elastic pmls resonator anchor loss simulation

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


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


Outline







Conclusions


How many MEMS?


Why resonant MEMS?

Microguitars from Cornell University (1997 and 2003)

Sensing elements (inertial, chemical)

Frequency references

Filter elements

Neural networks

Really high-pitch guitars


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)


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


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


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


Outline



Anchor losses and Perfectly Matched LayersAnchor losses and infinite domainsIdea of the perfectly matched layerElastic PMLs and finite elements




Conclusions


Example: Disk resonator

SiGe disk resonators built by E. Quévy


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


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)


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


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



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



Transformed domainx

σ

Regular domain

If solution clamped at x = L then

cin

cout= O(e−kγ) where γ =

∫ L

0σ(s) ds


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




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




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




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




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




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.


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


Elastic PMLs

∫

Ωε(w) : C : ε(u) dΩ − ω2

∫

Ωρw · u dΩ =

∫

Γw · tndΓ

ε(u) =

(

∂u

∂x

)s

=

(

∂u

∂xΛ−1

)s


Introduce transformed x with ∂x∂x = Λ


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



Map back to reference system (JΛ = det(Λ))


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



Map back to reference system (JΛ = det(Λ))

All terms are symmetric in w and u


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


Outline




Analysis of the discretized PMLsA two-dimensional model problemAnalysis of discrete reflectionChoice of PML parameters



Conclusions


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



k

L

λ(x) =

1 − iβ|x − L|p, x > L

1 x ≤ L.

1

λ

∂

∂x

(

1

λ

∂u

∂x

)

− k2yu + k2u = 0



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)


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)


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


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


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


Discrete reflection behavior


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


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


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


Outline





Complex symmetry and structured model reductionKrylov subspace projectionsStructure-preserving eigencomputationsStructure-preserving model reduction


Conclusions


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


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)


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?


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


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


Outline






Analysis of a disk resonatorAccuracy of the numericsDescription of the loss mechanismSensitivity to fabrication variations

Conclusions


Disk resonator simulations


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


Mesh convergence

Mesh density

Com

pute

dQ

Cubic

LinearQuadratic

1 2 3 4 5 6 7 80

1000

2000

3000

4000

5000

6000

7000


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


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


Response of the disk resonator


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


Q variation

Film thickness (µm)

Q

1.2 1.3 1.4 1.5 1.6 1.7 1.8100

102

104

106

108


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


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).


elastic pmls resonator anchor loss simulation

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