Structure-preserving model reduction for MEMSbindel/present/2011-03-cse.pdfStructure-preserving...
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Structure-preserving model reduction forMEMS
David Bindel
Department of Computer ScienceCornell University
SIAM CSE Meeting, 1 Mar 2011
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Collaborators
I Sunil BhaveI Emmanuel QuévyI Zhaojun BaiI Tsuyoshi KoyamaI Sanjay Govindjee
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Resonant MEMS
Microguitars from Cornell University (1997 and 2003)
I kHz-GHz mechanical resonatorsI Lots of applications:
I Inertial sensors (in phones, airbag systems, ...)I Chemical sensorsI Signal processing elementsI Really high-pitch guitars!
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Application: The Mechanical Cell Phone
filter
Mixer
Tuning
control
...RF amplifier /
preselector
Local
Oscillator
IF amplifier /
I Your cell phone has many moving parts!I What if we replace them with integrated MEMS?
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Ultimate Success
“Calling Dick Tracy!”
I Old dream: a Dick Trace watch phone!I New dream: long battery life for smart phones
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Example Resonant System
DC
AC
Vin
Vout
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Example Resonant System
AC
C0
Vout
Vin
Lx
Cx
Rx
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The Designer’s Dream
Ideally, would likeI Simple models for behavioral simulationI Interpretable degrees of freedomI Including all relevant physicsI Parameterized for design optimizationI With reasonably fast and accurate set-upI Backed by error analysis
We aren’t there yet.
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The Hero of the Hour
Major theme: use problem structure for better modelsI Algebraic
I Structure of ODEs (e.g. second-order structure)I Structure of matrices (e.g. complex symmetry)
I AnalyticI Perturbations of physics (thermoelastic damping)I Perturbations of geometry (near axisymmetry)I Perturbations of boundary conditions (clamping)
I GeometricI Simplified models: planar motion, axisymmetry, ...I Substructures
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SOAR and ODE structure
Damped second-order system:
Mu′′ + Cu′ + Ku = Pφy = V T u.
Projection basis Qn with Second Order ARnoldi (SOAR):
Mnu′′n + Cnu′n + Knun = Pnφ
y = V Tn u
where Pn = QTn P,Vn = QT
n V ,Mn = QTn MQn, . . .
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Checkerboard Resonator
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Checkerboard Resonator
D+
D−
D+
D−
S+ S+
S−
S−
I Anchored at outside cornersI Excited at northwest cornerI Sensed at southeast cornerI Surfaces move only a few nanometers
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Performance of SOAR vs Arnoldi
N = 2154→ n = 80
0.08 0.085 0.09 0.095 0.1 0.105
100
105
Mag
nitu
deBode plot
0.08 0.085 0.09 0.095 0.1 0.105−200
−100
0
100
200
Pha
se(d
egre
e)
ExactSOARArnoldi
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Aside: Next generation
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Complex Symmetry
Model with radiation damping (PML) gives complex problem:
(K − ω2M)u = f , where K = K T ,M = MT
Forced solution u is a stationary point of
I(u) =12
uT (K − ω2M)u − uT f .
Eigenvalues of (K ,M) are stationary points of
ρ(u) =uT KuuT Mu
First-order accurate vectors =⇒second-order accurate eigenvalues.
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Disk Resonator Simulations
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Disk Resonator Mesh
PML region
Wafer (unmodeled)
Electrode
Resonating disk
0 1 2 3 4
x 10−5
−4
−2
0
2x 10
−6
I Axisymmetric model with bicubic meshI About 10K nodal points in converged calculation
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Symmetric ROM 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)
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Symmetric ROM 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
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Aside: Model Expansion?
0 5 10 15 20 25 30
0
5
10
15
20
25
30
0 5 10 15 20 25 30
0
5
10
15
20
25
30
PML adds variables so that the Schur complement
A(k)ψ1 =(
K11 − k2M11 − C(k))ψ1 = 0
has a term C(k) to approximate a radiation boundary condition.
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Perturbative Structure
Dimensionless continuum equations for thermoelastic damping:
σ = Cε− ξθ1u = ∇ · σθ = η∇2θ − tr(ε)
Dimensionless coupling ξ and heat diffusivity η are 10−4 =⇒perturbation method (about ξ = 0).
Large, non-self-adjoint, first-order coupled problem→Smaller, self-adjoint, mechanical eigenproblem + symmetriclinear solve.
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Thermoelastic Damping Example
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Performance for Beam Example
10 20 30 40 50 60 70 80 90 1000
1
2
3
4
5
6
7
8
9
10
Beam length (microns)
Tim
e (s
)
Perturbation methodFirst−order form
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Aside: Effect of Nondimensionalization
100 µm beam example, first-order form.
Before nondimensionalizationI Time: 180 sI nnz(L) = 11M
After nondimensionalizationI Time: 10 sI nnz(L) = 380K
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Semi-Analytical Model Reduction
We work with hand-build model reduction all the time!I Circuit elements: Maxwell equation + field assumptionsI Beam theory: Elasticity + kinematic assumptionsI Axisymmetry: 3D problem + kinematic assumption
Idea: Provide global shapesI User defines shapes through a callbackI Mesh serves defines a quadrature ruleI Reduced equations fit known abstractions
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Global Shape Functions
Normally:u(X ) =
∑j
Nj(X )uj
Global shape functions:
u = ul + G(ug)
Then constrain values of some components of ul , ug .
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“Hello, World!”
Which mode shape comes from the reduced model (3 dof)?
Student Version of MATLABStudent Version of MATLAB
(Left: 28 MHz; Right: 31 MHz)
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Latest widgets
I This is a gyroscope!I HRG is widely usedI What about MEMS?
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Simplest model
Two degree of freedom model:
m[u1u2
]+ Ωg
[0 −11 0
] [u1u2
]+ k
[u1u2
]= f .
I Multiple eigenvalues when at rest (symmetry)I Rotation splits the eigenvalues — get a beat frequencyI Want dynamics of energy transfer between two modesI Except there are more than two modes!
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Structured models for wineglass gyros
We want to understand everything together!I Axisymmetry is critical
I Need to understand manufacturing defects!I Robustness through design and post-processing
I Low damping is criticalI Need thermoelastic effects (perturbation)I Need coupling to substrate (??)
I Need optical-mechanical coupling for drive/sense
The moral of the preceding:I Bad idea: 3D model in ANSYS + model reductionI Better idea: Do some reduction by hand first!
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Conclusions
Essentially, all models are wrong, but some are useful.– George Box
Questions?http://www.cs.cornell.edu/~bindel