Computer-Aided Design for Micro-Electro

MIT 08
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
Computer Aided Design of
Micro-Electro-Mechanical Systems
From Energy Losses to Dick Tracy Watches
D. Bindel
Courant Institute for Mathematical Sciences
New York University
MIT, 3 Nov 2008
The Computational Science Picture
MIT 08
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
Application modeling
Disk resonator
Beam resonator
Shear ring resonator, checkerboard, ...
Mathematical analysis
Physical modeling and finite element technology
Structured eigenproblems and reduced-order models
Parameter-dependent eigenproblems
Software engineering
HiQLab
SUGAR
FEAPMEX / MATFEAP
The Computational Science Picture
MIT 08
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
Application modeling
Disk resonator
Beam resonator
Shear ring resonator, checkerboard, ...
Mathematical analysis
Physical modeling and finite element technology
Structured eigenproblems and reduced-order models
Parameter-dependent eigenproblems
Software engineering
HiQLab
SUGAR
FEAPMEX / MATFEAP
Outline
MIT 08
1
Resonant MEMS and models
2
HiQLab
Thermoelastic
losses and
beam
resonators
3
Anchor losses and disk resonators
Conclusion
4
Thermoelastic losses and beam resonators
5
Conclusion
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Backup slides
Outline
MIT 08
1
Resonant MEMS and models
2
HiQLab
Thermoelastic
losses and
beam
resonators
3
Anchor losses and disk resonators
Conclusion
4
Thermoelastic losses and beam resonators
5
Conclusion
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Backup slides
What are MEMS?
MIT 08
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
MEMS Basics
MIT 08
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
Micro-Electro-Mechanical Systems
Chemical, fluid, thermal, optical (MECFTOMS?)
Applications:
Sensors (inertial, chemical, pressure)
Ink jet printers, biolab chips
Radio devices: cell phones, inventory tags, pico radio
Use integrated circuit (IC) fabrication technology
Tiny, but still classical physics
Resonant RF MEMS
MIT 08
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Microguitars from Cornell University (1997 and 2003)
Conclusion
MHz-GHz mechanical resonators
Backup slides
Favorite application: radio on chip
Close second: really high-pitch guitars
The Mechanical Cell Phone
MIT 08
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Mixer
RF amplifier /
preselector
Tuning
control
IF amplifier /
filter
Local
Oscillator
Backup slides
Your cell phone has many moving parts!
What if we replace them with integrated MEMS?
...
Ultimate Success
MIT 08
“Calling Dick Tracy!”
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
Disk Resonator
MIT 08
Vin
Resonant
MEMS and
models
HiQLab
AC
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
DC
Backup slides
Vout
Disk Resonator
MIT 08
Vin
Resonant
MEMS and
models
HiQLab
AC
Lx
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
C0
Cx
Rx
Conclusion
Backup slides
Vout
Electromechanical Model
MIT 08
Resonant
MEMS and
models
Assume time-harmonic steady state, no external forces:
iωC + G
iωB
δ Îexternal
δ V̂
=
−B T
K̃ − ω 2 M δ û
0
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Eliminate the mechanical terms:
Y (ω) δ V̂
= δ Îexternal
Y (ω) = iωC + G + iωH(ω)
H(ω) = B T (K̃ − ω 2 M)−1 B
Backup slides
Goal: Understand electromechanical piece (iωH(ω)).
As a function of geometry and operating point
Preferably as a simple circuit
Damping and Q
MIT 08
Resonant
MEMS and
models
Designers want high quality of resonance (Q)
Dimensionless damping in a one-dof system
HiQLab
d 2u
du
+ u = F (t)
+ Q −1
2
dt
dt
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
For a resonant mode with frequency ω ∈ C:
Q :=
|ω|
Stored energy
=
2 Im(ω)
Energy loss per radian
To understand Q, we need damping models!
The Designer’s Dream
MIT 08
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
Ideally, would like
Simple models for behavioral simulation
Parameterized for design optimization
Including all relevant physics
With reasonably fast and accurate set-up
We aren’t there yet.
Outline
MIT 08
1
Resonant MEMS and models
2
HiQLab
Thermoelastic
losses and
beam
resonators
3
Anchor losses and disk resonators
Conclusion
4
Thermoelastic losses and beam resonators
5
Conclusion
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Backup slides
Enter HiQLab
MIT 08
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
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
Heritage of HiQLab
MIT 08
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
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
Other Ingredients
MIT 08
Resonant
MEMS and
models
“Lesser artists borrow. Great artists steal.”
– Picasso, Dali, Stravinsky?
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
Lua: www.lua.org
Evolved from simulator data languages (DEL and SOL)
Pascal-like syntax fits on one page; complete language
description is 21 pages
Fast, freely available, widely used in game design
MATLAB: www.mathworks.com
“The Language of Technical Computing”
O CTAVE also works well
Standard numerical libraries: ARPACK, UMFPACK
M ATEXPR, MW RAP, and other utilities
HiQLab Structure
MIT 08
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
User interfaces
(MATLAB, Lua)
Solver library
(C, C++, Fortran, MATLAB)
Core libraries
(C++)
Problem description
(Lua)
Element library
(C++)
Standard finite element structures + some new ideas
Conclusion
Full scripting language for mesh input
Backup slides
Callbacks for boundary conditions, material properties
MATLAB interface for quick algorithm prototyping
Cross-language bindings are automatically generated
Outline
MIT 08
1
Resonant MEMS and models
2
HiQLab
Thermoelastic
losses and
beam
resonators
3
Anchor losses and disk resonators
Conclusion
4
Thermoelastic losses and beam resonators
5
Conclusion
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Backup slides
Damping Mechanisms
MIT 08
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Possible loss mechanisms:
Fluid damping
Material losses
Conclusion
Backup slides
Thermoelastic damping
Anchor loss
Model substrate as semi-infinite with a
Perfectly Matched Layer (PML).
Perfectly Matched Layers
MIT 08
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
Complex coordinate transformation
Generates a “perfectly matched” absorbing layer
Idea 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)
Model Problem
MIT 08
Resonant
MEMS and
models
Domain: x ∈ [0, ∞)
Governing eq:
HiQLab
1 ∂2u
∂2u
−
=0
∂x 2 c 2 ∂t 2
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Fourier transform:
d 2 û
+ k 2 û = 0
dx 2
Conclusion
Backup slides
Solution:
û = cout e−ikx + cin eikx
Model with Perfectly Matched Layer
MIT 08
σ
Resonant
MEMS and
models
HiQLab
Regular domain
x
Transformed domain
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
d x̃
= λ(x) where λ(s) = 1 − iσ(s)
dx
d 2 û
+ k 2 û = 0
d x̃ 2
û = cout e−ik x̃ + cin eik x̃
Model with Perfectly Matched Layer
MIT 08
σ
Resonant
MEMS and
models
HiQLab
x
Transformed domain
Regular domain
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
d x̃
= λ(x) where λ(s) = 1 − iσ(s),
dx
1 d 1 d û
+ k 2 û = 0
λ dx λ dx
û = cout e−ikx−k Σ(x) + cin eikx+k Σ(x)
Z x
Σ(x) =
σ(s) ds
0
Model with Perfectly Matched Layer
MIT 08
σ
Resonant
MEMS and
models
HiQLab
Regular domain
x
Transformed domain
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
If solution clamped at x = L then
Conclusion
Backup slides
cin
= O(e−k γ ) where γ = Σ(L) =
cout
Z
L
σ(s) ds
0
Model Problem Illustrated
MIT 08
Outgoing exp(−ix̃)
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
1
0.5
0.5
0
0
-0.5
-0.5
-1
0
5
10
15
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
Incoming exp(ix̃)
1
20
-1
0
10
5
15
20
Transformed coordinate
0
-2
-4
0
2
4
6
8
10
Re(x̃)
12
14
16
18
Model Problem Illustrated
MIT 08
Outgoing exp(−ix̃)
Incoming exp(ix̃)
1
Resonant
MEMS and
models
3
2
0.5
1
0
0
HiQLab
Anchor losses
and disk
resonators
-0.5
-1
-1
0
5
10
15
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
20
-2
0
5
15
10
20
Transformed coordinate
0
-2
-4
0
2
4
6
8
10
Re(x̃)
12
14
16
18
Model Problem Illustrated
MIT 08
Outgoing exp(−ix̃)
Incoming exp(ix̃)
1
Resonant
MEMS and
models
6
4
0.5
2
0
0
HiQLab
Anchor losses
and disk
resonators
-0.5
-2
-1
0
5
10
15
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
20
-4
0
5
15
10
20
Transformed coordinate
0
-2
-4
0
2
4
6
8
10
Re(x̃)
12
14
16
18
Model Problem Illustrated
MIT 08
Outgoing exp(−ix̃)
Incoming exp(ix̃)
1
Resonant
MEMS and
models
15
10
0.5
5
0
0
HiQLab
Anchor losses
and disk
resonators
-0.5
-5
-1
0
5
10
15
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
20
-10
0
5
15
10
20
Transformed coordinate
0
-2
-4
0
2
4
6
8
10
Re(x̃)
12
14
16
18
Model Problem Illustrated
MIT 08
Outgoing exp(−ix̃)
Incoming exp(ix̃)
1
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
40
0.5
20
0
0
-0.5
-1
0
5
10
15
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
20
-20
0
5
10
20
15
Transformed coordinate
0
-2
-4
0
2
4
6
8
10
Re(x̃)
12
14
16
18
Model Problem Illustrated
MIT 08
Outgoing exp(−ix̃)
Incoming exp(ix̃)
1
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
100
0.5
50
0
0
-0.5
-1
0
5
10
15
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
20
-50
0
5
10
20
15
Transformed coordinate
0
-2
-4
0
2
4
6
8
10
Re(x̃)
12
14
16
18
Finite Element Implementation
MIT 08
x2
ξ2
x(ξ)
Resonant
MEMS and
models
HiQLab
x̃2
ξ1
Ω
x̃(x)
Ω̃e
Ωe
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
x1
Combine PML and isoparametric mappings
Z
e
k =
B̃T DB̃J̃ dΩ
Ω
Z
e
m =
ρNT NJ̃ dΩ
Ω
Matrices are complex symmetric
x̃1
Eigenvalues and Model Reduction
MIT 08
Want to know about the transfer function H(ω):
Resonant
MEMS and
models
H(ω) = B T (K − ω 2 M)−1 B
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
Can either
Locate poles of H (eigenvalues of (K , M))
Plot H in a frequency range (Bode plot)
Usual tactic: subspace projection
Build an Arnoldi basis V for a Krylov subspace Kn
Compute with much smaller V ∗ KV and V ∗ MV
Can we do better?
Variational Principles
MIT 08
Variational form for complex symmetric eigenproblems:
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
Hermitian (Rayleigh quotient):
ρ(v ) =
v ∗ Kv
v ∗ Mv
Complex symmetric (modified Rayleigh quotient):
θ(v ) =
v T Kv
v T Mv
First-order accurate eigenvectors =⇒
Second-order accurate eigenvalues.
Key: relation between left and right eigenvectors.
Accurate Model Reduction
MIT 08
Resonant
MEMS and
models
Build new projection basis from V :
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
W = orth[Re(V ), Im(V )]
span(W ) contains both Kn and K̄n
=⇒ double digits correct vs. projection with V
W is a real-valued basis
=⇒ projected system is complex symmetric
Disk Resonator Simulations
MIT 08
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
Disk Resonator Mesh
MIT 08
Electrode
Resonant
MEMS and
models
Resonating disk
HiQLab
Wafer (unmodeled)
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
PML region
−6
x 10
2
0
Conclusion
−2
Backup slides
−4
0
1
2
3
4
−5
x 10
Axisymmetric model with bicubic mesh
About 10K nodal points in converged calculation
Mesh Convergence
MIT 08
7000
6000
Resonant
MEMS and
models
Linear
Quadratic
Cubic
5000
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Computed Q
HiQLab
4000
3000
2000
Conclusion
Backup slides
1000
0
1
2
3
5
4
Mesh density
6
7
8
Cubic elements converge with reasonable mesh density
Model Reduction Accuracy
MIT 08
Transfer (dB)
Resonant
MEMS and
models
0
-20
-40
-60
HiQLab
-80
47.2
Anchor losses
and disk
resonators
Conclusion
Backup slides
47.3
200
Phase (degrees)
Thermoelastic
losses and
beam
resonators
47.25
Frequency (MHz)
100
0
47.2
47.25
47.3
Frequency (MHz)
Results from ROM (solid and dotted lines) nearly
indistinguishable from full model (crosses)
Model Reduction Accuracy
MIT 08
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
10−2
|H(ω) − Hreduced(ω)|/H(ω)|
Resonant
MEMS and
models
10−4
10−6
Structure-preserving ROM
Arnoldi ROM
Backup slides
45
46
47
48
Frequency (MHz)
Preserve structure =⇒
get twice the correct digits
49
50
Response of the Disk Resonator
MIT 08
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
Variation in Quality of Resonance
MIT 08
108
Resonant
MEMS and
models
106
Anchor losses
and disk
resonators
Q
HiQLab
104
Thermoelastic
losses and
beam
resonators
Conclusion
102
Backup slides
100
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Film thickness (µm)
Simulation and lab measurements vs. disk thickness
Explanation of Q Variation
MIT 08
0.25
e
a
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
d
c
0.2
Imaginary frequency (MHz)
Resonant
MEMS and
models
b
0.15
0.1
a = 1.51 µm
b = 1.52 µm
c = 1.53 µm
0.05
d = 1.54 µm
Backup slides
e = 1.55 µm
0
46
46.5
d
c
e
b
a
47
47.5
Real frequency (MHz)
Interaction of two nearby eigenmodes
48
Outline
MIT 08
1
Resonant MEMS and models
2
HiQLab
Thermoelastic
losses and
beam
resonators
3
Anchor losses and disk resonators
Conclusion
4
Thermoelastic losses and beam resonators
5
Conclusion
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Backup slides
Thermoelastic Damping (TED)
MIT 08
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
Thermoelastic Damping (TED)
MIT 08
u is displacement and T = T0 + θ is temperature
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
σ = C − βθ1
ρü = ∇ · σ
˙
ρcv θ̇ = ∇ · (κ∇θ) − βT0 tr()
Coupling between temperature and volumetric strain:
Compression and expansion =⇒ heating and cooling
Heat diffusion =⇒ mechanical damping
Not often an important factor at the macro scale
Recognized source of damping in microresonators
Zener: semi-analytical approximation for TED in beams
We consider the fully coupled system
Nondimensionalized Equations
MIT 08
Continuum equations:
Resonant
MEMS and
models
σ = Ĉ − ξθ1
ü = ∇ · σ
HiQLab
˙
θ̇ = η∇2 θ − tr()
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
Discrete equations:
Muu ü + Kuu u = ξKuθ θ + f
Cθθ θ̈ + ηKθθ θ = −Cθu u̇
Micron-scale poly-Si devices: ξ and η are ∼ 10−4 .
Linearize about ξ = 0
Perturbative Mode Calculation
MIT 08
Discretized mode equation:
(−ω 2 Muu + Kuu )u = ξKuθ θ
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
(iωCθθ + ηKθθ )θ = −iωCθu u
First approximation about ξ = 0:
(−ω02 Muu + Kuu )u0 = 0
(iω0 Cθθ + ηKθθ )θ0 = −iω0 Cθu u0
First-order correction in ξ:
−δ(ω 2 )Muu u0 + (−ω02 Muu + Kuu )δu = ξKuθ θ0
Multiply by u0T :
2
δ(ω ) = −ξ
u0T Kuθ θ0
u0T Muu u0
!
Zener’s Model
MIT 08
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
1
Clarence Zener investigated TED in late 30s-early 40s.
2
Model for beams common in MEMS literature.
3
“Method of orthogonal thermodynamic potentials” ==
perturbation method + a variational method.
Comparison to Zener’s Model
MIT 08
−4
10
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Thermoelastic Damping Q
−1
Resonant
MEMS and
models
−5
10
Zener’s Formula
−6
HiQlab Results
10
−7
Conclusion
Backup slides
10
5
10
6
10
7
10
8
10
9
10
10
10
Frequency f(Hz)
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 with Arnoldi
Outline
MIT 08
1
Resonant MEMS and models
2
HiQLab
Thermoelastic
losses and
beam
resonators
3
Anchor losses and disk resonators
Conclusion
4
Thermoelastic losses and beam resonators
5
Conclusion
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Backup slides
Onward!
MIT 08
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
What about:
Modeling more geometrically complex devices?
Modeling general dependence on geometry?
Thermoelastic
losses and
beam
resonators
Modeling general dependence on operating point?
Conclusion
Digesting all this to help designers?
Backup slides
Computing nonlinear dynamics?
Future Work
MIT 08
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
Code development
Structural elements and elements for different physics
Design and implementation of parallelized version
Theoretical analysis
More damping mechanisms
Sensitivity 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
MIT 08
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
RF MEMS are a great source of problems
Interesting applications
Interesting physics (and not altogether understood)
Interesting computing challenges
http://www.cims.nyu.edu/~dbindel
Concluding Thoughts
MIT 08
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
The difference between art and science is that
science is what we understand well enough to
explain to a computer. Art is everything else.
Donald Knuth
The purpose of computing is insight, not numbers.
Richard Hamming
Checkerboard Resonator
MIT 08
D−
Resonant
MEMS and
models
D+
D+
HiQLab
D−
S−
Anchor losses
and disk
resonators
S+
S+
S−
Thermoelastic
losses and
beam
resonators
Anchored at outside corners
Conclusion
Backup slides
Checkerboard
resonators
Nonlinear eigenvalue
perturbation
Electromechanical
model
Hello world!
Reflection Analysis
Excited at northwest corner
Sensed at southeast corner
Surfaces move only a few nanometers
Checkerboard Model Reduction
MIT 08
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
Checkerboard
resonators
Nonlinear eigenvalue
perturbation
Electromechanical
model
Hello world!
Reflection Analysis
Finite element model: N = 2154
Expensive to solve for every H(ω) evaluation!
Build a reduced-order model to approximate behavior
Reduced system of 80 to 100 vectors
Evaluate H(ω) in milliseconds instead of seconds
Without damping: standard Arnoldi projection
With damping: Second-Order ARnoldi (SOAR)
Checkerboard Simulation
MIT 08
x 10
12
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
−100
−120
10
Amplitude (dB)
Resonant
MEMS and
models
8
−140
−160
−180
6
−200
9
9.2
9.4
9.6
Frequency (Hz)
x 10
9.4
9.6
Frequency (Hz)
x 10
4
9.8
7
4
2
Phase (rad)
3
Conclusion
0
Backup slides
Checkerboard
resonators
Nonlinear eigenvalue
perturbation
Electromechanical
model
Hello world!
Reflection Analysis
0
2
4
6
8
2
1
10
−5
x 10
0
9
9.2
9.8
7
Checkerboard Measurement
MIT 08
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
Checkerboard
resonators
Nonlinear eigenvalue
perturbation
Electromechanical
model
Hello world!
Reflection Analysis
S. Bhave, MEMS 05
Contributions
MIT 08
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
Checkerboard
resonators
Nonlinear eigenvalue
perturbation
Electromechanical
model
Hello world!
Reflection Analysis
Built predictive model used to design checkerboard
Used model reduction to get thousand-fold speedup
– fast enough for interactive use
General Picture
MIT 08
If w ∗ A = 0 and Av = 0 then
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
δ(w ∗ Av ) = w ∗ (δA)v
This implies
If A = A(λ) and w = w(v ), have
w ∗ (v )A(ρ(v ))v = 0.
Conclusion
ρ stationary when (ρ(v ), v ) is a nonlinear eigenpair.
Backup slides
If A(λ, ξ) and w0∗ and v0 are null vectors for A(λ0 , ξ0 ),
Checkerboard
resonators
Nonlinear eigenvalue
perturbation
Electromechanical
model
Hello world!
Reflection Analysis
w0∗ (Aλ δλ + Aξ δξ)v0 = 0.
Electromechanical Model
MIT 08
Kirchoff’s current law and balance of linear momentum:
d
(C(u)V ) + GV = Iexternal
dt 1 ∗
Mutt + Ku − ∇u
V C(u)V
= Fexternal
2
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Linearize about static equilibium (V0 , u0 ):
C(u0 ) δVt + G δV + (∇u C(u0 ) · δut ) V0 = δIexternal
Conclusion
M δutt + K̃ δu + ∇u (V0∗ C(u0 ) δV ) = δFexternal
Backup slides
Checkerboard
resonators
Nonlinear eigenvalue
perturbation
Electromechanical
model
Hello world!
Reflection Analysis
where
K̃ = K −
1 ∂2
(V ∗ C(u0 )V0 )
2 ∂u 2 0
4
HiQLab’s
Hello World
MIT 08
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
Checkerboard
resonators
Nonlinear eigenvalue
perturbation
Electromechanical
model
Hello world!
Reflection Analysis
2
0
−2
mesh = Mesh:new(2)
mat = −4make_material(’silicon2’, ’planestrain’)
mesh:blocks2d( { 0, l }, { -w/2.0, w/2.0 },
mat )
−6
0
2
4
6
8
10
−6
x 10
mesh:set_bc(function(x,y)
if x == 0 then return ’uu’, 0, 0; end
end)
HiQLab’s Hello World
MIT 08
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
Checkerboard
resonators
Nonlinear eigenvalue
perturbation
Electromechanical
model
Hello world!
Reflection Analysis
>>
>>
>>
>>
>>
>>
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);
Continuum 2D model problem
MIT 08
Resonant
MEMS and
models
k
HiQLab
Anchor losses
and disk
resonators
L
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
Checkerboard
resonators
Nonlinear eigenvalue
perturbation
Electromechanical
model
Hello world!
Reflection Analysis
1 − iβ|x − L|p , x > L
1
x ≤ L.
∂2u
1 ∂ 1 ∂u
+
+ k 2u = 0
λ ∂x λ ∂x
∂y 2
λ(x) =
Continuum 2D model problem
MIT 08
Resonant
MEMS and
models
k
HiQLab
Anchor losses
and disk
resonators
Backup slides
Checkerboard
resonators
Nonlinear eigenvalue
perturbation
Electromechanical
model
Hello world!
Reflection Analysis
L
Thermoelastic
losses and
beam
resonators
Conclusion
1 − iβ|x − L|p , x > L
1
x ≤ L.
1 ∂ 1 ∂u
− ky2 u + k 2 u = 0
λ ∂x λ ∂x
λ(x) =
Continuum 2D model problem
MIT 08
k
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
Checkerboard
resonators
Nonlinear eigenvalue
perturbation
L
1 − iβ|x − L|p , x > L
1
x ≤ L.
1 ∂ 1 ∂u
+ kx2 u = 0
λ ∂x λ ∂x
λ(x) =
Electromechanical
model
Hello world!
Reflection Analysis
1D problem, reflection of O(e−kx γ )
Discrete 2D model problem
MIT 08
Resonant
MEMS and
models
k
HiQLab
Anchor losses
and disk
resonators
L
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
Checkerboard
resonators
Nonlinear eigenvalue
perturbation
Electromechanical
model
Hello world!
Reflection Analysis
Discrete Fourier transform in y
Solve numerically in x
Project solution onto infinite space traveling modes
Extension of Collino and Monk (1998)
Nondimensionalization
MIT 08
k
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
L
λ(x) =
1 − iβ|x − L|p , x > L
1
x ≤ L.
Backup slides
Checkerboard
resonators
Nonlinear eigenvalue
perturbation
Electromechanical
model
Hello world!
Reflection Analysis
Rate of stretching:
Elements per wave:
Elements through the PML:
βhp
(kx h)−1 and (ky h)−1
N
Nondimensionalization
MIT 08
k
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
L
λ(x) =
1 − iβ|x − L|p , x > L
1
x ≤ L.
Backup slides
Checkerboard
resonators
Nonlinear eigenvalue
perturbation
Electromechanical
model
Hello world!
Reflection Analysis
Rate of stretching:
Elements per wave:
Elements through the PML:
βhp
(kx h)−1 and (ky h)−1
N
Discrete reflection behavior
MIT 08
− log10 (r) at (kh)−1 = 10
1
Resonant
MEMS and
models
0.5
log10 (βh)
0
-0.5
3
2
4
-1
1
-1.5
Checkerboard
resonators
Nonlinear eigenvalue
perturbation
Electromechanical
model
Hello world!
Reflection Analysis
4
4
3
2
Conclusion
Backup slides
3
3
1
Thermoelastic
losses and
beam
resonators
2
2
3
Anchor losses
and disk
resonators
2
2
HiQLab
4
3
5
10
15
4
2
1
20
25
3
30
Number of PML elements
Quadratic elements, p = 1, (kx h)−1 = 10
Discrete reflection decomposition
MIT 08
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
Checkerboard
resonators
Nonlinear eigenvalue
perturbation
Electromechanical
model
Hello world!
Reflection Analysis
Model discrete reflection as two parts:
Far-end reflection (clamping reflection)
Approximated well by continuum calculation
Grows as (kx h)−1 grows
Interface reflection
Discrete effect: mesh does not resolve decay
Does not depend on N
Grows as (kx h)−1 shrinks
Discrete reflection behavior
MIT 08
− log10 (r) at (kh)−1 = 10
− log10 (rinterface + rnominal ) at (kh)−1 = 10
1
2
4
2
4
3
10
15
20
25
Number of PML elements
3
30
2
-1
2
3
3
-0.5
3
3
4
2 3 4
1
-1.5
4
2
1
5
2
4
4
1
-1.5
4
4
3
0
1
-1
log10 (βh)
3
3
2
log10 (βh)
-0.5
3
1
Conclusion
0.5
2
Thermoelastic
losses and
beam
resonators
0
1
Anchor losses
and disk
resonators
2
2
3
HiQLab
2
2
Resonant
MEMS and
models
1
0.5
5
10
15
2
20
3 4
25
30
Number of PML elements
Quadratic elements, p = 1, (kx h)−1 = 10
Backup slides
Checkerboard
resonators
Nonlinear eigenvalue
perturbation
Electromechanical
model
Hello world!
Reflection Analysis
Model does well at predicting actual reflection
Similar picture for other wavelengths, element types,
stretch functions
Choosing PML parameters
MIT 08
Resonant
MEMS and
models
HiQLab
Anchor losses
and disk
resonators
Thermoelastic
losses and
beam
resonators
Conclusion
Backup slides
Checkerboard
resonators
Nonlinear eigenvalue
perturbation
Electromechanical
model
Hello world!
Reflection Analysis
Discrete reflection dominated by
Interface reflection when kx large
Far-end reflection when kx small
Heuristic for PML parameter choice
Choose an acceptable reflection level
Choose β based on interface reflection at kxmax
Choose length based on far-end reflection at kxmin