Computer-Aided Design for Micro-Electro-Mechanical Systemsbindel/present/2006-02-psu.pdf · Computer-Aided Design for Micro-Electro-Mechanical Systems Eigenvalues, ... Software engineering

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry

DiskResonatorAnalysis

Conclusions

Backup Slides

Computer-Aided Design forMicro-Electro-Mechanical Systems

Eigenvalues, Energy Losses, and Dick Tracy Watches

D. Bindel

Computer Science DivisionDepartment of EECS

University of California, Berkeley

Penn State, 27 Feb 2006

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

The Computational Science Picture

Application modelingCheckerboard resonatorDisk resonatorShear ring resonator

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

Software engineeringHiQLabSUGARFEAPMEX

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

The Computational Science Picture

Application modelingCheckerboard resonatorDisk resonatorShear ring resonator

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

Software engineeringHiQLabSUGARFEAPMEX

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Outline

1 Resonant MEMS

2 Anchor Losses

3 Complex Symmetry

4 Disk Resonator Analysis

CAD forMEMS

ResonantMEMSMEMS Basics

RF MEMS

Checkerboard Model

HiQLab

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Outline

1 Resonant MEMS

2 Anchor Losses

3 Complex Symmetry


CAD forMEMS


RF MEMS

Checkerboard Model

HiQLab

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

What are MEMS?

CAD forMEMS


RF MEMS

Checkerboard Model

HiQLab

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

MEMS Basics

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

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

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

CAD forMEMS


RF MEMS

Checkerboard Model

HiQLab

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Radio-Frequency MEMS

Microguitars from Cornell University (1997 and 2003)

MHz-GHz mechanical resonatorsImpact: smaller, lower-power cell phonesOther uses:

Sensing elements (e.g. chemical sensors)Really high-pitch guitars

CAD forMEMS


RF MEMS

Checkerboard Model

HiQLab

AnchorLosses

ComplexSymmetry


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

CAD forMEMS


RF MEMS

Checkerboard Model

HiQLab

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Ultimate Success

“Calling Dick Tracy!”

CAD forMEMS


RF MEMS

Checkerboard Model

HiQLab

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Designing Transfer Functions

Time domain:

Mu′′ + Cu′ + Ku = bφ(t)y(t) = pT u

Frequency domain:

−ω2Mu + iωCu + K u = bφ(ω)

y(ω) = pT u

Transfer function:

H(ω) = pT (−ω2M + iωC + K )−1by(ω) = H(ω)φ(ω)

CAD forMEMS


RF MEMS

Checkerboard Model

HiQLab

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Narrowband Filter Needs

20 log10 |H(ω)|

ω

Want “sharp” poles for narrowband filters=⇒ Want to minimize damping

CAD forMEMS


RF MEMS

Checkerboard Model

HiQLab

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Checkerboard Resonator

D+

D−

D+

D−

S+ S+

S−

S−

Anchored at outside cornersExcited at northwest cornerSensed at southeast cornerSurfaces move only a few nanometers

CAD forMEMS


RF MEMS

Checkerboard Model

HiQLab

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Checkerboard Model Reduction

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

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

CAD forMEMS


RF MEMS

Checkerboard Model

HiQLab

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Checkerboard Simulation

0 2 4 6 8 10

x 10−5

0

2

4

6

8

10

12

x 10

9 9.2 9.4 9.6 9.8

x 107

−200

−180

−160

−140

−120

−100

Frequency (Hz)

Am

plitu

de (

dB)

9 9.2 9.4 9.6 9.8

x 107

0

1

2

3

4

Frequency (Hz)

Pha

se (

rad)

CAD forMEMS


RF MEMS

Checkerboard Model

HiQLab

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Checkerboard Measurement

S. Bhave, MEMS 05

CAD forMEMS


RF MEMS

Checkerboard Model

HiQLab

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Damping and Q

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

d2udt2 + Q−1 du

dt+ u = F (t)

For a resonant mode with frequency ω ∈ C:

Q :=|ω|

2 Im(ω)=

Stored energyEnergy loss per radian

CAD forMEMS


RF MEMS

Checkerboard Model

HiQLab

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Enter HiQLab

Existing codes do not compute quality factors... and awkward to prototype new solvers... and awkward to programmatically define meshesSo I wrote a new finite element code: HiQLab

CAD forMEMS


RF MEMS

Checkerboard Model

HiQLab

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

HiQLab Structure

User interfaces

(C++)Core libraries

Solver library(C, C++, Fortran, MATLAB)

Element library(C++)

Problem description(Lua)

(MATLAB, Lua)

Full scripting language for mesh inputCallbacks for boundary conditions, material propertiesMATLAB interface for quick algorithm prototypingCross-language bindings are automatically generated

CAD forMEMS


RF MEMS

Checkerboard Model

HiQLab

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Contributions

Built predictive model used to design checkerboardUsed model reduction to get thousand-fold speedup– fast enough for interactive useWrote FEAPMEX to script parameter studiesWrote a new code, HiQLab, to study damping

CAD forMEMS

ResonantMEMS

AnchorLossesSubstrate modeling

1-D Example

Finite Elements

ComplexSymmetry


Conclusions

Backup Slides

Outline

1 Resonant MEMS

2 Anchor Losses

3 Complex Symmetry


CAD forMEMS

ResonantMEMS


1-D Example

Finite Elements

ComplexSymmetry


Conclusions

Backup Slides

Example: Disk Resonator

SiGe disk resonators built by E. Quévy

CAD forMEMS

ResonantMEMS


1-D Example

Finite Elements

ComplexSymmetry


Conclusions

Backup Slides

Damping Mechanisms

Possible loss mechanisms:Fluid dampingMaterial lossesThermoelastic dampingAnchor loss

Model substrate as semi-infinite with a

Perfectly Matched Layer (PML).

CAD forMEMS

ResonantMEMS


1-D Example

Finite Elements

ComplexSymmetry


Conclusions

Backup Slides

Perfectly Matched Layers

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

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

CAD forMEMS

ResonantMEMS


1-D Example

Finite Elements

ComplexSymmetry


Conclusions

Backup Slides

Model Problem

Domain: x ∈ [0,∞)

Governing eq:

∂2u∂x2 −

1c2

∂2u∂t2 = 0

Fourier transform:

d2udx2 + k2u = 0

Solution:u = coute−ikx + cineikx

CAD forMEMS

ResonantMEMS


1-D Example

Finite Elements

ComplexSymmetry


Conclusions

Backup Slides

Model with Perfectly Matched Layer

Transformed domainx

σ

Regular domain

dxdx

= λ(x) where λ(s) = 1− iσ(s)

d2udx2 + k2u = 0

u = coute−ik x + cineik x

CAD forMEMS

ResonantMEMS


1-D Example

Finite Elements

ComplexSymmetry


Conclusions

Backup Slides


Transformed domainx

σ

Regular domain

dxdx

= λ(x) where λ(s) = 1− iσ(s),

1λ

ddx

(1λ

dudx

)+ k2u = 0

u = coute−ikx−kΣ(x) + cineikx+kΣ(x)

Σ(x) =

∫ x

0σ(s) ds

CAD forMEMS

ResonantMEMS


1-D Example

Finite Elements

ComplexSymmetry


Conclusions

Backup Slides


Transformed domainx

σ

Regular domain

If solution clamped at x = L then

cin

cout= O(e−kγ) where γ = Σ(L) =

∫ L

0σ(s) ds

CAD forMEMS

ResonantMEMS


1-D Example

Finite Elements

ComplexSymmetry


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

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

CAD forMEMS

ResonantMEMS


1-D Example

Finite Elements

ComplexSymmetry


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

-2

-1

0

1

2

3

-1

-0.5

0

0.5

1

CAD forMEMS

ResonantMEMS


1-D Example

Finite Elements

ComplexSymmetry


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

-4

-2

0

2

4

6

-1

-0.5

0

0.5

1

CAD forMEMS

ResonantMEMS


1-D Example

Finite Elements

ComplexSymmetry


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

CAD forMEMS

ResonantMEMS


1-D Example

Finite Elements

ComplexSymmetry


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

CAD forMEMS

ResonantMEMS


1-D Example

Finite Elements

ComplexSymmetry


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

CAD forMEMS

ResonantMEMS


1-D Example

Finite Elements

ComplexSymmetry


Conclusions

Backup Slides

Finite Element Implementation

x(ξ)

ξ2

ξ1

x1

x2 x2

x1

Ωe Ωe

Ω

x(x)

Combine PML and isoparametric mappings

ke =

∫Ω

BT DBJ dΩ

me =

∫Ω

ρNT NJ dΩ

Matrices are complex symmetric

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Outline

1 Resonant MEMS

2 Anchor Losses

3 Complex Symmetry


CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Eigenvalues and Model Reduction

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

H(ω) = pT (K − ω2M)−1b

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

Usual tactic: subspace projectionBuild an Arnoldi basis VCompute with much smaller V ∗KV and V ∗MV

Can we do better?

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Variational Principles

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

ρ(v) =v∗Kvv∗Mv

Complex symmetric (modified Rayleigh quotient):

θ(v) =vT KvvT Mv

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

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Accurate Model Reduction

Build new projection basis from V :

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

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

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Contributions

New formulation of perfectly matched layersEasy to apply PML to axisymmetric, 2D, or 3D modelsSame formulation applies to electromagnetics, etc.

Analysis of discretization error for perfectly matchedlayers

Results in cheap, automatic parameter optimizationStructure-preserving model reduction for complexsymmetric systems

Double accuracy for same work as standard method

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Outline

1 Resonant MEMS

2 Anchor Losses

3 Complex Symmetry


CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Disk Resonator Simulations

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Disk Resonator Mesh

PML region

Wafer (unmodeled)

Electrode

Resonating disk

0 1 2 3 4

x 10−5

−4

−2

0

2x 10

−6

Axisymmetric model with bicubic meshAbout 10K nodal points in converged calculation

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Mesh Convergence

Mesh density

Com

pute

dQ

Cubic

LinearQuadratic

1 2 3 4 5 6 7 80

1000

2000

3000

4000

5000

6000

7000

Cubic elements converge with reasonable mesh density

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Model Reduction Accuracy

Frequency (MHz)

Tra

nsf

er(d

B)

Frequency (MHz)

Phase

(deg

rees

)

47.2 47.25 47.3

47.2 47.25 47.3

0

100

200

-80

-60

-40

-20

0

Results from ROM (solid and dotted lines) nearindistinguishable from full model (crosses)

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Model Reduction Accuracy

Frequency (MHz)

|H(ω

)−

Hreduced(ω

)|/H

(ω)|

Arnoldi ROM

Structure-preserving ROM

45 46 47 48 49 50

10−6

10−4

10−2

Preserve structure =⇒get twice the correct digits

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Response of the Disk Resonator

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Time-Averaged Energy Flux

0 0.5 1 1.5 2 2.5 3 3.5

x 10−5

−2

0

2

x 10−6

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10−6

−1

−0.5

0

0.5

1

1.5

2

2.5x 10

−6

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Variation in Quality of Resonance

Film thickness (µm)

Q

1.2 1.3 1.4 1.5 1.6 1.7 1.8100

102

104

106

108

Simulation and lab measurements vs. disk thickness

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Explanation of Q Variation

Real frequency (MHz)

Imagin

ary

freq

uen

cy(M

Hz)

ab

cdd

e

a b

cdd

e

a = 1.51 µm

b = 1.52 µm

c = 1.53 µm

d = 1.54 µm

e = 1.55 µm

46 46.5 47 47.5 480

0.05

0.1

0.15

0.2

0.25

Interaction of two nearby eigenmodes

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Contributions

Built disk model in HiQLab and verified against labmeasurementsDemonstrated dominance of anchor loss for this devicePredicted geometric sensitivity of quality factor Q(which was subsequently verified in the lab)Explained Q sensitivity in terms of mode interference

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Contributions Summary (1)

Application modelingFinite element models of several devicesDiscovery of effects of mode interferenceImportance of anchor loss vs thermoelastic damping

Mathematical analysisReformulation of perfectly-matched layersAnalysis of discretization and parameter choice in PMLsComplex symmetry-preserving model reductionPerturbation solution for thermoelastic damping

Software: HiQLab, FEAPMEX, SUGAR

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Contributions Summary (2)

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

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Other Contributions

Lowered complexity of roots from O(n3) to O(n2)

Code to go into next LAPACK releaseDeveloped first sparse subspace continuation code

Going into the next Matcont release

Developed new network tomography methodDesigned initial security model for OceanStoreServed as IEEE 754R secretaryResponsible for last CLAPACK version

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Future Work

Code developmentStructural elements and elements for different physicsDesign and implementation of parallelized version

Theoretical analysisMore damping mechanismsSensitivity analysis and variational model reduction

Application collaborationsUse of nonlinear effects (quasi-static and dynamic)New designs (e.g. internal dielectric drives)Continued experimental comparisons

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions

Backup Slides

Conclusions

RF MEMS are a great source of problemsInteresting applicationsInteresting physics (and not altogether understood)Interesting numerical mathematics

http://www.cs.berkeley.edu/~dbindel

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions

Backup SlidesDamping

MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis

Perturbations andTED

SubspaceContinuation

Sources of Damping

Fluid dampingAir 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 dampingVolume changes induce temperature changeDiffusion of heat leads to mechanical loss

Anchor lossElastic waves radiate from structure

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Sources of Damping

Fluid dampingAir 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 dampingVolume changes induce temperature changeDiffusion of heat leads to mechanical loss

Anchor lossElastic waves radiate from structure

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Fabrication Outline

300 um

1 Si wafer2 Deposit 2 microns SiO23 Pattern and etch SiO2 layer4 Deposit 2 microns polycrystalline Si5 Pattern and etch Si layer6 Release etch remaining SiO2

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Fabrication Outline

300 um

2 um


CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Fabrication Outline

300 um

2 um


CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Fabrication Outline

300 um

2 um

2 um


CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Fabrication Outline

300 um

2 um

2 um


CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Fabrication Outline

300 um

2 um

2 um


CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Fabrication Result

300 um

2 um

2 um

(C. Nguyen, iMEMS 02)

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Role of simulation

HiQLab: Modeling RF MEMSExplore fundamental device physics

Particularly details of damping

Detailed finite element modelingReduced models eventually to go into SUGAR

SUGAR: “Be SPICE to the MEMS world”Fast enough for early design stagesSimple enough to attract usersSupport design, analysis, optimization, synthesisVerify models by comparison to measurement

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Why simulate?

“Build and break” is too expensiveWafer processing costs months, thousands of dollarsFabrication is impreciseDays or weeks to take good measurements

Good experiments need good hypothesesEven when device behavior is understood, still need tounderstand system behavior

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



From simulation to synthesis

Computer can assist at many levels:Fundamental physicsDetailed device modelsSystem-level models and macromodelsMetrologyDesign optimizationDesign synthesis

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Research thrusts

Model development (e.g. new finite elements)Numerical algorithms (e.g. model reduction)Numerical software engineering (SUGAR, HiQLab)Metrology and comparison to measurementOptimization and design synthesis

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Research group

Faculty StudentsA. 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 forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



SUGAR

Goal: “Be SPICE to the MEMS world”Fast enough for early design stagesSimple enough to attract usersSupport design, analysis, optimization, synthesisVerify models by comparison to measurement

System assembly

Models

Solvers

Matlab Web Library

Sensitivity analysis

Static analysis

Steady−state analysis

Transient analysis

Results

Netlist

Interfaces

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



SUGAR: Analysis of a micromirror

(Mirror design by M. Last)

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



SUGAR: Design synthesis

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



SUGAR: Comparison to measurement

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Elastic PMLs

∫Ω

ε(w) : C : ε(u) dΩ− ω2∫

Ωρw · u dΩ =

∫Γ

w · tndΓ

ε(u) =

(∂u∂x

)s

Start from standard weak formIntroduce transformed x with ∂x

∂x = Λ

Map back to reference system (JΛ = det(Λ))All terms are symmetric in w and u

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Elastic PMLs

∫Ω

ε(w) : C : ε(u) dΩ− ω2∫

Ωρw · u dΩ =

∫Γ

w · tndΓ

ε(u) =

(∂u∂x

)s

=

(∂u∂x

Λ−1)s


∂x = Λ


CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Elastic PMLs

∫Ω

ε(w) : C : ε(u) JΛ dΩ− ω2∫

Ωρw · u JΛ dΩ =

∫Γ

w · tn dΓ

ε(u) =

(∂u∂x

)s

=

(∂u∂x

Λ−1)s


∂x = Λ


CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Elastic PMLs

∫Ω

ε(w) : C : ε(u) JΛ dΩ− ω2∫

Ωρw · u JΛ dΩ =

∫Γ

w · tn dΓ

ε(u) =

(∂u∂x

)s

=

(∂u∂x

Λ−1)s


∂x = Λ


CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Continuum 2D model problem

k

L

λ(x) =

1− iβ|x − L|p, x > L1 x ≤ L.

1λ

∂

∂x

(1λ

∂u∂x

)+

∂2u∂y2 + k2u = 0

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis




k

L

λ(x) =

1− iβ|x − L|p, x > L1 x ≤ L.

1λ

∂

∂x

(1λ

∂u∂x

)− k2

y u + k2u = 0

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis




k

L

λ(x) =

1− iβ|x − L|p, x > L1 x ≤ L.

1λ

∂

∂x

(1λ

∂u∂x

)+ k2

x u = 0

1D problem, reflection of O(e−kxγ)

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Discrete 2D model problem

k

L

Discrete Fourier transform in ySolve numerically in xProject solution onto infinite space traveling modesExtension of Collino and Monk (1998)

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Nondimensionalization

k

L

λ(x) =

1− iβ|x − L|p, x > L1 x ≤ L.

Rate of stretching: βhp

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

Elements through the PML: N

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis




k

L

λ(x) =

1− iβ|x − L|p, x > L1 x ≤ L.

Rate of stretching: βhp

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

Elements through the PML: N

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Discrete reflection behavior

Number of PML elements

log10(β

h)

− log10

(r) at (kh)−1 = 10

1

1

1

2

2

2

2 2 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

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Discrete reflection decomposition

Model discrete reflection as two parts:Far-end reflection (clamping reflection)

Approximated well by continuum calculationGrows as (kxh)−1 grows

Interface reflectionDiscrete effect: mesh does not resolve decayDoes not depend on NGrows as (kxh)−1 shrinks

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Discrete reflection behavior


log10(β

h)

− log10

(r) at (kh)−1 = 10

1

1

1

2

2

2

2 2 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(β

h)

− log10(rinterface + rnominal) at (kh)−1 = 10

1

11

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 reflectionSimilar picture for other wavelengths, element types,stretch functions

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Choosing PML parameters

Discrete reflection dominated byInterface reflection when kx largeFar-end reflection when kx small

Heuristic for PML parameter choiceChoose an acceptable reflection levelChoose β based on interface reflection at kmax

xChoose length based on far-end reflection at kmin

x

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Thermoelastic damping (TED)

u is displacement and T = T0 + θ is temperature

σ = Cε− βθ1ρutt = ∇ · σ

ρcvθt = ∇ · (κ∇θ)− βT0 tr(εt)

Volumetric strain rate drives energy transfer frommechanical to thermal domain

Irreversible diffusion =⇒ mechanical dampingNot often an important factor at the macro scaleRecognized source of damping in microresonators

Zener: semi-analytical approximation for TED in beamsWe consider the fully coupled system

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis




σ = Cε− ξθ1utt = ∇ · σθt = η∇2θ − tr(εt)

ξ :=

(β

ρc

)2 T0

cvand η :=

κ

ρcv cL

Length ∼ LTime ∼ L/c, where c =

√E/ρ

Temperature ∼ T0β

ρcv

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Scaling analysis


ξ :=

(β

ρc

)2 T0

cvand η :=

κ

ρcv cL

Micron-scale poly-Si devices: ξ and η are ∼ 10−4.Small η leads to thermal boundary layersLinearize about ξ = 0

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Discrete mode equations


σ = Cε− ξθ1−ω2u = ∇ · σ

iωθ = η∇2θ − iω tr(ε)

−ω2Muuu + Kuuu + Kutθ = 0iωDttθ + Kttθ + iωDtuu = 0

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Perturbation computation

−ω2Muuu + Kuuu + Kutθ = 0iωDttθ + Kttθ + iωDtuu = 0

Approximate ω by perturbation about Kut = 0:

−ω20Muuu0 + Kuuu0 = 0

iω0Dttθ0 + Kttθ0 + iω0Dtuu0 = 0

Choose v : vT u0 6= 0 and compute[(−ω2

0Muu + Kuu) −2ω0Muuu0vT 0

] [δuδω

]=

[−Kutθ0

0

]

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Comparison to Zener’s model

105

106

107

108

109

1010

10−7

10−6

10−5

10−4

The

rmoe

last

ic D

ampi

ng Q

−1

Frequency f(Hz)

Zener’s Formula

HiQlab Results

Comparison of fully coupled simulation to Zenerapproximation over a range of frequenciesReal and imaginary parts after first-order correctionagree to about three digits with Arnoldi

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Thermoelastic boundary layer

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 abar)Fixed temperature and displacement at leftFree at right

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Shear ring resonator

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

0.9hw = 5.000000e−05

Value = 2.66E+07 Hz.

Ring is driven in a shearing motionCan couple ring to other resonatorsHow do we track the desired mode?

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Mode tracking

Find a continuous solution to(K (s)− ω(s)2M(s)

)u(s) = 0.

K and M are symmetric and M > 0Eigenvectors are M-orthogonalPerturbation theory gives good shiftsLook if u(s + h) and u(s) are on the same path bylooking at u(s + h)T M(s + h)u(s)

Many more subtleties in the nonsymmetric caseFocus of the CIS algorithm

CAD forMEMS

ResonantMEMS

AnchorLosses

ComplexSymmetry


Conclusions


MEMS Fabrication

SUGAR and HiQLab

Elastic PMLs

Reflection Analysis



Mode tracking in a shear resonator

2 2.5 3 3.5 4 4.5 5 5.5 6

x 10−5

2.5

3

3.5

4

4.5

5

5.5x 107

Half width of annulus

Fre

quen

cy (

Hz)

Finite elementAnalytic result

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