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CSE07
Model Reduction and Mode Computation forDamped Resonant MEMS
David Bindel
SIAM CSE 07, 19 Feb 2007
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CSE07
Collaborators
Tsuyoshi KoyamaSanjay GovindjeeSunil BhaveEmmanuel QuévyZhaojun Bai
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CSE07
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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CSE07
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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CSE07
Ultimate Success
“Calling Dick Tracy!”
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CSE07
Example Resonant System
DC
AC
Vin
Vout
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CSE07
Example Resonant System
AC
C0
Vout
Vin
Lx
Cx
Rx
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CSE07
The Designer’s Dream
Ideally, would likeSimple models for behavioral simulationParameterized for design optimizationIncluding all relevant physicsWith reasonably fast and accurate set-up
We aren’t there yet. Today, some progress on the last two.
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CSE07
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
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CSE07
Damping Mechanisms
Possible loss mechanisms:Anchor lossThermoelastic dampingOther material lossesFluid damping
Our goal: Reduced models that include these effects.
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CSE07
Perfectly Matched Layers
Model substrate as semi-infinite with a
Perfectly Matched Layer (PML).
Complex coordinate transformationGenerates a “perfectly matched” absorbing layerIdea works with general linear wave equations
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CSE07
Perfectly Matched Layers by Picture
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
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CSE07
Perfectly Matched Layers by Picture
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
-2
-1
0
1
2
3
-1
-0.5
0
0.5
1
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CSE07
Perfectly Matched Layers by Picture
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
-4
-2
0
2
4
6
-1
-0.5
0
0.5
1
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CSE07
Perfectly Matched Layers by Picture
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
-10
-5
0
5
10
15
-1
-0.5
0
0.5
1
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CSE07
Perfectly Matched Layers by Picture
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
-20
0
20
40
-1
-0.5
0
0.5
1
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CSE07
Perfectly Matched Layers by Picture
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
-50
0
50
100
-1
-0.5
0
0.5
1
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CSE07
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
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CSE07
Complex Symmetry
Discretized (forced) problem + fixed PML take the form:
(K − ω2M)u = f , where K = K T , M = MT
Can still characterize u as 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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CSE07
Accurate Model Reduction
Usual: Orthogonal projection onto Arnoldi basis V .Us: 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
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CSE07
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)
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CSE07
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
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CSE07
Thermoelastic Damping (TED)
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CSE07
Thermoelastic Damping (TED)
u is displacement and T = T0 + θ is temperature
σ = Cε− βθ1ρu = ∇ · σ
ρcv θ = ∇ · (κ∇θ)− βT0 tr(ε)
Coupling between temperature and volumetric strain:Compression and expansion =⇒ heating and coolingHeat 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
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CSE07
Nondimensionalized Equations
Continuum equations:
σ = Cε− ξθ1u = ∇ · σθ = η∇2θ − tr(ε)
Discrete equations:
Muuu + Kuuu = ξKuθθ + fCθθθ + ηKθθθ = −Cθuu
Micron-scale poly-Si devices: ξ and η are ∼ 10−4.Linearize about ξ = 0
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CSE07
Perturbative Mode Calculation
Discretized mode equation:
(−ω2Muu + Kuu)u = ξKuθθ
(iωCθθ + ηKθθ)θ = −iωCθuu
First approximation about ξ = 0:
(−ω20Muu + Kuu)u0 = 0
(iω0Cθθ + ηKθθ)θ0 = −iω0Cθuu0
First-order correction in ξ:
−δ(ω2)Muuu0 + (−ω20Muu + Kuu)δu = ξKuθθ0
Multiply by uT0 :
δ(ω2) = −ξ
(uT
0 Kuθθ0
uT0 Muuu0
)
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CSE07
Zener’s Model
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.
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CSE07
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
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CSE07
General Picture
If w∗A = 0 and Av = 0 then
δ(w∗Av) = w∗(δA)v
This impliesIf A = A(λ) and w = w(v), have
w∗(v)A(ρ(v))v = 0.
ρ stationary when (ρ(v), v) is a nonlinear eigenpair.If A(λ, ξ) and w∗
0 and v0 are null vectors for A(λ0, ξ0),
w∗0 (Aλδλ + Aξδξ)v0 = 0.
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CSE07
Conclusions
Resonant MEMS have lots of interesting applicationsDesigners want reduced models with relevant physicsDamping is crucial, but not well handled in generalOur work: use equation structure in making reducedmodels with damping (modal or more general)