Computer Aided Design of Micro-Electro-Mechanical Systemsbindel/present/2012-12-fudan.pdf ·...

Computer Aided Design ofMicro-Electro-Mechanical Systems

From Eigenvalues to Devices

David Bindel

Department of Computer ScienceCornell University

Fudan University, 13 Dec 2012

(Cornell University) Fudan 1 / 56

David Bindel

Text

The Computational Science & Engineering Picture

Application

Analysis Computation


A Favorite Application: MEMS

I’ve worked on this for a while:SUGAR (early 2000s) – SPICE for MEMSHiQLab (2006) – high-Q mechanical resonator device modelingAxFEM (2012) – solid-wave gyro device modeling

Goal today: two illustrative snapshots.


Resonant MEMS

Outline

1 Resonant MEMS

2 Anchor losses and disk resonators

3 Elastic wave gyros

4 Conclusion


Resonant 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 MEMS

Where are MEMS used?


Resonant MEMS

My favorite applications


Resonant MEMS

Why you should care, too!


Resonant MEMS

The Mechanical Cell Phone

filter

Mixer

Tuning

control

...RF amplifier /

preselector

Local

Oscillator

IF amplifier /

...and lots of mechanical sensors, too!


Resonant MEMS

Ultimate Success

“Calling Dick Tracy!”


Resonant MEMS

Computational Challenges

Devices are fun – but I’m not a device designer.Why am I in this?


Resonant MEMS

Model System

DC

AC

V

in

V

out


Resonant MEMS

The Circuit Designer View

AC

C

0

V

out

V

in

L

x

C

x

R

x


Resonant MEMS

Electromechanical Model

Balance laws ( KCL and BLM ):

d

dt

(C(u)V ) +GV = Iexternal

Mutt

+Ku �ru

✓1

2

V ⇤C(u)V

◆= F

external

Linearize about static equilibium (V0

, u0

):

C(u0

) �Vt

+G �V + (ru

C(u0

) · �ut

)V0

= �Iexternal

M �utt

+

˜K �u +ru

(V ⇤0

C(u0

) �V ) = �Fexternal

where˜K = K � 1

2

@2

@u2(V ⇤

0

C(u0

)V0

)


Resonant MEMS

Electromechanical Model

Assume time-harmonic steady state, no external forces:"i!C +G i!B

�BT

˜K � !2M

# � ˆV�u

�=

� ˆI

external

0

�

Eliminate the mechanical terms:

Y (!) � ˆV = � ˆIexternal

Y (!) = i!C +G + i!H(!)

H(!) = BT

(

˜K � !2M )

�1B

Goal: Understand electromechanical piece (i!H(!)).As a function of geometry and operating pointPreferably as a simple circuit


Resonant MEMS

Damping and Q

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

d2u

dt2+Q�1

du

dt+ u = F (t)

For a resonant mode with frequency ! 2 C:

Q :=

|!|2 Im(!)

=

Stored energyEnergy loss per radian

To understand Q, we need damping models!


Resonant MEMS

The Designer’s Dream

Reality is messy:Coupled physics... some poorly understood (damping)... subject to fabrication errors

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

We aren’t there yet.


Anchor losses and disk resonators

Outline

1 Resonant MEMS



4 Conclusion



Disk Resonator Simulations



Damping Mechanisms

Possible loss mechanisms:Fluid dampingMaterial lossesThermoelastic dampingAnchor loss

Model substrate as semi-infinite =) resonances!



Resonances in Physics



Resonances and Literature

In bells of frost I heard the resonance die.– Li Bai (translated by Vikram Seth)



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)Works great for MEMS, too!(Bindel and Govindjee, 2005)



Model Problem IllustratedOutgoing 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





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





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





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





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





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



Finite Element Implementation

x(⇠)

⇠

2

⇠

1

x

1

x

2

x

2

x

1

⌦e ⌦e

⌦⇤

x(x)

Matrices are complex symmetric



Eigenvalues and Model Reduction

Frequency (MHz)

Tra

nsf

er(d

B)

Frequency (MHz)

Phas

e(d

egre

es)

47.2 47.25 47.3

47.2 47.25 47.3

0

100

200

-80

-60

-40

-20

0

Goal: understand H(!):

H(!) = BT

(K � !2M)

�1B

Look atPoles of H (eigenvalues)Bode plots of H

Model reduction: Replace H(!) by cheaper ˆH(!).



Approximation from Subspaces

A general recipe for large-scale numerical approximation:1 A subspace V containing good approximations.2 A criterion for “optimal” approximations in V.

Basic building block for eigensolvers and model reduction!

Better subspaces, better criteria, better answers.



Variational Principles

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

⇢(v) =v⇤Kv

v⇤Mv

Complex symmetric (modified Rayleigh quotient):

✓(v) =vTKv

vTMv

First-order accurate eigenvectors =)Second-order accurate eigenvalues.Good for model reduction, too!



Disk Resonator Simulations



Disk Resonator Mesh

PML region

Wafer (unmodeled)

Electrode

Resonating disk

0 1 2 3 4x 10−5

−4−2

02

x 10−6

Axisymmetric model, bicubic, ⇡ 10

4 nodal points at convergence



Model Reduction Accuracy

Frequency (MHz)

Tra

nsf

er(d

B)

Frequency (MHz)

Phas

e(d

egre

es)

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



Explanation of Q Variation

Film thickness (µm)

Q

1.2 1.3 1.4 1.5 1.6 1.7 1.8100

102

104

106

108



Explanation of Q Variation

Real frequency (MHz)

Imag

inar

yfr

equen

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


Elastic wave gyros

Outline

1 Resonant MEMS



4 Conclusion


Elastic wave gyros

Bryan’s Experiment

“On the beats in the vibrations of a revolving cylinder or bell”by G. H. Bryan, 1890


Elastic wave gyros

A Small Application

Northrup-Grummond HRG


Elastic wave gyros

Current example: Micro-HRG / GOBLiT / OMG

Goal: Cheap, small (1mm) HRGCollaborator roles:

Basic designFabricationMeasurement

Our part:Detailed physicsFast softwareSensitivityDesign optimization


Elastic wave gyros

How It Works


Elastic wave gyros

Goal state

We want to compute:GeometryFundamental frequenciesAngular gain (Bryan’s factor)Damping (thermoelastic, radiation, material)Sensitivities of everythingEffects of symmetry breaking

For speed and accuracy: use structure!Axisymmetric geometry =) 3D to 2D via FourierPerturbed geometry =) interactions for different wave numbers


Elastic wave gyros

Getting the Geometry

Simple isotropic etch modeling fails – 1mm is huge!Working on better simulator (reaction-diffusion).For now, take idealized geometries on faith...


Elastic wave gyros

Full Dynamics


Elastic wave gyros

Essential Dynamics

Dynamics in 2D subspace of degenerate modes:��!2mI + 2i!⌦gJ + kI

�c = 0

Scaled gain g is Bryan’s factor

BF =

Angular rate of pattern relative to bodyAngular rate of vibrating body


Elastic wave gyros

If no parameters in the world were very large or very small,science would reduce to an exhaustive list of everything.

– Nick Trefethen


Elastic wave gyros

Fourier Picture

Write displacement fields as Fourier series:

u =

1X

m=0

0

@

2

4umr

(r, z) cos(m✓)um✓

(r, z) sin(m✓)umz

(r, z) cos(m✓)

3

5+

2

4�u0

mr

(r, z) sin(m✓)u0m✓

(r, z) cos(m✓)�u0

mz

(r, z) sin(m✓)

3

5

1

A

Works whenever geometry is axisymmetricTreat non-axisymmetric geometries as mapped axisymmetric

Now coefficients in PDEs are non-axisymmetric

Problems with different m decouple if everything axisymmetric


Elastic wave gyros

Fourier Picture

Perfect axisymmetry:2

6664

K11

K22

K33

. . .

3

7775� !2

2

6664

M11

M22

M33

. . .

3

7775


Elastic wave gyros

Fourier Picture

Broken symmetry (via coefficients):2

6664

K11

✏ ✏ ✏✏ K

22

✏ ✏✏ ✏ K

33

✏

✏ ✏ ✏. . .

3

7775� !2

2

6664

M11

✏ ✏ ✏✏ M

22

✏ ✏✏ ✏ M

33

✏

✏ ✏ ✏. . .

3

7775


Elastic wave gyros

Perturbing FourierModes “near” azimuthal number m = nonlinear eigenvalues

⇣K

mm

� !2Mmm

+ Emm

(!)⌘u = 0.

Need:Control on E

mm

Depends on frequency spacingDepends on Fourier analysis of perturbation

Perturbation theory for nonlinearly perturbed eigenproblemsFor self-adjoint case, results similar to Lehmann intervals

First-order estimate: (Kmm

� !2

0

Mmm

) u0

= 0; then

�(!2

) =

uT0

Emm

(!0

) u0

uT0

Mmm

u0

.


Elastic wave gyros

Perturbation and Radiation

Incorporating numerical radiation BCs gives:⇣K � !2M + G(!)

⌘u = 0.

Perturbation approach: ignore G to get (!0

, u0

). Then

�(!2

) =

uT0

G(!0

) u0

uT0

Mmm

u0

.

Works when BC has small influence (coefficients aren’t small).

Also an approach to understanding sensitivity to BC!... explains why PML works okay despite being inappropriate?


Conclusion

Outline

1 Resonant MEMS



4 Conclusion


Conclusion

The Computational Science & Engineering Picture

Application

Analysis Computation


Conclusion

Conclusions

The difference between art and science is that science iswhat we understand well enough to explain to a computer. Artis everything else.

Donald Knuth

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

Collaborators:Disk: S. Govindjee, T. Koyama, S. Bhave, E. QuevyHRG: S. Bhave, L. Fegely, E. Yilmaz

Funding: DARPA MTO, Sloan Foundation


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