Computer Aided Design of Micro-Electro-Mechanical Systemsbindel/present/2008-06-householder-b.pdf0 2 4 6 8 10 12 14 16 18 0 5 10 15 20 0 5 10 15 20-4-2 0-50 0 50 100-1-0.5 0 0.5 1

Computer Aided Design ofMicro-Electro-Mechanical SystemsFrom Energy Losses to Dick Tracy Watches

D. Bindel

Courant Institute for Mathematical SciencesNew York University

Householder XVIII, 6 June 2008

(Courant Institute) Householder 08 1 / 65

Collaborators

Jim DemmelSanjay GovindjeeTsuyoshi KoyamaZhaojun BaiSunil BhaveEmmanuel Quévy...


The Computational Science Picture

Application modelingCheckerboard filterDisk resonator and radiation lossBeam resonator and thermoelastic lossShear ring resonator

Mathematical analysisPhysical modeling and finite element technologyStructured eigenproblems and reduced-order modelsParameter-dependent eigenproblems

Software engineeringHiQLabSUGARFEAPMEX / MATFEAP


What are MEMS?


MEMS Basics

Micro-Electro-Mechanical SystemsChemical, fluid, thermal, optical (MECFTOMS?)

Applications:Sensors (inertial, chemical, pressure)Ink jet printers, biolab chipsRadio devices: cell phones, inventory tags, pico radio

Use integrated circuit (IC) fabrication technologyTiny, but still classical physics


Resonant RF MEMS

Microguitars from Cornell University (1997 and 2003)

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


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?


Ultimate Success

“Calling Dick Tracy!”


Disk Resonator

DC

AC

Vin

Vout


Disk Resonator

AC

C0

Vout

Vin

Lx

Cx

Rx


The Designer’s Dream

Want a reduced model for the electromechanical admittance

Y (ω) = iωC + G + iωH(ω)H(ω) = BT (K̃ − ω2M)−1B

Ideally, would likeSimple models for behavioral simulationParameterized for design optimizationWith reasonably fast and accurate set-upIncluding all relevant physics

Lots of linear algebra problems left!


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


Checkerboard Model Reduction

Finite element model: N = 2154Expensive to solve for every H(ω) evaluation!

Build a reduced-order model to approximate behaviorReduced system of 80 to 100 vectorsEvaluate H(ω) in milliseconds instead of secondsWithout damping: standard Arnoldi projectionWith damping: Second-Order ARnoldi (SOAR)


Checkerboard Simulation

0 2 4 6 8 10

x 10−5

0

2

4

6

8

10

12

x 10

9 9.2 9.4 9.6 9.8

x 107

−200

−180

−160

−140

−120

−100

Frequency (Hz)

Am

plitu

de (

dB)

9 9.2 9.4 9.6 9.8

x 107

0

1

2

3

4

Frequency (Hz)

Pha

se (

rad)


Damping and Q

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

d2udt2

+ Q−1dudt

+ u = F (t)

For a resonant mode with frequency ω ∈ C:

Q :=|ω|

2 Im(ω)=

Stored energyEnergy loss per radian

To understand Q, we need damping models!


Damping Mechanisms

Possible loss mechanisms:Fluid dampingMaterial lossesThermoelastic dampingAnchor loss

Model substrate as semi-infinite with a

Perfectly Matched Layer (PML).


Perfectly Matched Layers

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

Electromagnetics (Berengér, 1994)Quantum mechanics – exterior complex scaling(Simon, 1979)Elasticity in standard finite element framework(Basu and Chopra, 2003)


Model Problem IllustratedOutgoing exp(−ix̃) Incoming exp(ix̃)

Transformed coordinate

Re(x̃)

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




Re(x̃)

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




Re(x̃)

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




Re(x̃)

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




Re(x̃)

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




Re(x̃)

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


Finite Element Implementation

x(ξ)

ξ2

ξ1

x1

x2 x̃2

x̃1

Ωe Ω̃e

Ω�

x̃(x)

Combine PML and isoparametric mappings

ke =∫

Ω�B̃T DB̃J̃ dΩ�

me =∫

Ω�ρNT NJ̃ dΩ�

Matrices are complex symmetric


Eigenvalues and Model Reduction

Want to know about the transfer function H(ω):

H(ω) = BT (K − ω2M)−1B

Can eitherLocate poles of H (eigenvalues of (K , M))Plot H in a frequency range (Bode plot)

Usual tactic: subspace projectionBuild an Arnoldi basis V for a Krylov subspace KnCompute with much smaller V ∗KV and V ∗MV

Can we do better?


Variational Principles

Variational form for complex symmetric eigenproblems:Hermitian (Rayleigh quotient):

ρ(v) =v∗Kvv∗Mv

Complex symmetric (modified Rayleigh quotient):

θ(v) =vT KvvT Mv

(Hochstenbach and Arbenz, 2004)

First-order accurate eigenvectors =⇒Second-order accurate eigenvalues.Key: relation between left and right eigenvectors.


Accurate Model Reduction

Build new projection basis from V :

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

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


Disk Resonator Simulations


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


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) nearly indistinguishablefrom full model (crosses)


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


Response of the Disk Resonator


Variation in Quality of Resonance

Film thickness (µm)

Q

1.2 1.3 1.4 1.5 1.6 1.7 1.8100

102

104

106

108

Simulation and lab measurements vs. disk thickness(Courant Institute) Householder 08 28 / 65

Explanation of Q Variation

Real frequency (MHz)

Imagin

ary

freq

uen

cy(M

Hz)

ab

cdd

e

a b

cdd

e

a = 1.51 µm

b = 1.52 µm

c = 1.53 µm

d = 1.54 µm

e = 1.55 µm

46 46.5 47 47.5 480

0.05

0.1

0.15

0.2

0.25

Interaction of two nearby eigenmodes(Courant Institute) Householder 08 29 / 65

Thermoelastic Damping (TED)


Thermoelastic Damping (TED)

u is displacement and T = T0 + θ is temperature

σ = C�− βθ1ρü = ∇ · σ

ρcv θ̇ = ∇ · (κ∇θ)− βT0 tr(�̇)

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


Nondimensionalized Equations

Continuum equations:

σ = Ĉ�− ξθ1ü = ∇ · σθ̇ = η∇2θ − tr(�̇)

Discrete equations:

Muuü + Kuuu = ξKuθθ + fCθθθ̇ + ηKθθθ = −Cθuu̇

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


Perturbative Mode Calculation

Discretized mode equation:

(−ω2Muu + Kuu)u = ξKuθθ(iωCθθ + ηKθθ)θ = −iωCθuu


Perturbative Mode Calculation

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

(uT0 Kuθθ0uT0 Muuu0

)


The Power of Perturbation

Full method: nonsymmetric eigensolve of sizePerturbation method:

1 Purely mechanical symmetric eigensolve2 Linear solve for corresponding thermal field3 A couple dot products for frequency correction


Zener’s Model

Clarence Zener investigated TED in late 30s-early 40s.Model for beams common in MEMS literature.“Method of orthogonal thermodynamic potentials” == perturbationmethod + a variational method.


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 Zener approximationover a range of frequenciesReal and imaginary parts after first-order correction agree toabout three digits with Arnoldi


Onward!

The purpose of computing is insight, not numbers.Richard Hamming

What about:Modeling more geometrically complex devices?Modeling general dependence on geometry?Modeling general dependence on operating point?Computing nonlinear dynamics?Digesting all this to help designers?


Conclusions

RF MEMS are still a great source of problemsStill many unresolved physical modeling problemsNeed better parameterized and nonlinear model reduction methodsSystem scale simulations mean parallel computing challenges

http://www.cims.nyu.edu/~dbindel


Resonant MEMS and modelsCheckerboard resonator exampleAnchor losses and disk resonatorsThermoelastic losses and beam resonatorsConclusionBackup slidesMesh convergence1D PMLHiQLabCheckerboard resonatorsNonlinear eigenvalue perturbationElectromechanical modelHello world!Reflection Analysis

Documents

Computer Aided Design of Micro-Electro-Mechanical Systemsbindel/present/2008-06-householder-b.pdf0 2 4 6 8 10 12 14 16 18 0 5 10 15 20 0 5 10 15 20-4-2 0-50 0 50 100-1-0.5 0 0.5 1