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USNCCM07
Structure Preserving Model Reduction forDamped Resonant MEMS
David Bindel
USNCCM 07, 23 Jul 2007
USNCCM07
Collaborators
Tsuyoshi KoyamaSanjay GovindjeeSunil BhaveEmmanuel QuévyZhaojun Bai
USNCCM07
Resonant MEMS
Microguitars from Cornell University (1997 and 2003)
MHz-GHz mechanical resonatorsFavorite application: radio on chipClose second: really high-pitch guitars
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?
USNCCM07
Ultimate Success
“Calling Dick Tracy!”
USNCCM07
Example Resonant System
DC
AC
Vin
Vout
USNCCM07
Example Resonant System
AC
C0
Vout
Vin
Lx
Cx
Rx
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)
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).
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.
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.
USNCCM07
The Hero of the Hour
Major theme: use problem structure for better reducedmodels
ODE structureComplex symmetric structurePerturbative structureGeometric structure
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, . . .
USNCCM07
Checkerboard Resonator
USNCCM07
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
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
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.
USNCCM07
Disk Resonator Simulations
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
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)
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
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.
USNCCM07
Thermoelastic Damping Example
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
USNCCM07
Aside: Effect of Nondimensionalization
100 µm beam example, first-order form.
Before nondimensionalizationTime: 180 snnz(L) = 11M
After nondimensionalizationTime: 10 snnz(L) = 380K
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
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 .
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)
USNCCM07
Conclusions
Respecting problem structure is a Good Thing!ODE structureComplex symmetric structurePerturbative structureGeometric structure
Result:Better accuracy, faster set-up, better understanding.