Computer-Aided Design for Micro-Electro-Mechanical Systems
Transcript of Computer-Aided Design for Micro-Electro-Mechanical Systems
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CAD for
MEMS
ResonantMEMS
HiQLab
Anchor
Losses
DiskResonatorAnalysis
Conclusions
Backup Slides
Computer-Aided Design for
Micro-Electro-Mechanical Systems
Eigenvalues, Energy Losses, and Dick Tracy Watches
D. Bindel
Computer Science DivisionDepartment of EECSUniversity of California, Berkeley
Purdue, 17 Apr 2006
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CAD for
MEMS
ResonantMEMS
HiQLab
Anchor
Losses
DiskResonatorAnalysis
Conclusions
Backup Slides
The Computational Science Picture
Application modeling
Checkerboard resonatorDisk resonatorShear ring resonator
Mathematical analysis
Physical modeling and finite element technologyStructured eigenproblems and reduced-order modelsParameter-dependent eigenproblems
Software engineeringHiQLabSUGARFEAPMEX
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CAD for
MEMS
ResonantMEMS
HiQLab
Anchor
Losses
DiskResonatorAnalysis
Conclusions
Backup Slides
The Computational Science Picture
Application modeling
Checkerboard resonatorDisk resonatorShear ring resonator
Mathematical analysis
Physical modeling and finite element technologyStructured eigenproblems and reduced-order modelsParameter-dependent eigenproblems
Software engineeringHiQLabSUGARFEAPMEX
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CAD for
MEMS
ResonantMEMS
HiQLab
Anchor
Losses
DiskResonatorAnalysis
Conclusions
Backup Slides
Outline
1 Resonant MEMS
2 HiQLab
3 Anchor Losses
4 Disk Resonator Analysis
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CAD for
MEMS
ResonantMEMS
MEMS Basics
RF MEMS
Checkerboard Model
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
Outline
1 Resonant MEMS
2 HiQLab
3 Anchor Losses
4 Disk Resonator Analysis
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CAD for
MEMS
ResonantMEMS
MEMS Basics
RF MEMS
Checkerboard Model
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
What are MEMS?
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CAD for
MEMS
ResonantMEMS
MEMS Basics
RF MEMS
Checkerboard Model
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
MEMS Basics
Micro-Electro-Mechanical Systems
Chemical, 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 technology
Tiny, but still classical physics
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CAD for
MEMS
ResonantMEMS
MEMS Basics
RF MEMS
Checkerboard Model
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
Radio-Frequency MEMS
Microguitars from Cornell University (1997 and 2003)
MHz-GHz mechanical resonators
Impact: smaller, lower-power cell phonesOther uses:
Sensing elements (e.g. chemical sensors)Really high-pitch guitars
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CAD for
MEMS
ResonantMEMS
MEMS Basics
RF MEMS
Checkerboard Model
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
Micromechanical Filters
Mechanical filter
Capacitive senseCapacitive drive
Radio signal
Filtered signal
Mechanical high-frequency (high MHz-GHz) filterYour cell phone is mechanical!New MEMS filters can be integrated with circuitry= smaller and lower power
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CAD for
MEMS
ResonantMEMS
MEMS Basics
RF MEMS
Checkerboard Model
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
Ultimate Success
Calling Dick Tracy!
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CAD for
MEMS
ResonantMEMS
MEMS Basics
RF MEMS
Checkerboard Model
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
Designing Transfer Functions
Time domain:
Mu + Cu + Ku = b(t)
y(t) = pTu
Frequency domain:
2Mu+ iCu+ Ku = b()
y() = pTu
Transfer function:
H() = pT(2M+ iC+ K)1b
y() = H()()
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CAD for
MEMS
ResonantMEMS
MEMS Basics
RF MEMS
Checkerboard Model
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
Narrowband Filter Needs
20log10 |H()|
Want sharp poles for narrowband filters
= Want to minimize damping
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CAD for
MEMS
ResonantMEMS
MEMS Basics
RF MEMS
Checkerboard Model
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
Checkerboard Resonator
D+
D
D+
D
S+ S+
S
S
Anchored at outside corners
Excited atnorthwestcorner
Sensed atsoutheastcorner
Surfaces move only a few nanometers
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CAD for
MEMS
ResonantMEMS
MEMS Basics
RF MEMS
Checkerboard Model
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
Checkerboard Model Reduction
Finite element model: N=2154
Expensive to solve for everyH()evaluation!
Build areduced-order modelto approximate behavior
Reduced system of 80 to 100 vectorsEvaluateH()in milliseconds instead of secondsWithout damping: standard Arnoldi projectionWith damping: Second-Order ARnoldi (SOAR)
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CAD for
MEMS
ResonantMEMS
MEMS Basics
RF MEMS
Checkerboard Model
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
Checkerboard Simulation
0 2 4 6 8 10
x 105
9 9.2 9.4 9.6 9.8
x 107
200
180
160
140
120
100
Frequency (Hz)
Amplitude(dB)
9 9.2 9.4 9.6 9.8
x 107
0
1
2
3
4
Frequency (Hz)
Phase
(rad)
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CAD for
MEMS
ResonantMEMS
MEMS Basics
RF MEMS
Checkerboard Model
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
Checkerboard Measurement
S. Bhave, MEMS 05
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CAD for
MEMS
ResonantMEMS
MEMS Basics
RF MEMS
Checkerboard Model
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
Contributions
Built predictive model used to design checkerboardUsed model reduction to get thousand-fold speedup
fast enough for interactive use
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CAD for
MEMS
ResonantMEMS
HiQLab
Anchor
LossesDiskResonatorAnalysis
Conclusions
Backup Slides
Outline
1 Resonant MEMS
2 HiQLab
3 Anchor Losses
4 Disk Resonator Analysis
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CAD forMEMS
ResonantMEMS
HiQLab
Anchor
LossesDiskResonatorAnalysis
Conclusions
Backup Slides
Damping andQ
Designers want highquality of resonance(Q)
Dimensionless damping in a one-dof system
d2udt2
+ Q1 dudt
+ u=F(t)
For a resonant mode with frequency C:
Q:= ||2 Im() = Stored energy
Energy loss per radian
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
Enter HiQLab
Existing codes do not compute quality factors
... and awkward to prototype new solvers
... and awkward to programmatically define meshes
So I wrote a new finite element code: HiQLab
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
Heritage of HiQLab
SUGAR: SPICE for the MEMS world
System-level simulation using modified nodal analysis
Flexible device description language
C core with MATLAB interfaces and numerical routines
FEAPMEX: MATLAB + a finite element code
MATLAB interfaces for steering, testing solvers, running
parameter studies
Time-tested finite element architecture
But old F77, brittle in places
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
Other Ingredients
Lesser artists borrow. Great artists steal.
Picasso, Dali, Stravinsky?
Lua: www.lua.orgEvolved from simulator data languages (DEL and SOL)Pascal-like syntax fits on one page; complete languagedescription is 21 pagesFast, freely available, widely used in game design
MATLAB: www.mathworks.comThe Language of Technical ComputingGood sparse matrix supportStar-P: http://www.interactivesupercomputing.com/
Standard numerical libraries: ARPACK, UMFPACK
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
HiQLab Structure
User interfaces
(C++)Core libraries
Solver library(C, C++, Fortran, MATLAB)
Element library
(C++)
Problem description
(Lua)
(MATLAB, Lua)
Standard finite element structures + some new ideas
Full scripting language for mesh input
Callbacks for boundary conditions, material properties
MATLAB interface for quick algorithm prototyping
Cross-language bindings are automatically generated
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
HiQLabs Hello World
mesh = Mesh:new(2)
mat = make_material(silicon2, planestrain)
mesh:blocks2d( { 0, l }, { -w/2.0, w/2.0 },
mat )
mesh:set_bc(function(x,y)
if x == 0 then return uu, 0, 0; end
end)
HiQL b H ll W ld
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
HiQLabs Hello World
>> mesh = Mesh_load(beammesh.lua);
>> [M,K] = Mesh_assemble_mk(mesh);
>> [V,D] = eigs(K,M, 5, sm);
>> opt.axequal = 1; opt.deform = 1;
>> Mesh_scale_u(mesh, V(:,1), 2, 1e-6);
>> plotfield2d(mesh, opt);
C t ib ti
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
Contributions
Wrote a new code, HiQLab, to study damping
HiQLab is based on my earlier simulators:
SUGAR, for system-level MEMS simulationFEAPMEX, for scripting parameter studies
O tli
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
Substrate modeling
1-D Example
Finite Elements
Complex Symmetry
DiskResonatorAnalysis
Conclusions
Backup Slides
Outline
1 Resonant MEMS
2 HiQLab
3 Anchor Losses
4 Disk Resonator Analysis
E l Di k R t
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
Substrate modeling
1-D Example
Finite Elements
Complex Symmetry
DiskResonatorAnalysis
Conclusions
Backup Slides
Example: Disk Resonator
SiGe disk resonators built by E. Quvy
Damping Mechanisms
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
Substrate modeling
1-D Example
Finite Elements
Complex Symmetry
DiskResonatorAnalysis
Conclusions
Backup Slides
Damping Mechanisms
Possible loss mechanisms:
Fluid damping
Material losses
Thermoelastic dampingAnchor loss
Model substrate as semi-infinite with a
Perfectly Matched Layer (PML).
Perfectly Matched Layers
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
Substrate modeling
1-D Example
Finite Elements
Complex Symmetry
DiskResonatorAnalysis
Conclusions
Backup Slides
Perfectly Matched Layers
Complex coordinate transformation
Generates a perfectly matched absorbing layer
Idea works with general linear wave equationsElectromagnetics (Berengr, 1994)Quantum mechanics exterior complex scaling(Simon, 1979)Elasticity in standard finite element framework
(Basu and Chopra, 2003)
Model Problem
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
Substrate modeling
1-D Example
Finite Elements
Complex Symmetry
DiskResonatorAnalysis
ConclusionsBackup Slides
Model Problem
Domain: x [0,)
Governing eq:
2u
x2
1
c22u
t2 =0
Fourier transform:
d2u
dx2 + k2u=0
Solution:
u=couteikx + cine
ikx
Model Problem Illustrated
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
Substrate modeling
1-D Example
Finite Elements
Complex Symmetry
DiskResonatorAnalysis
ConclusionsBackup Slides
Model Problem Illustrated
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
Model Problem Illustrated
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
Substrate modeling
1-D Example
Finite Elements
Complex Symmetry
DiskResonatorAnalysis
ConclusionsBackup Slides
Model Problem Illustrated
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
Model Problem Illustrated
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
Substrate modeling
1-D Example
Finite Elements
Complex Symmetry
DiskResonatorAnalysis
ConclusionsBackup Slides
Model Problem Illustrated
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
Model Problem Illustrated
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
Substrate modeling
1-D Example
Finite Elements
Complex Symmetry
DiskResonatorAnalysis
Conclusions
Backup Slides
Model Problem Illustrated
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
Model Problem Illustrated
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
Substrate modeling
1-D Example
Finite Elements
Complex Symmetry
DiskResonatorAnalysis
Conclusions
Backup Slides
Model Problem Illustrated
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
Model Problem Illustrated
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
Substrate modeling
1-D Example
Finite Elements
Complex Symmetry
DiskResonatorAnalysis
Conclusions
Backup Slides
Model Problem Illustrated
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
Finite Element Implementation
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
Substrate modeling
1-D Example
Finite Elements
Complex Symmetry
DiskResonatorAnalysis
Conclusions
Backup Slides
Finite Element Implementation
x()
2
1
x1
x2
x2
x1
e e
x(x)
Combine PML and isoparametric mappings
k
e
= B
T
DBJ d
me =
NTNJ d
Matrices arecomplex symmetric
Eigenvalues and Model Reduction
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
Substrate modeling
1-D Example
Finite Elements
Complex Symmetry
DiskResonatorAnalysis
Conclusions
Backup Slides
Eigenvalues and Model Reduction
Want to know about the transfer functionH():
H() =pT(K 2M)1b
Can either
Locate poles ofH(eigenvalues of(K,M))
PlotHin a frequency range (Bode plot)
Usual tactic: subspace projection
Build an Arnoldi basisV
Compute with much smallerVKV andVMV
Can we do better?
Variational Principles
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
Substrate modeling
1-D Example
Finite Elements
Complex Symmetry
DiskResonatorAnalysis
Conclusions
Backup Slides
p
Variational form for complex symmetric eigenproblems:
Hermitian (Rayleigh quotient):
(v) = vKv
vMv
Complex symmetric (modified Rayleigh quotient):
(v) = vTKv
vTMv
First-order accurate eigenvectors =Second-order accurate eigenvalues.
Key: relation between left and right eigenvectors.
Accurate Model Reduction
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
Substrate modeling
1-D Example
Finite Elements
Complex Symmetry
DiskResonatorAnalysis
Conclusions
Backup Slides
Build new projection basis fromV:
W =orth[Re(V), Im(V)]
span(W)contains both Knand Kn= double digits correct vs. projection with V
Wis a real-valued basis
= projected system is complex symmetric
Contributions
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
Substrate modeling
1-D Example
Finite Elements
Complex Symmetry
DiskResonatorAnalysis
Conclusions
Backup Slides
New formulation of perfectly matched layers
Easy to apply PML to axisymmetric, 2D, or 3D modelsSame formulation applies to electromagnetics, etc.
Analysis of discretization error for PMLs
Results in cheap, automatic parameter optimization
Structure-preserving model reduction for complexsymmetric systems
Double accuracy for same work as standard method
Outline
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
1 Resonant MEMS
2 HiQLab
3 Anchor Losses
4 Disk Resonator Analysis
Disk Resonator Simulations
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
Disk Resonator Mesh
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
PML region
Wafer (unmodeled)
Electrode
Resonating disk
0 1 2 3 4
x 105
4
2
0
2
x 106
Axisymmetric model with bicubic mesh
About 10K nodal points in converged calculation
Mesh Convergence
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
Mesh density
Computed
Q
Cubic
LinearQuadratic
1 2 3 4 5 6 7 80
1000
2000
3000
4000
5000
6000
7000
Cubic elements converge with reasonable mesh density
Model Reduction Accuracy
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
Frequency (MHz)
Transfer(dB)
Frequency (MHz)
Phase(degrees)
47.2 47.25 47.3
47.2 47.25 47.3
0
100
200
-80
-60
-40
-20
Results from ROM (solid and dotted lines) near
indistinguishable from full model(crosses)
Model Reduction Accuracy
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
Frequency (MHz)
|H()
Hredu
ced
()|/H()|
Arnoldi ROMStructure-preserving ROM
45 46 47 48 49 50
106
104
102
Preserve structure =get twice the correct digits
Response of the Disk Resonator
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
Variation in Quality of Resonance
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
Film thickness (m)
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 ofQVariation
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
Real frequency (MHz)
Imaginaryfreque
ncy(MHz)
ab
cdd
e
a b
cdd
e
a =1.51 m
b =1.52 m
c =1.53 m
d =1.54 m
e =1.55 m
46 46.5 47 47.5 480
0.05
0.1
0.15
0.2
. 5
Interaction of two nearby eigenmodes
Contributions
http://movies/sweep35s.movhttp://movies/sweep35s.movhttp://movies/sweep35s.movhttp://movies/sweep35s.movhttp://movies/sweep35s.movhttp://movies/sweep35s.movhttp://movies/sweep35s.movhttp://movies/sweep35s.movhttp://movies/sweep35s.movhttp://movies/sweep35s.movhttp://movies/sweep35s.movhttp://movies/sweep35s.movhttp://movies/sweep35s.movhttp://movies/sweep35s.movhttp://movies/sweep35s.movhttp://movies/sweep35s.movhttp://movies/sweep35s.movhttp://find/ -
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
Built disk model in HiQLab and verified against lab
measurements
Demonstrated dominance of anchor loss for this device
Predicted geometric sensitivity of quality factor Q
(which was subsequently verified in the lab)
ExplainedQsensitivity in terms of mode interference
Contributions Summary (1)
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
Application modeling
Finite element models of several devicesDiscovery of effects of mode interference
Importance of anchor loss vs thermoelastic dampingMathematical analysis
Reformulation of perfectly-matched layersAnalysis of discretization and parameter choice in PMLsComplex symmetry-preserving model reduction
Perturbation solution for thermoelastic dampingSoftware: HiQLab, FEAPMEX, SUGAR
Contributions Summary (2)
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
HiQLab (about 33000 lines of code)
Collaborations at Berkeley, Cornell, Stanford, Bosch
FEAPMEX (about 5000 lines of code)
2400+ page viewsUsed for instrument models, stochastic structuralanalysis, ultrasonic nondestructive evaluation problems,material parameter identification problems
SUGAR (about 18000 lines of code)
2000+ downloadsUsed in classes at Berkeley, Cornell, Johns HopkinsContinued research use for design optimization
Other Contributions
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
Lowered complexity of rootsfromO(n3)toO(n2)
Code to go into next LAPACK release
Developed first sparse subspace continuation code
Going into the next Matcont release
Developed new network tomography method
Designed initial security model for OceanStore
Served as IEEE 754R secretary
Responsible for last CLAPACK version
Future Work
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
Code development
Structural elements and elements for different physicsDesign and implementation of parallelized version
Theoretical analysisMore damping mechanismsSensitivity analysis and variational model reduction
Application collaborations
Use of nonlinear effects (quasi-static and dynamic)
New designs (e.g. internal dielectric drives)Continued experimental comparisons
Conclusions
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup Slides
RF MEMS are a great source of problems
Interesting applications
Interesting physics (and not altogether understood)Interesting numerical mathematics
http://www.cs.berkeley.edu/~dbindel
Application: Network monitoring
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex SymmetryDamping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
Set ofnhosts in a large network
n(n 1)directed paths between them
Want latency and packet loss rates for each path
Use info to choose servers, route around faults
Network distance metrics
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex SymmetryDamping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
Linear relation between path lengthyijand link length xl:
yij=
lpath
xl=
l
gijlxl
wheregijlindicates if linklis used on path i j.Write in matrix form (one path per row): y=Gx.
Examples:
LatencyLog probability of transmission success
Jitter (variance in latency)
Path dependencies
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex SymmetryDamping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
A3
B3
B2
B1
A1
A2
(Ai Bi) = (A1 Bi) + (Ai B1) (A1 B1)
Many linear dependencies among paths!
k :=rank(G) n2
Measure onlykpaths to infer properties for all paths
Rank ofG: Lucent scan (bound)
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex SymmetryDamping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
100 200 300 400 500 600 700 800 900 10000
1
2
3
4
5
6
7x 10
4
Number of end hosts ontheoverlay (n)
Rankofpathspace(k)
original measurementregression on nregression on nlogn
regressino on n1.25
regression on n1.5
regression on n1.75
Rank ofG: AS-level Albert-Barabasi
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex SymmetryDamping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation100 200 300 400 500 600 700 800 900 10000
1000
2000
3000
4000
5000
6000
7000
8000
Number of endhostson the overlay(n)
Rankofpathspace
(k)
original measurementregression on nregression on nlogn
regressino on n1.25
regression on n1.5
regression on n1.75
Rank ofG: AS Barabasi + RT Waxman
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex SymmetryDamping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation100 200 300 400 500 600 700 800 900 10000
1000
2000
3000
4000
5000
6000
7000
8000
9000
Number of endhostson the overlay(n)
Rankofpathspace
(k)
original measurementregression on nregression on nlogn
regression on n1.25
regression on n1.5
regression on n1.75
Contributions
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex SymmetryDamping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
Accurate predictions onO(n2)paths from a smallnumber of measurements
Prototype monitoring system on PlanetLab
Ongoing work on fault diagnosis using similar ideas
See: http://list.cs.northwestern.edu/
Model with Perfectly Matched Layer
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex SymmetryDamping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
Transformed domainx
Regular domain
dx
dx =(x)where(s) =1 i(s)
d2u
dx2
+ k2u=0
u=couteikx + cine
ikx
Model with Perfectly Matched Layer
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex SymmetryDamping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
Transformed domainx
Regular domain
dx
dx =(x)where(s) =1 i(s),
1
d
dx
1
du
dx+ k
2u=0
u=couteikxk(x) + cine
ikx+k(x)
(x) = x
0
(s) ds
Model with Perfectly Matched Layer
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex SymmetryDamping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
Transformed domainx
Regular domain
If solution clamped atx=Lthen
cincout
=O(ek)where= (L) =
L
0
(s) ds
Eigenvalues and Model Reduction
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex SymmetryDamping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
Want to know about the transfer functionH():
H() =pT(K 2M)1b
Can either
Locate poles ofH(eigenvalues of(K,M))PlotHin a frequency range (Bode plot)
Usual tactic: subspace projection
Build an Arnoldi basisV
Compute with much smallerVKV andVMV
Can we do better?
Variational Principles
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
Variational form for complex symmetric eigenproblems:
Hermitian (Rayleigh quotient):
(v) = vKv
vMv
Complex symmetric (modified Rayleigh quotient):
(v) = vTKv
vTMv
First-order accurate eigenvectors =Second-order accurate eigenvalues.
Key: relation between left and right eigenvectors.
Accurate Model Reduction
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
Build new projection basis fromV:
W =orth[Re(V), Im(V)]
span(W)contains both Knand Kn= double digits correct vs. projection with V
Wis a real-valued basis
= projected system is complex symmetric
Model Reduction Accuracy
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
Frequency (MHz)
Transfer(dB)
Frequency (MHz)
Phase(degrees)
47.2 47.25 47.3
47.2 47.25 47.3
0
100
200
-80
-60
-40
-20
Results from ROM (solid and dotted lines) near
indistinguishable from full model (crosses)
Model Reduction Accuracy
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
Frequency (MHz)
|H()
Hred
uced
()|/H()|
Arnoldi ROM
Structure-preserving ROM
45 46 47 48 49 50
106
10
4
102
Preserve structure =get twice the correct digits
Time-Averaged Energy Flux
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
0 0.5 1 1.5 2 2.5 3 3.5
x 105
2
0
2
x 106
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 106
1
0.5
0
0.5
1
1.5
2
2.5x 10
6
Sources of Damping
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
Fluid damping
Air is a viscous fluid (Re 1)Can operate in a vacuumShown not to dominate in many RF designs
Material lossesLow intrinsic losses in silicon, diamond, germaniumTerrible material losses in metals
Thermoelastic damping
Volume changes induce temperature change
Diffusion of heat leads to mechanical lossAnchor loss
Elastic waves radiate from structure
Sources of Damping
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
Fluid damping
Air is a viscous fluid (Re 1)Can operate in a vacuumShown not to dominate in many RF designs
Material losses
Low intrinsic losses in silicon, diamond, germaniumTerrible material losses in metals
Thermoelastic damping
Volume changes induce temperature change
Diffusion of heat leads to mechanical lossAnchor loss
Elastic waves radiate from structure
Fabrication Outline
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
300 um
1 Si wafer
2 Deposit 2 microns SiO2
3 Pattern and etch SiO2 layer
4 Deposit 2 microns polycrystalline Si5 Pattern and etch Si layer
6 Release etch remaining SiO2
Fabrication Outline
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
300 um
2 um
1 Si wafer
2 Deposit 2 microns SiO2
3 Pattern and etch SiO2 layer
4 Deposit 2 microns polycrystalline Si5 Pattern and etch Si layer
6 Release etch remaining SiO2
Fabrication Outline
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
300 um
2 um
1 Si wafer
2 Deposit 2 microns SiO2
3 Pattern and etch SiO2 layer
4 Deposit 2 microns polycrystalline Si5 Pattern and etch Si layer
6 Release etch remaining SiO2
Fabrication Outline
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
300 um
2 um
2 um
1 Si wafer
2 Deposit 2 microns SiO2
3 Pattern and etch SiO2 layer
4 Deposit 2 microns polycrystalline Si5 Pattern and etch Si layer
6 Release etch remaining SiO2
Fabrication Outline
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
300 um
2 um
2 um
1 Si wafer
2 Deposit 2 microns SiO2
3 Pattern and etch SiO2 layer
4 Deposit 2 microns polycrystalline Si5 Pattern and etch Si layer
6 Release etch remaining SiO2
CAD f
Fabrication Outline
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
300 um
2 um
2 um
1 Si wafer
2 Deposit 2 microns SiO2
3 Pattern and etch SiO2 layer
4 Deposit 2 microns polycrystalline Si
5 Pattern and etch Si layer
6 Release etch remaining SiO2
CAD for
Fabrication Result
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
300 um
2 um
2 um
(C. Nguyen, iMEMS 02)
CAD for
Role of simulation
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
HiQLab: Modeling RF MEMSExplore fundamental device physics
Particularly details of damping
Detailed finite element modeling
Reduced models eventually to go into SUGAR
SUGAR: Be SPICE to the MEMS world
Fast enough for early design stages
Simple enough to attract users
Support design, analysis, optimization, synthesis
Verify models by comparison to measurement
CAD for
Why simulate?
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
Build and break is too expensive
Wafer processing costs months, thousands of dollarsFabrication is impreciseDays or weeks to take good measurements
Good experiments need good hypotheses
Even when device behavior is understood, still need to
understand system behavior
CAD for
From simulation to synthesis
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
Computer can assist at many levels:
Fundamental physics
Detailed device models
System-level models and macromodels
Metrology
Design optimization
Design synthesis
CAD for
Research thrusts
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CAD forMEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
Model development (e.g. new finite elements)
Numerical algorithms (e.g. model reduction)
Numerical software engineering (SUGAR, HiQLab)
Metrology and comparison to measurement
Optimization and design synthesis
CAD for
Research group
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MEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
Faculty Students
A. Agogino (ME) D. Bindel (CS)
Z. Bai (Math,CS,UCD) J.V. Clark (AS&T)
J. Demmel (Math,CS) C. Cobb (ME)S. Govindjee (CEE) D. Garmire (CS)
R. Howe (EE,ME) T. Koyama (CEE)
K.S.J. Pister (EE) J. Nie (Math)
C. Sequin (CS) H. Wei (CEE)
Y. Zhang (CEE)
CAD for
SUGAR
Goal: Be SPICE to the MEMS world
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MEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
Goal: Be SPICE to the MEMS world
Fast enough for early design stagesSimple enough to attract users
Support design, analysis, optimization, synthesis
Verify models by comparison to measurement
System assembly
Models
Solvers
Matlab Web Library
Sensitivity analysis
Static analysis
Steadystate analysis
Transient analysis
Results
Netlist
Interfaces
CAD for
SUGAR: Analysis of a micromirror
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MEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation (Mirror design by M. Last)
CAD for
SUGAR: Design synthesis
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MEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
CAD forMEMS
SUGAR: Comparison to measurement
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MEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
CAD forMEMS
Elastic PMLs
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MEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
(w) :C : (u) d 2
w u d =
w tnd
(u) =uxs
Start from standard weak form
Introduce transformed xwith xx =
Map back to reference system (J
=det())
All terms are symmetric inwandu
CAD forMEMS
Elastic PMLs
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MEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
(w) :C : (u) d 2
w u d =
w tnd
(u) =uxs
=ux
1s
Start from standard weak form
Introduce transformed xwith xx =
Map back to reference system (J
=det())
All terms are symmetric inwandu
CAD forMEMS
Elastic PMLs
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MEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
(w) :C: (u) J d
2
w u J d =
w tnd
(u) =uxs
=ux
1s
Start from standard weak form
Introduce transformed xwith xx =
Map back to reference system (J
=det())
All terms are symmetric inwandu
CAD forMEMS
Elastic PMLs
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MEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
(w) :C: (u) J d
2
w u J d =
w tnd
(u) =uxs
=ux
1s
Start from standard weak form
Introduce transformed xwith xx =
Map back to reference system (J
=det())
All terms are symmetric inwandu
CAD forMEMS
Continuum 2D model problem
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MEMS
ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
k
L
(x) =
1 i|x L|p, x>L1 x L.
1
x
1
u
x
+
2u
y2+ k2u=0
CAD forMEMS
Continuum 2D model problem
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ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
k
L
(x) =
1 i|x L|p, x>L1 x L.
1
x
1
u
x
k2yu+ k
2u=0
CAD forMEMS
Continuum 2D model problem
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ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
k
L
(x) =
1 i|x L|p, x>L1 x L.
1
x
1ux
+ k2xu=0
1D problem, reflection ofO(ekx)
CAD forMEMS
Discrete 2D model problem
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ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
k
L
Discrete Fourier transform iny
Solve numerically inx
Project solution onto infinite space traveling modes
Extension of Collino and Monk (1998)
CAD forMEMS
Nondimensionalization
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ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
k
L
(x) =
1 i|x L|p, x>L1 x L.
Rate of stretching: hp
Elements per wave: (kxh)1 and(kyh)
1
Elements through the PML: N
CAD forMEMS
Nondimensionalization
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ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
DampingMEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
k
L
(x) =
1 i|x L|p, x>L1 x L.
Rate of stretching: hp
Elements per wave: (kxh)1 and(kyh)
1
Elements through the PML: N
CAD forMEMS
Discrete reflection behavior
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ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
DampingMEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
Number of PML elements
log10
(h)
log10
(r) at (kh) = 10
1
1
1
2
2
2
22 2 2
333
3
3
3
3
444
4
4
5 10 15 20 25 30
-1.5
-1
-0.5
0
0.5
1
Quadratic elements,p=1,(kxh)1 =10
CAD forMEMS
Discrete reflection decomposition
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ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
DampingMEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
Model discrete reflection as two parts:
Far-end reflection (clamping reflection)
Approximated well by continuum calculation
Grows as(kxh)1
growsInterface reflection
Discrete effect: mesh does not resolve decayDoes not depend onNGrows as(kxh)
1 shrinks
CAD forMEMS
Discrete reflection behavior
log10
(r) at (kh) = 10
1 log10(rinterface+ rnominal) at (kh)
= 10
1
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ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
DampingMEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
Number of PML elements
log10
(h)
1
1
1
2
2
2
22 2 2
333
3
3
3
3
444
4
4
5 10 15 20 25 30
-1.5
-1
-0.5
0
0.5
1
Number of PML elements
log10
(h)
1
1
1
2
2
2
2 2 2 2
333
3
3
3
444
4
4
5 10 15 20 25 30
-1.5
-1
-0.5
0
0.5
1
Quadratic elements,p=1,(kxh)1 =10
Model does well at predicting actual reflection
Similar picture for other wavelengths, element types,
stretch functions
CAD forMEMS
Choosing PML parameters
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ResonantMEMS
HiQLab
AnchorLosses
DiskResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
DampingMEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
Discrete reflection dominated by
Interface reflection whenkx largeFar-end reflection whenkxsmall
Heuristic for PML parameter choice
Choose an acceptable reflection levelChoosebased on interface reflection atkmaxxChoose length based on far-end reflection at kminx
CAD forMEMS
Thermoelastic damping (TED)
u is displacement and T = T0 + is temperature
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ResonantMEMS
HiQLab
AnchorLosses
Disk
ResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
DampingMEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
u is displacement andT T0+ is temperature
= C 1
utt =
cvt = () T0 tr(t)
Volumetric strain rate drives energy transfer frommechanical to thermal domain
Irreversible diffusion = mechanical dampingNot often an important factor at the macro scale
Recognized source of damping in microresonatorsZener: semi-analytical approximation for TED in beams
We consider the fully coupled system
CAD forMEMS
Nondimensionalization
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ResonantMEMS
HiQLab
AnchorLosses
Disk
ResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
DampingMEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
= C 1utt =
t = 2 tr(t)
:=
c
2 T0
cvand :=
cvcL
Length L
Time L/c, wherec=
E/
Temperature T0
cv
CAD forMEMS
Scaling analysis
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ResonantMEMS
HiQLab
AnchorLosses
Disk
ResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
DampingMEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
= C 1
utt =
t = 2 tr(t)
:=
c
2 T0cv
and :=
cvcL
Micron-scale poly-Si devices: and are 104
.Smallleads to thermal boundary layers
Linearize about=0
CAD forMEMS
Discrete mode equations
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ResonantMEMS
HiQLab
AnchorLosses
Disk
ResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
DampingMEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
= C 1utt =
t = 2 tr(t)
= C 1
2u =
i = 2 i tr()
2Muuu+ Kuuu+ Kut = 0
iDtt+ Ktt+ iDtuu = 0
CAD forMEMS
Perturbation computation
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ResonantMEMS
HiQLab
AnchorLosses
Disk
ResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
DampingMEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
2Muuu+ Kuuu+ Kut = 0
iDtt+ Ktt+ iDtuu = 0
Approximateby perturbation aboutKut=0:
20Muuu0+ Kuuu0 = 0
i0Dtt0+ Ktt0+ i0Dtuu0 = 0
Choosev :vTu0 =0 and compute
(20Muu+ Kuu) 20Muuu0
vT 0
u
=
Kut0
0
CAD forMEMS
Comparison to Zeners model
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ResonantMEMS
HiQLab
AnchorLosses
Disk
ResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
DampingMEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
105
106
107
108
109
1010
107
106
105
104
Thermo
elasticDampingQ
1
Frequency f(Hz)
Zeners Formula
HiQlab Results
Comparison of fully coupled simulation to Zener
approximation over a range of frequencies
Real and imaginary parts after first-order correction
agree to about three digits withArnoldi
CAD forMEMS
Thermoelastic boundary layer
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ResonantMEMS
HiQLab
AnchorLosses
Disk
ResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
0 20 40 60 80 100 1200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 20 40 60 80 100 1200
0.5
1x 10
4
One-dimensional test problem (longitudinal mode in a
bar)
Fixed temperature and displacement at left
Free at right
CAD forMEMS
Shear ring resonator
0.9
hw = 5.000000e05
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ResonantMEMS
HiQLab
AnchorLosses
Disk
ResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Value = 2.66E+07 Hz.
Ring is driven in a shearing motion
Can couple ring to other resonators
How do we track the desired mode?
CAD forMEMS
Mode tracking
Find a continuous solution to
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ResonantMEMS
HiQLab
AnchorLosses
Disk
ResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
Find a continuous solution toK(s) (s)2M(s)
u(s) =0.
K andMare symmetric andM>0Eigenvectors areM-orthogonal
Perturbation theory gives good shifts
Look ifu(s+ h)andu(s)are on the same path by
looking atu(s+ h)
T
M(s+ h)u(s)Many more subtleties in the nonsymmetric case
Focus of theCIS algorithm
CAD forMEMS
Mode tracking in a shear resonator
5 5x 10
7
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ResonantMEMS
HiQLab
AnchorLosses
Disk
ResonatorAnalysis
Conclusions
Backup SlidesNetwork tomography
PML model problem
Complex Symmetry
Damping
MEMS Fabrication
SUGAR and HiQLab
Elastic PMLs
Reflection Analysis
TED
Subspace
Continuation
2 2.5 3 3.5 4 4.5 5 5.5 6
x 105
2.5
3
3.5
4
4.5
5
5.5
Halfwidthof annulus
Frequency(Hz)
Finite elementAnalytic result
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