elastic pmls resonator anchor loss simulation
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Elastic PMLsfor
Resonator Anchor Loss SimulationDavid Bindel1 and Sanjay Govindjee 2
1 Department of Electrical Engineering and Computer Science2 Department of Civil Engineering
University of California at Berkeley
D. Bindel, USNCCM8 – p.1/45
Contributors
Tsuyoshi Koyama – PhD Student, Civil EngineeringWei He – PhD Student, Civil EngineeringEmmanuel Quévy – Postdoc, Electrical EngineeringRoger Howe – Professor, Electrical EngineeringJames Demmel – Professor, Computer Science
D. Bindel, USNCCM8 – p.2/45
Outline
Electromechanical resonators and RF MEMS
Damping and quality of resonance
Anchor losses and Perfectly Matched Layers
Analysis of the discretized PMLs
Complex symmetry and structured model reduction
Analysis of a disk resonator
Conclusions
D. Bindel, USNCCM8 – p.3/45
Outline
Electromechanical resonators and RF MEMS
Damping and quality of resonance
Anchor losses and Perfectly Matched Layers
Analysis of the discretized PMLs
Complex symmetry and structured model reduction
Analysis of a disk resonator
Conclusions
D. Bindel, USNCCM8 – p.4/45
How many MEMS?
D. Bindel, USNCCM8 – p.5/45
Why resonant MEMS?
Microguitars from Cornell University (1997 and 2003)
Sensing elements (inertial, chemical)
Frequency references
Filter elements
Neural networks
Really high-pitch guitars
D. Bindel, USNCCM8 – p.6/45
Micromechanical filters
Mechanical filter
Capacitive senseCapacitive drive
Radio signal
Filtered signal
Mechanical high-frequency (high MHz-GHz) filterYour cell phone is mechanical!
Advantage over quartz surface acoustic wave filtersIntegrated into chipLow power
Success =⇒ “Calling Dick Tracy!”D. Bindel, USNCCM8 – p.7/45
Designing transfer functions
Time domain:
Mu′′ + Cu′ + Ku = bφ(t)
y(t) = pT u
Frequency domain:
−ω2Mu + iωCu + Ku = bφ(ω)
y(ω) = pT u
Transfer function:
H(ω) = pT (−ω2M + iωC + K)−1b
y(ω) = H(ω)φ(ω)
D. Bindel, USNCCM8 – p.8/45
Damping and filters20 log10 |H(ω)|
ω
Want “sharp” poles for narrowband filters
=⇒ Want to minimize dampingElectronic filters have too muchUnderstanding of damping in MEMS is lacking
D. Bindel, USNCCM8 – p.9/45
Damping andQ
Designers want high quality of resonance (Q)Dimensionless damping in a one-dof system:
d2u
dt2+ Q−1du
dt+ u = F (t)
For a resonant mode with frequency ω ∈ C:
Q :=|ω|
2 Im(ω)=
Stored energyEnergy loss per radian
D. Bindel, USNCCM8 – p.10/45
Sources of damping
Fluid dampingAir is a viscous fluid (Re 1)Can operate in a vacuumShown not to dominate in many RF designs
Material lossesLow intrinsic losses in silicon, diamond, germaniumTerrible material losses in metals
Thermoelastic dampingVolume changes induce temperature changeDiffusion of heat leads to mechanical loss
Anchor lossElastic waves radiate from structure
D. Bindel, USNCCM8 – p.11/45
Sources of damping
Fluid dampingAir is a viscous fluid (Re 1)Can operate in a vacuumShown not to dominate in many RF designs
Material lossesLow intrinsic losses in silicon, diamond, germaniumTerrible material losses in metals
Thermoelastic dampingVolume changes induce temperature changeDiffusion of heat leads to mechanical loss
Anchor lossElastic waves radiate from structure
D. Bindel, USNCCM8 – p.11/45
Outline
Electromechanical resonators and RF MEMS
Damping and quality of resonance
Anchor losses and Perfectly Matched LayersAnchor losses and infinite domainsIdea of the perfectly matched layerElastic PMLs and finite elements
Analysis of the discretized PMLs
Complex symmetry and structured model reduction
Analysis of a disk resonator
Conclusions
D. Bindel, USNCCM8 – p.12/45
Example: Disk resonator
SiGe disk resonators built by E. Quévy
D. Bindel, USNCCM8 – p.13/45
Substrate model
Goal: Understand energy loss in disk resonator
Dominant loss is elastic radiation from anchor
Resonator size substrate sizeSubstrate appears semi-infinite
Possible semi-infinite modelsMatched asymptotic modesDirichlet-to-Neumann mapsBoundary dampersHigher-order local ABCsPerfectly matched layers
D. Bindel, USNCCM8 – p.14/45
Perfectly matched layers
Apply a complex coordinate transformation
Generates a non-physical absorbing layer
No impedance mismatch between the computationaldomain and the absorbing layer
Idea works with general linear wave equationsFirst applied to Maxwell’s equations (Berengér 95)Similar idea earlier in quantum mechanics(exterior complex scaling, Simon 79)Applies to elasticity in standard FEM framework(Basu and Chopra, 2003)
D. Bindel, USNCCM8 – p.15/45
1-D model problem
Domain: x ∈ [0,∞)
Governing eq:∂2u
∂x2−
1
c2
∂2u
∂t2= 0
Fourier transform:
d2u
dx2+ k2u = 0
Solution:u = coute
−ikx + cineikx
D. Bindel, USNCCM8 – p.16/45
1-D model problem with PML
Transformed domainx
σ
Regular domain
dx
dx= λ(x) where λ(s) = 1 − iσ(s)
d2u
dx2+ k2u = 0
u = coute−ikx + cine
ikx
D. Bindel, USNCCM8 – p.17/45
1-D model problem with PML
Transformed domainx
σ
Regular domain
dx
dx= λ(x) where λ(s) = 1 − iσ(s)
1
λ
d
dx
(
1
λ
du
dx
)
+ k2u = 0
u = cout exp
(
−k
∫ x
0σ(s) ds
)
e−ikx+cin exp
(
k
∫ x
0σ(s) ds
)
eikx
D. Bindel, USNCCM8 – p.17/45
1-D model problem with PML
Transformed domainx
σ
Regular domain
If solution clamped at x = L then
cin
cout= O(e−kγ) where γ =
∫ L
0σ(s) ds
D. Bindel, USNCCM8 – p.17/45
1-D model problem illustratedOutgoing exp(−ix) Incoming exp(ix)
Transformed coordinate
Re(x)
Im(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
D. Bindel, USNCCM8 – p.18/45
1-D model problem illustratedOutgoing exp(−ix) Incoming exp(ix)
Transformed coordinate
Re(x)
Im(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
D. Bindel, USNCCM8 – p.18/45
1-D model problem illustratedOutgoing exp(−ix) Incoming exp(ix)
Transformed coordinate
Re(x)
Im(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
D. Bindel, USNCCM8 – p.18/45
1-D model problem illustratedOutgoing exp(−ix) Incoming exp(ix)
Transformed coordinate
Re(x)
Im(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
D. Bindel, USNCCM8 – p.18/45
1-D model problem illustratedOutgoing exp(−ix) Incoming exp(ix)
Transformed coordinate
Re(x)
Im(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
D. Bindel, USNCCM8 – p.18/45
1-D model problem illustratedOutgoing exp(−ix) Incoming exp(ix)
Transformed coordinate
Re(x)
Im(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
Clamp solution at transformed end to isolate outgoing wave.
D. Bindel, USNCCM8 – p.18/45
Elastic PMLs
∫
Ωε(w) : C : ε(u) dΩ − ω2
∫
Ωρw · u dΩ =
∫
Γw · tndΓ
ε(u) =
(
∂u
∂x
)s
=
(
∂u
∂xΛ−1
)s
Start from standard weak form
D. Bindel, USNCCM8 – p.19/45
Elastic PMLs
∫
Ωε(w) : C : ε(u) dΩ − ω2
∫
Ωρw · u dΩ =
∫
Γw · tndΓ
ε(u) =
(
∂u
∂x
)s
=
(
∂u
∂xΛ−1
)s
Start from standard weak form
Introduce transformed x with ∂x∂x = Λ
D. Bindel, USNCCM8 – p.19/45
Elastic PMLs
∫
Ωε(w) : C : ε(u) JΛ dΩ − ω2
∫
Ωρw · u JΛ dΩ =
∫
Γw · tn dΓ
ε(u) =
(
∂u
∂x
)s
=
(
∂u
∂xΛ−1
)s
Start from standard weak form
Introduce transformed x with ∂x∂x = Λ
Map back to reference system (JΛ = det(Λ))
D. Bindel, USNCCM8 – p.19/45
Elastic PMLs
∫
Ωε(w) : C : ε(u) JΛ dΩ − ω2
∫
Ωρw · u JΛ dΩ =
∫
Γw · tn dΓ
ε(u) =
(
∂u
∂x
)s
=
(
∂u
∂xΛ−1
)s
Start from standard weak form
Introduce transformed x with ∂x∂x = Λ
Map back to reference system (JΛ = det(Λ))
All terms are symmetric in w and u
D. Bindel, USNCCM8 – p.19/45
Finite element implementation
x(ξ)
ξ2
ξ1
x1
x2 x2
x1
Ωe Ωe
Ω
x(x)
Combine PML and isoparametric mappings
ke =
∫
Ω
BTDBJdΩ
me =
(∫
Ω
ρNTNJdΩ
)
Matrices are complex symmetric
D. Bindel, USNCCM8 – p.20/45
Outline
Electromechanical resonators and RF MEMS
Damping and quality of resonance
Anchor losses and Perfectly Matched Layers
Analysis of the discretized PMLsA two-dimensional model problemAnalysis of discrete reflectionChoice of PML parameters
Complex symmetry and structured model reduction
Analysis of a disk resonator
Conclusions
D. Bindel, USNCCM8 – p.21/45
Continuum 2D model problem
k
L
λ(x) =
1 − iβ|x − L|p, x > L
1 x ≤ L.
1
λ
∂
∂x
(
1
λ
∂u
∂x
)
+∂2u
∂y2+ k2u = 0
D. Bindel, USNCCM8 – p.22/45
Continuum 2D model problem
k
L
λ(x) =
1 − iβ|x − L|p, x > L
1 x ≤ L.
1
λ
∂
∂x
(
1
λ
∂u
∂x
)
− k2yu + k2u = 0
D. Bindel, USNCCM8 – p.22/45
Continuum 2D model problem
k
L
λ(x) =
1 − iβ|x − L|p, x > L
1 x ≤ L.
1
λ
∂
∂x
(
1
λ
∂u
∂x
)
+ k2xu = 0
1D problem, reflection of O(e−kxγ)
D. Bindel, USNCCM8 – p.22/45
Discrete 2D model problem
k
L
Discrete Fourier transform in y
Solve numerically in x
Project solution onto infinite space traveling modes
Extension of Collino and Monk (1998)
D. Bindel, USNCCM8 – p.23/45
Nondimensionalization
k
L
λ(x) =
1 − iβ|x − L|p, x > L
1 x ≤ L.
Rate of stretching: βhp
Elements per wave: (kxh)−1 and (kyh)−1
Elements through the PML: N
D. Bindel, USNCCM8 – p.24/45
Nondimensionalization
k
L
λ(x) =
1 − iβ|x − L|p, x > L
1 x ≤ L.
Rate of stretching: βhp
Elements per wave: (kxh)−1 and (kyh)−1
Elements through the PML: N
D. Bindel, USNCCM8 – p.24/45
Discrete reflection behavior
Number of PML elements
log10(β
h)
− log10(r) at (kh)−1 = 10
1
1
1
2
2
2
2 2 2 2
333
3
3
3
3
444
4
4
5 10 15 20 25 30
-1.5
-1
-0.5
0
0.5
1
Quadratic elements, p = 1, (kxh)−1 = 10
D. Bindel, USNCCM8 – p.25/45
Discrete reflection decomposition
Model discrete reflection as two parts:
Far-end reflection (clamping reflection)Approximated well by continuum calculationGrows as (kxh)−1 grows
Interface reflectionDiscrete effect: mesh does not resolve decayDoes not depend on N
Grows as (kxh)−1 shrinks
D. Bindel, USNCCM8 – p.26/45
Discrete reflection behavior
Number of PML elements
log10(β
h)
− log10(r) at (kh)−1 = 10
1
1
1
2
2
2
2 2 2 2
333
3
3
3
3
444
4
4
5 10 15 20 25 30
-1.5
-1
-0.5
0
0.5
1
Number of PML elements
log10(β
h)
− log10(rinterface + rnominal) at (kh)−1 = 10
11
1
2
22
2 2 2 2
333
3
3
3
444
4
4
5 10 15 20 25 30
-1.5
-1
-0.5
0
0.5
1
Quadratic elements, p = 1, (kxh)−1 = 10
Model does well at predicting actual reflection
Similar picture for other wavelengths, element types,stretch functions
D. Bindel, USNCCM8 – p.27/45
Choosing PML parameters
Discrete reflection dominated byInterface reflection when kx largeFar-end reflection when kx small
Heuristic for PML parameter choiceChoose an acceptable reflection levelChoose β based on interface reflection at kmax
x
Choose length based on far-end reflection at kminx
D. Bindel, USNCCM8 – p.28/45
Outline
Electromechanical resonators and RF MEMS
Damping and quality of resonance
Anchor losses and Perfectly Matched Layers
Analysis of the discretized PMLs
Complex symmetry and structured model reductionKrylov subspace projectionsStructure-preserving eigencomputationsStructure-preserving model reduction
Analysis of a disk resonator
Conclusions
D. Bindel, USNCCM8 – p.29/45
Eigenvalues and model reduction
Want to know about the transfer function H(ω):
H(ω) = pT (K − ω2M)−1b
Can either
Locate poles of H (eigenvalues of (K,M))
Determine Q = |ω|2 Im(ω)
Plot H in a frequency range (Bode plot)
Solve both problems with the same tool:Krylov subspace projections
D. Bindel, USNCCM8 – p.30/45
Projecting via Arnoldi
Build a Krylov subspace basis by shift-invert Arnoldi
Choose shift σ in frequency range of interest
Form and factor Kshift = K − σ2M
Use Arnoldi to build an orthonormal basis V for
Kn = spanu0, K−1shiftu0, . . . , K
−(n−1)shift u0
Compute eigenvalues and reduced models from projection
Compute eigenvalues from (V ∗KV, V ∗MV )
Approximate H(ω) by Galerkin projection
H(ω) ≈ (V ∗p)∗(V ∗KV − ω2V ∗MV )−1(V ∗b)
D. Bindel, USNCCM8 – p.31/45
Accurate eigenvalues
Hermitian systems: Rayleigh-Ritz is optimalRaleigh quotient is stationary at eigenvectors
ρ(v) =v∗Kv
v∗Mv
First-order accurate eigenvectors =⇒second-order accurate eigenvalues
Can we obtain optimal accuracy for PML eigenvalues?
D. Bindel, USNCCM8 – p.32/45
Accurate eigenvalues
PML matrices are complex symmetricModified RQ is stationary at eigenvectors
θ(v) =vT Kv
vT Mv
=⇒ second-order accurate eigenvaluesHochstenbach and Arbenz, 2004
D. Bindel, USNCCM8 – p.33/45
Accurate model reduction
Accurate eigenvalues from v and v together
Accurate model reduction in the same wayBuild new projection basis from V :
W = orth[Re(V ), Im(V )]
span(W ) contains both Kn and Kn
Double convergence vs projection with V
W is a real-valued basisProjected system remains complex symmetric
D. Bindel, USNCCM8 – p.34/45
Outline
Electromechanical resonators and RF MEMS
Damping and quality of resonance
Anchor losses and Perfectly Matched Layers
Analysis of the discretized PMLs
Complex symmetry and structured model reduction
Analysis of a disk resonatorAccuracy of the numericsDescription of the loss mechanismSensitivity to fabrication variations
Conclusions
D. Bindel, USNCCM8 – p.35/45
Disk resonator simulations
D. Bindel, USNCCM8 – p.36/45
Disk resonator mesh
WaferV−
V+V+
PML region
DiskElectrode
0 1 2 3 4
x 10−5
−4
−2
0
2x 10
−6
Axisymmetric model with bicubic mesh
About 10K nodal points in converged calculation
D. Bindel, USNCCM8 – p.37/45
Mesh convergence
Mesh density
Com
pute
dQ
Cubic
LinearQuadratic
1 2 3 4 5 6 7 80
1000
2000
3000
4000
5000
6000
7000
D. Bindel, USNCCM8 – p.38/45
Model reduction performance
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
D. Bindel, USNCCM8 – p.39/45
Model reduction performance
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
D. Bindel, USNCCM8 – p.40/45
Response of the disk resonator
D. Bindel, USNCCM8 – p.41/45
Time-averaged energy flux
0 0.5 1 1.5 2 2.5 3 3.5
x 10−5
−2
0
2
x 10−6
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10−6
−1
−0.5
0
0.5
1
1.5
2
2.5x 10
−6
D. Bindel, USNCCM8 – p.42/45
Q variation
Film thickness (µm)
Q
1.2 1.3 1.4 1.5 1.6 1.7 1.8100
102
104
106
108
D. Bindel, USNCCM8 – p.43/45
Explanation of Q variation
Real frequency (MHz)
Imagin
ary
freq
uen
cy(M
Hz)
ab
cdd
e
a b
cdd
e
a = 1.51 µm
b = 1.52 µm
c = 1.53 µm
d = 1.54 µm
e = 1.55 µm
46 46.5 47 47.5 480
0.05
0.1
0.15
0.2
0.25
D. Bindel, USNCCM8 – p.44/45
Conclusions
MEMS damping is important and non-trivial
Elastic PMLs work well for modeling anchor lossFormulation fits naturally with mapped elementsEstimate multi-D performance with simple models
Use complex symmetry to compute eigenvalues andreduced models
Simulations show effects that hand analysis misses
Reference:Bindel and Govindjee, “Elastic PMLs for resonator anchorloss simulation,” IJNME (to appear).
D. Bindel, USNCCM8 – p.45/45
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