Computer-Aided Design for Micro-Electro-Mechanical Systems

8/21/2019 Computer-Aided Design for Micro-Electro-Mechanical Systems

1/115

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
http://find/


2/115

CAD for

MEMS

ResonantMEMS

HiQLab

Anchor

Losses


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
http://find/


3/115

CAD for

MEMS

ResonantMEMS

HiQLab

Anchor

Losses


Conclusions

Backup Slides

The Computational Science Picture


Checkerboard resonatorDisk resonatorShear ring resonator

Mathematical analysis

Physical modeling and finite element technologyStructured eigenproblems and reduced-order modelsParameter-dependent eigenproblems

Software engineeringHiQLabSUGARFEAPMEX
http://goforward/http://find/http://goback/


4/115

CAD for

MEMS

ResonantMEMS

HiQLab

Anchor

Losses


Conclusions

Backup Slides

Outline

1 Resonant MEMS

2 HiQLab

3 Anchor Losses

4 Disk Resonator Analysis
http://find/http://goback/


5/115

CAD for

MEMS

ResonantMEMS

MEMS Basics

RF MEMS

Checkerboard Model

HiQLab

AnchorLosses


Conclusions

Backup Slides

Outline

1 Resonant MEMS

2 HiQLab

3 Anchor Losses

http://find/


6/115

CAD for

MEMS

ResonantMEMS

MEMS Basics

RF MEMS

Checkerboard Model

HiQLab

AnchorLosses


Conclusions

Backup Slides

What are MEMS?
http://applications/AMSVisualizer.apphttp://find/http://goback/


7/115

CAD for

MEMS

ResonantMEMS

MEMS Basics

RF MEMS

Checkerboard Model

HiQLab

AnchorLosses


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
http://find/


8/115

CAD for

MEMS

ResonantMEMS

MEMS Basics

RF MEMS

Checkerboard Model

HiQLab

AnchorLosses


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
http://find/


9/115

CAD for

MEMS

ResonantMEMS

MEMS Basics

RF MEMS

Checkerboard Model

HiQLab

AnchorLosses


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


10/115

CAD for

MEMS

ResonantMEMS

MEMS Basics

RF MEMS

Checkerboard Model

HiQLab

AnchorLosses


Conclusions

Backup Slides

Ultimate Success

Calling Dick Tracy!
http://find/


11/115

CAD for

MEMS

ResonantMEMS

MEMS Basics

RF MEMS

Checkerboard Model

HiQLab

AnchorLosses


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()()


12/115

CAD for

MEMS

ResonantMEMS

MEMS Basics

RF MEMS

Checkerboard Model

HiQLab

AnchorLosses


Conclusions

Backup Slides

Narrowband Filter Needs

20log10 |H()|

Want sharp poles for narrowband filters

= Want to minimize damping
http://find/


13/115

CAD for

MEMS

ResonantMEMS

MEMS Basics

RF MEMS

Checkerboard Model

HiQLab

AnchorLosses


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
http://movies/corner95.avihttp://find/


14/115

CAD for

MEMS

ResonantMEMS

MEMS Basics

RF MEMS

Checkerboard Model

HiQLab

AnchorLosses


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)
http://find/


15/115

CAD for

MEMS

ResonantMEMS

MEMS Basics

RF MEMS

Checkerboard Model

HiQLab

AnchorLosses


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)
http://find/


16/115

CAD for

MEMS

ResonantMEMS

MEMS Basics

RF MEMS

Checkerboard Model

HiQLab

AnchorLosses


Conclusions

Backup Slides

Checkerboard Measurement

S. Bhave, MEMS 05
http://find/


17/115

CAD for

MEMS

ResonantMEMS

MEMS Basics

RF MEMS

Checkerboard Model

HiQLab

AnchorLosses


Conclusions

Backup Slides

Contributions

Built predictive model used to design checkerboardUsed model reduction to get thousand-fold speedup

fast enough for interactive use
http://find/


18/115

CAD for

MEMS

ResonantMEMS

HiQLab

Anchor

LossesDiskResonatorAnalysis

Conclusions

Backup Slides

Outline

1 Resonant MEMS

2 HiQLab

3 Anchor Losses

http://find/


19/115

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
http://find/


20/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


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
http://find/


21/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


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
http://find/


22/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


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
http://find/


23/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


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
http://find/


24/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


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
http://goback/http://find/http://goback/


25/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


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
http://find/


26/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


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


27/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses

Substrate modeling

1-D Example

Finite Elements

Complex Symmetry


Conclusions

Backup Slides

Outline

1 Resonant MEMS

2 HiQLab

3 Anchor Losses


E l Di k R t
http://find/


28/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses

Substrate modeling

1-D Example

Finite Elements

Complex Symmetry


Conclusions

Backup Slides

Example: Disk Resonator

SiGe disk resonators built by E. Quvy

Damping Mechanisms
http://find/


29/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses

Substrate modeling

1-D Example

Finite Elements

Complex Symmetry


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
http://find/


30/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses

Substrate modeling

1-D Example

Finite Elements

Complex Symmetry


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
http://find/


31/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses

Substrate modeling

1-D Example

Finite Elements

Complex Symmetry


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


32/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses

Substrate modeling

1-D Example

Finite Elements

Complex Symmetry




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

http://find/


33/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses

Substrate modeling

1-D Example

Finite Elements

Complex Symmetry






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

http://find/


34/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses

Substrate modeling

1-D Example

Finite Elements

Complex Symmetry






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

http://find/


35/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses

Substrate modeling

1-D Example

Finite Elements

Complex Symmetry


Conclusions

Backup Slides




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

http://find/


36/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses

Substrate modeling

1-D Example

Finite Elements

Complex Symmetry


Conclusions

Backup Slides




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

http://find/


37/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses

Substrate modeling

1-D Example

Finite Elements

Complex Symmetry


Conclusions

Backup Slides




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
http://find/


38/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses

Substrate modeling

1-D Example

Finite Elements

Complex Symmetry


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
http://find/


39/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses

Substrate modeling

1-D Example

Finite Elements

Complex Symmetry


Conclusions

Backup Slides


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
http://find/


40/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses

Substrate modeling

1-D Example

Finite Elements

Complex Symmetry


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
http://find/


41/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses

Substrate modeling

1-D Example

Finite Elements

Complex Symmetry


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
http://find/


42/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses

Substrate modeling

1-D Example

Finite Elements

Complex Symmetry


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
http://find/


43/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions

Backup Slides

1 Resonant MEMS

2 HiQLab

3 Anchor Losses


Disk Resonator Simulations
http://find/


44/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions

Backup Slides

Disk Resonator Mesh


45/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


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


46/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


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
http://find/


47/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


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)

http://find/


48/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


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
http://find/


49/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions

Backup Slides

Variation in Quality of Resonance
http://movies/disk40cycle.movhttp://find/


50/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


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
http://movies/sweep35s.movhttp://find/


51/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


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/


52/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


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)
http://find/


53/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions

Backup Slides


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)
http://find/


54/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


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
http://find/


55/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


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
http://find/


56/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


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
http://find/


57/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


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
http://find/


58/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


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
http://find/


59/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


PML model problem


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
http://find/


60/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


PML model problem


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)
http://find/


61/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


PML model problem


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
http://find/


62/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


PML model problem


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)


regressino on n1.25

regression on n1.5

regression on n1.75

Rank ofG: AS Barabasi + RT Waxman
http://find/http://find/


63/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


PML model problem


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)


regression on n1.25

regression on n1.5

regression on n1.75

Contributions
http://find/http://find/


64/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


PML model problem


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
http://find/


65/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


PML model problem


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

http://find/


66/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


PML model problem


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

http://find/


67/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


PML model problem


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

http://find/


68/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


PML model problem


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
http://find/


69/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


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
http://find/


70/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


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

http://find/


71/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


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)

http://find/


72/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


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
http://find/


73/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


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
http://find/


74/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


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
http://find/


75/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


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
http://find/


76/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


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
http://find/


77/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


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





Fabrication Outline


78/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


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





Fabrication Outline
http://find/


79/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


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





Fabrication Outline


80/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


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





CAD f

Fabrication Outline
http://find/


81/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


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



4 Deposit 2 microns polycrystalline Si

5 Pattern and etch Si layer


CAD for

Fabrication Result
http://find/


82/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


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
http://find/


83/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


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?
http://find/


84/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


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
http://find/


85/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


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
http://find/


86/115

CAD forMEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


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
http://find/


87/115

MEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


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
http://find/


88/115

MEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


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
http://find/


89/115

MEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


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


90/115

MEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


PML model problem

Complex Symmetry

Damping

MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis

TED

Subspace

Continuation

CAD forMEMS

SUGAR: Comparison to measurement
http://find/


91/115

MEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


PML model problem

Complex Symmetry

Damping

MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis

TED

Subspace

Continuation

CAD forMEMS

Elastic PMLs
http://find/


92/115

MEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


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
http://find/


93/115

MEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


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




=det())


CAD forMEMS

Elastic PMLs


94/115

MEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


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




=det())


CAD forMEMS

Elastic PMLs
http://find/


95/115

MEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


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




=det())


CAD forMEMS

Continuum 2D model problem
http://find/


96/115

MEMS

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


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

http://find/


97/115

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


PML model problem

Complex Symmetry

Damping

MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis

TED

Subspace

Continuation

k

L

(x) =


1

x

1

u

x

k2yu+ k

2u=0

CAD forMEMS

http://find/


98/115

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


PML model problem

Complex Symmetry

Damping

MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis

TED

Subspace

Continuation

k

L

(x) =


1

x

1ux

+ k2xu=0

1D problem, reflection ofO(ekx)

CAD forMEMS

Discrete 2D model problem
http://find/


99/115

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


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
http://find/


100/115

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


PML model problem

Complex Symmetry

Damping

MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis

TED

Subspace

Continuation

k

L

(x) =


Rate of stretching: hp

Elements per wave: (kxh)1 and(kyh)

1

Elements through the PML: N

CAD forMEMS

http://find/


101/115

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


PML model problem

Complex Symmetry

DampingMEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis

TED

Subspace

Continuation

k

L

(x) =


Rate of stretching: hp

Elements per wave: (kxh)1 and(kyh)

1

Elements through the PML: N

CAD forMEMS

Discrete reflection behavior
http://find/


102/115

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


PML model problem

Complex Symmetry


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
http://find/


103/115

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


PML model problem

Complex Symmetry


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
http://find/


104/115

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


PML model problem

Complex Symmetry


SUGAR and HiQLab

Elastic PMLs

Reflection Analysis

TED

Subspace

Continuation


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


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


105/115

ResonantMEMS

HiQLab

AnchorLosses


Conclusions


PML model problem

Complex Symmetry


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
http://find/


106/115

ResonantMEMS

HiQLab

AnchorLosses

Disk

ResonatorAnalysis

Conclusions


PML model problem

Complex Symmetry


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

http://find/


107/115

ResonantMEMS

HiQLab

AnchorLosses

Disk

ResonatorAnalysis

Conclusions


PML model problem

Complex Symmetry


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
http://find/


108/115

ResonantMEMS

HiQLab

AnchorLosses

Disk

ResonatorAnalysis

Conclusions


PML model problem

Complex Symmetry


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
http://find/


109/115

ResonantMEMS

HiQLab

AnchorLosses

Disk

ResonatorAnalysis

Conclusions


PML model problem

Complex Symmetry


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
http://find/


110/115

ResonantMEMS

HiQLab

AnchorLosses

Disk

ResonatorAnalysis

Conclusions


PML model problem

Complex Symmetry


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
http://find/


111/115

ResonantMEMS

HiQLab

AnchorLosses

Disk

ResonatorAnalysis

Conclusions


PML model problem

Complex Symmetry


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
http://find/


112/115

ResonantMEMS

HiQLab

AnchorLosses

Disk

ResonatorAnalysis

Conclusions


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
http://find/


113/115

ResonantMEMS

HiQLab

AnchorLosses

Disk

ResonatorAnalysis

Conclusions


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
http://find/


114/115

ResonantMEMS

HiQLab

AnchorLosses

Disk

ResonatorAnalysis

Conclusions


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
http://find/


115/115

ResonantMEMS

HiQLab

AnchorLosses

Disk

ResonatorAnalysis

Conclusions


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

Documents

Computer-Aided Design for Micro-Electro-Mechanical Systems