TD25_vg1.7_ccsd … CLIC_G damped symmetric disk with compact damped magnetic couplers
Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose...
Transcript of Structure Preserving Model Reduction for Damped Resonant …bindel/present/2007-07-usnccm.pdfClose...
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USNCCM07
Structure Preserving Model Reduction forDamped Resonant MEMS
David Bindel
USNCCM 07, 23 Jul 2007
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USNCCM07
Collaborators
Tsuyoshi KoyamaSanjay GovindjeeSunil BhaveEmmanuel QuévyZhaojun Bai
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USNCCM07
Resonant MEMS
Microguitars from Cornell University (1997 and 2003)
MHz-GHz mechanical resonatorsFavorite application: radio on chipClose second: really high-pitch guitars
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USNCCM07
The Mechanical Cell Phone
filter
Mixer
Tuning
control
...RF amplifier /
preselector
Local
Oscillator
IF amplifier /
Your cell phone has many moving parts!What if we replace them with integrated MEMS?
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USNCCM07
Ultimate Success
“Calling Dick Tracy!”
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USNCCM07
Example Resonant System
DC
AC
Vin
Vout
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USNCCM07
Example Resonant System
AC
C0
Vout
Vin
Lx
Cx
Rx
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USNCCM07
Model Reduction: Basic set-up
Linear time-invariant system:
Mu′′ + Ku = bφ(t)y(t) = pT u
Frequency domain:
−ω2Mu + K u = bφ(ω)
y(ω) = pT u
Transfer function:
H(ω) = pT (−ω2M + K )−1by(ω) = H(ω)φ(ω)
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USNCCM07
Model Reduction: Basic set-up
Have a rational transfer function relating input and output:
H(ω) = pT (−ω2M + K )−1b
Can approximate H by Galerkin projection:
H(ω) = (Vp)T (−ω2V T MV + V T KV )−1(Vb)
Could also try to approximate H directly (often equivalent).
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USNCCM07
The Designer’s Dream
Ideally, would likeCompact models for behavioral simulationParameterized for design optimizationIncluding all relevant physicsWith reasonably fast and accurate set-up
We aren’t there yet.
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USNCCM07
Standard Projection Model Reduction
Approximate H by Galerkin projection:
H(iω) = (Vp)T (−ω2V T MV + V T KV )−1(Vb)
1 Define Kσ := K − σ2M; build a Krylov subspace
span(V ) = Kn(K−1σ M, K−1
σ b) = span{(K−1σ M)jK−1
σ b}nj=0
Has the moment-matching property:
H(k)(iσ) = H(k)(iσ), k = 0, . . . , n
Get 2n moments for symmetric systems (or forseparate left and right subspaces).
2 Project onto one or more modal vectors.
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USNCCM07
The Hero of the Hour
Major theme: use problem structure for better reducedmodels
ODE structureComplex symmetric structurePerturbative structureGeometric structure
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USNCCM07
SOAR and ODE structure
Damped second-order system:
Mu′′ + Cu′ + Ku = Pφ
y = 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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USNCCM07
Checkerboard Resonator
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USNCCM07
Checkerboard Resonator
� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �
� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � � � � � � � � � � �
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D+
D−
D+
D−
S+ S+
S−
S−
Anchored at outside cornersExcited at northwest cornerSensed at southeast cornerSurfaces move only a few nanometers
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USNCCM07
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
de
Bode 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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USNCCM07
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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USNCCM07
Disk Resonator Simulations
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USNCCM07
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
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USNCCM07
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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USNCCM07
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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USNCCM07
Perturbative Structure
Dimensionless continuum equations for thermoelasticdamping:
σ = Cε− ξθ1u = ∇ · σθ = η∇2θ − tr(ε)
Dimensionless coupling ξ and heat diffusivity η are 10−4
=⇒ perturubation method (about ξ = 0).
Large, non-self-adjoint, first-order coupled problem →Smaller, self-adjoint, mechanical eigenproblem + symmetriclinear solve.
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USNCCM07
Thermoelastic Damping Example
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USNCCM07
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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USNCCM07
Aside: Effect of Nondimensionalization
100 µm beam example, first-order form.
Before nondimensionalizationTime: 180 snnz(L) = 11M
After nondimensionalizationTime: 10 snnz(L) = 380K
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USNCCM07
Semi-Analytical Model Reduction
We work with hand-build model reduction all the time!Circuit elements: Maxwell equation + field assumptionsBeam theory: Elasticity + kinematic assumptionsAxisymmetry: 3D problem + kinematic assumption
Idea: Provide global shapesUser defines shapes through a callbackMesh serves defines a quadrature ruleReduced equations fit known abstractions
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USNCCM07
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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USNCCM07
“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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USNCCM07
Conclusions
Respecting problem structure is a Good Thing!ODE structureComplex symmetric structurePerturbative structureGeometric structure
Result:Better accuracy, faster set-up, better understanding.