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Model Reduction and Mode Computation forDamped Resonant MEMS

David Bindel

SIAM CSE 07, 19 Feb 2007

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Collaborators

Tsuyoshi KoyamaSanjay GovindjeeSunil BhaveEmmanuel QuévyZhaojun Bai

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Resonant MEMS

Microguitars from Cornell University (1997 and 2003)

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

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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?

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Ultimate Success

“Calling Dick Tracy!”

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Example Resonant System

DC

AC

Vin

Vout

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Example Resonant System

AC

C0

Vout

Vin

Lx

Cx

Rx

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The Designer’s Dream

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

We aren’t there yet. Today, some progress on the last two.

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Damping and Q

Designers want high quality of resonance (Q)Dimensionless damping in a one-dof system

d2udt2 + Q−1 du

dt+ u = F (t)

For a resonant mode with frequency ω ∈ C:

Q :=|ω|

2 Im(ω)=

Stored energyEnergy loss per radian

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Damping Mechanisms

Possible loss mechanisms:Anchor lossThermoelastic dampingOther material lossesFluid damping

Our goal: Reduced models that include these effects.

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Perfectly Matched Layers

Model substrate as semi-infinite with a

Perfectly Matched Layer (PML).

Complex coordinate transformationGenerates a “perfectly matched” absorbing layerIdea works with general linear wave equations

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Perfectly Matched Layers by Picture

Outgoing exp(−ix) Incoming exp(ix)

Transformed coordinate

Re(x)

0 2 4 6 8 10 12 14 16 18

0 5 10 15 200 5 10 15 20

-4

-2

0

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

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Perfectly Matched Layers by Picture

Outgoing exp(−ix) Incoming exp(ix)

Transformed coordinate

Re(x)

0 2 4 6 8 10 12 14 16 18

0 5 10 15 200 5 10 15 20

-4

-2

0

-2

-1

0

1

2

3

-1

-0.5

0

0.5

1

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Perfectly Matched Layers by Picture

Outgoing exp(−ix) Incoming exp(ix)

Transformed coordinate

Re(x)

0 2 4 6 8 10 12 14 16 18

0 5 10 15 200 5 10 15 20

-4

-2

0

-4

-2

0

2

4

6

-1

-0.5

0

0.5

1

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Perfectly Matched Layers by Picture

Outgoing exp(−ix) Incoming exp(ix)

Transformed coordinate

Re(x)

0 2 4 6 8 10 12 14 16 18

0 5 10 15 200 5 10 15 20

-4

-2

0

-10

-5

0

5

10

15

-1

-0.5

0

0.5

1

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Perfectly Matched Layers by Picture

Outgoing exp(−ix) Incoming exp(ix)

Transformed coordinate

Re(x)

0 2 4 6 8 10 12 14 16 18

0 5 10 15 200 5 10 15 20

-4

-2

0

-20

0

20

40

-1

-0.5

0

0.5

1

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Perfectly Matched Layers by Picture

Outgoing exp(−ix) Incoming exp(ix)

Transformed coordinate

Re(x)

0 2 4 6 8 10 12 14 16 18

0 5 10 15 200 5 10 15 20

-4

-2

0

-50

0

50

100

-1

-0.5

0

0.5

1

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Finite Element Implementation

x(ξ)

ξ2

ξ1

x1

x2 x2

x1

Ωe Ωe

Ω

x(x)

Combine PML and isoparametric mappings

ke =

∫Ω

BT DBJ dΩ

me =

∫Ω

ρNT NJ dΩ

Matrices are complex symmetric

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Complex Symmetry

Discretized (forced) problem + fixed PML take the form:

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

Can still characterize u as 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.

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Accurate Model Reduction

Usual: Orthogonal projection onto Arnoldi basis V .Us: Build new projection basis from V :

W = orth[Re(V ), Im(V )]

span(W ) contains both Kn and Kn=⇒ double digits correct vs. projection with VW is a real-valued basis=⇒ projected system is complex symmetric

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Model Reduction 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)

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Model Reduction 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

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Thermoelastic Damping (TED)

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Thermoelastic Damping (TED)

u is displacement and T = T0 + θ is temperature

σ = Cε− βθ1ρu = ∇ · σ

ρcv θ = ∇ · (κ∇θ)− βT0 tr(ε)

Coupling between temperature and volumetric strain:Compression and expansion =⇒ heating and coolingHeat diffusion =⇒ mechanical dampingNot often an important factor at the macro scaleRecognized source of damping in microresonators

Zener: semi-analytical approximation for TED in beamsWe consider the fully coupled system

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Nondimensionalized Equations

Continuum equations:

σ = Cε− ξθ1u = ∇ · σθ = η∇2θ − tr(ε)

Discrete equations:

Muuu + Kuuu = ξKuθθ + fCθθθ + ηKθθθ = −Cθuu

Micron-scale poly-Si devices: ξ and η are ∼ 10−4.Linearize about ξ = 0

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Perturbative Mode Calculation

Discretized mode equation:

(−ω2Muu + Kuu)u = ξKuθθ

(iωCθθ + ηKθθ)θ = −iωCθuu

First approximation about ξ = 0:

(−ω20Muu + Kuu)u0 = 0

(iω0Cθθ + ηKθθ)θ0 = −iω0Cθuu0

First-order correction in ξ:

−δ(ω2)Muuu0 + (−ω20Muu + Kuu)δu = ξKuθθ0

Multiply by uT0 :

δ(ω2) = −ξ

(uT

0 Kuθθ0

uT0 Muuu0

)

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Zener’s Model

1 Clarence Zener investigated TED in late 30s-early 40s.2 Model for beams common in MEMS literature.3 “Method of orthogonal thermodynamic potentials” ==

perturbation method + a variational method.

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Comparison to Zener’s Model

105

106

107

108

109

1010

10−7

10−6

10−5

10−4

The

rmoe

last

ic D

ampi

ng Q

−1

Frequency f(Hz)

Zener’s Formula

HiQlab Results

Comparison of fully coupled simulation to Zenerapproximation over a range of frequenciesReal and imaginary parts after first-order correctionagree to about three digits with Arnoldi

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General Picture

If w∗A = 0 and Av = 0 then

δ(w∗Av) = w∗(δA)v

This impliesIf A = A(λ) and w = w(v), have

w∗(v)A(ρ(v))v = 0.

ρ stationary when (ρ(v), v) is a nonlinear eigenpair.If A(λ, ξ) and w∗

0 and v0 are null vectors for A(λ0, ξ0),

w∗0 (Aλδλ + Aξδξ)v0 = 0.

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Conclusions

Resonant MEMS have lots of interesting applicationsDesigners want reduced models with relevant physicsDamping is crucial, but not well handled in generalOur work: use equation structure in making reducedmodels with damping (modal or more general)