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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
The Computational Science & Engineering Picture
Application
Analysis Computation
(Cornell University) Fudan 2 / 56
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.
(Cornell University) Fudan 3 / 56
Resonant MEMS
Outline
1 Resonant MEMS
2 Anchor losses and disk resonators
3 Elastic wave gyros
4 Conclusion
(Cornell University) Fudan 4 / 56
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
(Cornell University) Fudan 5 / 56
Resonant MEMS
Where are MEMS used?
(Cornell University) Fudan 6 / 56
Resonant MEMS
My favorite applications
(Cornell University) Fudan 7 / 56
Resonant MEMS
Why you should care, too!
(Cornell University) Fudan 8 / 56
Resonant MEMS
The Mechanical Cell Phone
filter
Mixer
Tuning
control
...RF amplifier /
preselector
Local
Oscillator
IF amplifier /
...and lots of mechanical sensors, too!
(Cornell University) Fudan 9 / 56
Resonant MEMS
Ultimate Success
“Calling Dick Tracy!”
(Cornell University) Fudan 10 / 56
Resonant MEMS
Computational Challenges
Devices are fun – but I’m not a device designer.Why am I in this?
(Cornell University) Fudan 11 / 56
Resonant MEMS
Model System
DC
AC
V
in
V
out
(Cornell University) Fudan 12 / 56
Resonant MEMS
The Circuit Designer View
AC
C
0
V
out
V
in
L
x
C
x
R
x
(Cornell University) Fudan 13 / 56
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
)
(Cornell University) Fudan 14 / 56
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
(Cornell University) Fudan 15 / 56
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!
(Cornell University) Fudan 16 / 56
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.
(Cornell University) Fudan 17 / 56
Anchor losses and disk resonators
Outline
1 Resonant MEMS
2 Anchor losses and disk resonators
3 Elastic wave gyros
4 Conclusion
(Cornell University) Fudan 18 / 56
Anchor losses and disk resonators
Disk Resonator Simulations
(Cornell University) Fudan 19 / 56
Anchor losses and disk resonators
Damping Mechanisms
Possible loss mechanisms:Fluid dampingMaterial lossesThermoelastic dampingAnchor loss
Model substrate as semi-infinite =) resonances!
(Cornell University) Fudan 20 / 56
Anchor losses and disk resonators
Resonances in Physics
(Cornell University) Fudan 21 / 56
Anchor losses and disk resonators
Resonances and Literature
In bells of frost I heard the resonance die.– Li Bai (translated by Vikram Seth)
(Cornell University) Fudan 22 / 56
Anchor losses and disk resonators
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)
(Cornell University) Fudan 23 / 56
Anchor losses and disk resonators
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
(Cornell University) Fudan 24 / 56
Anchor losses and disk resonators
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
-2
-1
0
1
2
3
-1
-0.5
0
0.5
1
(Cornell University) Fudan 24 / 56
Anchor losses and disk resonators
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
-4
-2
0
2
4
6
-1
-0.5
0
0.5
1
(Cornell University) Fudan 24 / 56
Anchor losses and disk resonators
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
-10
-5
0
5
10
15
-1
-0.5
0
0.5
1
(Cornell University) Fudan 24 / 56
Anchor losses and disk resonators
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
-20
0
20
40
-1
-0.5
0
0.5
1
(Cornell University) Fudan 24 / 56
Anchor losses and disk resonators
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
-50
0
50
100
-1
-0.5
0
0.5
1
(Cornell University) Fudan 24 / 56
Anchor losses and disk resonators
Finite Element Implementation
x(⇠)
⇠
2
⇠
1
x
1
x
2
x
2
x
1
⌦e ⌦e
⌦⇤
x(x)
Matrices are complex symmetric
(Cornell University) Fudan 25 / 56
Anchor losses and disk resonators
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(!).
(Cornell University) Fudan 26 / 56
Anchor losses and disk resonators
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.
(Cornell University) Fudan 27 / 56
Anchor losses and disk resonators
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!
(Cornell University) Fudan 28 / 56
Anchor losses and disk resonators
Disk Resonator Simulations
(Cornell University) Fudan 29 / 56
Anchor losses and disk resonators
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
(Cornell University) Fudan 30 / 56
Anchor losses and disk resonators
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)
(Cornell University) Fudan 31 / 56
Anchor losses and disk resonators
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
(Cornell University) Fudan 32 / 56
Anchor losses and disk resonators
Response of the Disk Resonator
(Cornell University) Fudan 33 / 56
Anchor losses and disk resonators
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
(Cornell University) Fudan 34 / 56
Anchor losses and disk resonators
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
(Cornell University) Fudan 35 / 56
Anchor losses and disk resonators
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
(Cornell University) Fudan 36 / 56
Elastic wave gyros
Outline
1 Resonant MEMS
2 Anchor losses and disk resonators
3 Elastic wave gyros
4 Conclusion
(Cornell University) Fudan 37 / 56
Elastic wave gyros
Bryan’s Experiment
“On the beats in the vibrations of a revolving cylinder or bell”by G. H. Bryan, 1890
(Cornell University) Fudan 38 / 56
Elastic wave gyros
A Small Application
Northrup-Grummond HRG
(Cornell University) Fudan 39 / 56
Elastic wave gyros
Current example: Micro-HRG / GOBLiT / OMG
Goal: Cheap, small (1mm) HRGCollaborator roles:
Basic designFabricationMeasurement
Our part:Detailed physicsFast softwareSensitivityDesign optimization
(Cornell University) Fudan 40 / 56
Elastic wave gyros
How It Works
(Cornell University) Fudan 41 / 56
Elastic wave gyros
How It Works
(Cornell University) Fudan 42 / 56
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
(Cornell University) Fudan 43 / 56
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...
(Cornell University) Fudan 44 / 56
Elastic wave gyros
Full Dynamics
(Cornell University) Fudan 45 / 56
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
(Cornell University) Fudan 46 / 56
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
(Cornell University) Fudan 47 / 56
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
(Cornell University) Fudan 48 / 56
Elastic wave gyros
Fourier Picture
Perfect axisymmetry:2
6664
K11
K22
K33
. . .
3
7775� !2
2
6664
M11
M22
M33
. . .
3
7775
(Cornell University) Fudan 49 / 56
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
(Cornell University) Fudan 50 / 56
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
.
(Cornell University) Fudan 51 / 56
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?
(Cornell University) Fudan 52 / 56
Conclusion
Outline
1 Resonant MEMS
2 Anchor losses and disk resonators
3 Elastic wave gyros
4 Conclusion
(Cornell University) Fudan 53 / 56
Conclusion
The Computational Science & Engineering Picture
Application
Analysis Computation
(Cornell University) Fudan 54 / 56
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
(Cornell University) Fudan 55 / 56
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