/2010-07-siam.pdfStructure-preserving model reduction for MEMS modeling David Bindel Department of Computer Science

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 23: Structure-preserving model reduction for MEMS modelingbindel/present/2010-07-siam.pdfStructure-preserving model reduction for MEMS modeling David Bindel Department of Computer Science

Aside: Effect of Nondimensionalization

100 µm beam example, first-order form.

Before nondimensionalizationI Time: 180 sI nnz(L) = 11M

After nondimensionalizationI Time: 10 sI nnz(L) = 380K

Page 24: Structure-preserving model reduction for MEMS modelingbindel/present/2010-07-siam.pdfStructure-preserving model reduction for MEMS modeling David Bindel Department of Computer Science

Semi-Analytical Model Reduction

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

Idea: Provide global shapesI User defines shapes through a callbackI Mesh serves defines a quadrature ruleI Reduced equations fit known abstractions

Page 25: Structure-preserving model reduction for MEMS modelingbindel/present/2010-07-siam.pdfStructure-preserving model reduction for MEMS modeling David Bindel Department of Computer Science

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 26: Structure-preserving model reduction for MEMS modelingbindel/present/2010-07-siam.pdfStructure-preserving model reduction for MEMS modeling David Bindel Department of Computer Science

“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 27: Structure-preserving model reduction for MEMS modelingbindel/present/2010-07-siam.pdfStructure-preserving model reduction for MEMS modeling David Bindel Department of Computer Science

The latest

http://www.cs.cornell.edu/~bindel