Reduced Order Models in Microsystems and RF MEMS
Reduced Order Models in Microsystems and RF MEMS David Bindel UC Berkeley, CS Division
Reduced Order Models in Microsystems and RF MEMS – p.1/14
Collaborators Faculty
Grad students
A. Agogino (ME)
D. Bindel (CS)
Z. Bai (Math/CS)
J.V. Clark (AS&T)
J. Demmel (Math/CS) D. Garmire (CS) S. Govindjee (CEE)
T. Koyama (CEE)
R. Howe (EE)
R. Kamalian (ME) J. Nie (Math) S. Bhave (EE)
Reduced Order Models in Microsystems and RF MEMS – p.2/14
MEMS Basics Micro-electro-mechanical systems Chemical, fluid, thermal, optical (MECFTOMS?) Applications: Sensors (inertial, chemical, pressure) Ink jet printers, biolab chips RF devices Use IC fabrication technology Large surface area / volume ratio Still mostly classical (vs. nanosystems)
Reduced Order Models in Microsystems and RF MEMS – p.3/14
RF MEMS
Microguitars from Cornell University (1997 and 2003)
MHz-GHz mechanical resonators Impact: smaller, lower-power cell phones Replace quartz freq references, filter elements Integrate into CMOS stack Other uses: Sensing elements Really high-pitch guitars
Reduced Order Models in Microsystems and RF MEMS – p.4/14
Second-order system equations Time domain mechanical equations with damping: M u00 + Cu0 + Ku = P φ y = V Tu
Linearized: #" # " # " # " #0 " C K u0 P M 0 u0 + = φ(t) −I 0 u u 0 0 I y = V Tu
Usual tactic: Model reduction on linearized system. Goal: Work with second-order system directly. Reduced Order Models in Microsystems and RF MEMS – p.5/14
Second-order Krylov subspaces Let A, B ∈ CN ×N , r0 ∈ CN , and define the sequence r1 = Ar0 , rj = Arj−1 + Brj−2
for j ≥ 2
Now define a second-order Krylov subspace Gn (A, B; r0 ) = span{r0 , r1 , r2 , . . . , rn−1 }.
Note "
rj rj−1
#
j
= H v,
"
A B where H = I 0
#
" # r0 and v = 0
=⇒ Gn (A, B; r0 ) provides the same info as Kn (H; v)! (T.-J. Su + Craig Jr. ’91, cited by White + Ramaswamy ’00) Reduced Order Models in Microsystems and RF MEMS – p.6/14
SOAR-based model reduction Let A = −K −1 C , B = −K −1 M , r0 = −K −1 P . Form an orthonormal basis Qk for Gn (A, B; r0 ) by a second-order Arnoldi (SOAR) iteration (Bai). Reduced model: Mn u00n + Cn u0n + Kn un = Pn φ y = VnT u
where Pn = QTn P, Vn = QTn V, Mn = QTn M Qn , . . . Basic structure (and properties) are preserved.
Reduced Order Models in Microsystems and RF MEMS – p.7/14
Example: Checkerboard resonator
(Design and layout by S. Bhave)
Array of coupled resonators Anchored at corners Excited at northwest Sensed at southeast Surfaces move a few nm Reduced Order Models in Microsystems and RF MEMS – p.8/14
Performance of SOAR vs Arnoldi N = 2154 → n = 80 Bode plot
5
Magnitude
10
0
10
0.08
Phase(degree)
200 100
0.085
0.09
0.095
0.1
0.105
0.09
0.095
0.1
0.105
Exact SOAR Arnoldi
0 −100 −200 0.08
0.085
Reduced Order Models in Microsystems and RF MEMS – p.9/14
Example: Disk resonator anchor loss Electrode
Disk
V− Wafer
V+
V+
Goal: Understand energy lost through radiation from post. Treat substrate as a half-space Use a complex-valued change of coordinates to build an absorbing Perfectly Matched Layer (PML) (Basu and Chopra ’03) M and K are complex symmetric Need a fine mesh to resolve quality of resonant peaks Reduced Order Models in Microsystems and RF MEMS – p.10/14
PML ROM: Preserving structure Weak form of the PML equation (frequency domain) is Z Z Z s2 ρw · u JdΩ + ˜(w) : C˜(u) JdΩ = w · p˜ n JdΓ Ω
Ω
Γ
Finite element discretization (Bubnov-Galerkin) yields complex-symmetric mass and stiffness. Reduction procedure: ˆn Run Arnoldi on (s20 M + K)−1 to get basis Q ˆ n ), Im(Q ˆ n )]) Qn = orth([Re(Q Mn = QTn M Qn and Kn = QTn KQn are symmetric
ROM is a Bubnov-Galerkin projection of original equation. Reduced Order Models in Microsystems and RF MEMS – p.11/14
Anchor loss N = 57475 → n = 3 Magnitude (dB)
−10 −20 −30 −40 −50 4.53
4.531
4.532
4.533
4.534 4.535 4.536 Frequency (Hz)
4.537
4.538
4.539
4.54 x 108
4.531
4.532
4.533
4.534 4.535 4.536 Frequency (Hz)
4.537
4.538
4.539
4.54 x 108
Phase(degree)
200 150 100 50 0 4.53
Reduced Order Models in Microsystems and RF MEMS – p.12/14
Why n = 3? 0
x 10 5
−2 −4 −6 −8 −10 −12 −14 −16 4.524 4.526 4.528
4.53
4.532 4.534 4.536 4.538
4.54 x 10 8
Behavior in this region is dominated by two poles Mode “mixing” dramatically affects predicted resonance! Impact: Eliminate most anchor loss for disk resonator? Reduced Order Models in Microsystems and RF MEMS – p.13/14
Ongoing and future work Continued development of predictive CAD for RF MEMS Parameter-dependent model reduction Structure-preserving model reduction for thermoelasticity: # " #00 " " # " #0 " #" # " # Muu 0 u u P 0 0 Kuu Kut u = φ(t) + + 0 0 θ 0 Ktt θ 0 Dtu Dtt θ Integration into the SUGAR framework for MEMS simulation
Reduced Order Models in Microsystems and RF MEMS – p.14/14
Reduced Order Models in Microsystems and RF MEMS – p.1/14
Collaborators Faculty
Grad students
A. Agogino (ME)
D. Bindel (CS)
Z. Bai (Math/CS)
J.V. Clark (AS&T)
J. Demmel (Math/CS) D. Garmire (CS) S. Govindjee (CEE)
T. Koyama (CEE)
R. Howe (EE)
R. Kamalian (ME) J. Nie (Math) S. Bhave (EE)
Reduced Order Models in Microsystems and RF MEMS – p.2/14
MEMS Basics Micro-electro-mechanical systems Chemical, fluid, thermal, optical (MECFTOMS?) Applications: Sensors (inertial, chemical, pressure) Ink jet printers, biolab chips RF devices Use IC fabrication technology Large surface area / volume ratio Still mostly classical (vs. nanosystems)
Reduced Order Models in Microsystems and RF MEMS – p.3/14
RF MEMS
Microguitars from Cornell University (1997 and 2003)
MHz-GHz mechanical resonators Impact: smaller, lower-power cell phones Replace quartz freq references, filter elements Integrate into CMOS stack Other uses: Sensing elements Really high-pitch guitars
Reduced Order Models in Microsystems and RF MEMS – p.4/14
Second-order system equations Time domain mechanical equations with damping: M u00 + Cu0 + Ku = P φ y = V Tu
Linearized: #" # " # " # " #0 " C K u0 P M 0 u0 + = φ(t) −I 0 u u 0 0 I y = V Tu
Usual tactic: Model reduction on linearized system. Goal: Work with second-order system directly. Reduced Order Models in Microsystems and RF MEMS – p.5/14
Second-order Krylov subspaces Let A, B ∈ CN ×N , r0 ∈ CN , and define the sequence r1 = Ar0 , rj = Arj−1 + Brj−2
for j ≥ 2
Now define a second-order Krylov subspace Gn (A, B; r0 ) = span{r0 , r1 , r2 , . . . , rn−1 }.
Note "
rj rj−1
#
j
= H v,
"
A B where H = I 0
#
" # r0 and v = 0
=⇒ Gn (A, B; r0 ) provides the same info as Kn (H; v)! (T.-J. Su + Craig Jr. ’91, cited by White + Ramaswamy ’00) Reduced Order Models in Microsystems and RF MEMS – p.6/14
SOAR-based model reduction Let A = −K −1 C , B = −K −1 M , r0 = −K −1 P . Form an orthonormal basis Qk for Gn (A, B; r0 ) by a second-order Arnoldi (SOAR) iteration (Bai). Reduced model: Mn u00n + Cn u0n + Kn un = Pn φ y = VnT u
where Pn = QTn P, Vn = QTn V, Mn = QTn M Qn , . . . Basic structure (and properties) are preserved.
Reduced Order Models in Microsystems and RF MEMS – p.7/14
Example: Checkerboard resonator
(Design and layout by S. Bhave)
Array of coupled resonators Anchored at corners Excited at northwest Sensed at southeast Surfaces move a few nm Reduced Order Models in Microsystems and RF MEMS – p.8/14
Performance of SOAR vs Arnoldi N = 2154 → n = 80 Bode plot
5
Magnitude
10
0
10
0.08
Phase(degree)
200 100
0.085
0.09
0.095
0.1
0.105
0.09
0.095
0.1
0.105
Exact SOAR Arnoldi
0 −100 −200 0.08
0.085
Reduced Order Models in Microsystems and RF MEMS – p.9/14
Example: Disk resonator anchor loss Electrode
Disk
V− Wafer
V+
V+
Goal: Understand energy lost through radiation from post. Treat substrate as a half-space Use a complex-valued change of coordinates to build an absorbing Perfectly Matched Layer (PML) (Basu and Chopra ’03) M and K are complex symmetric Need a fine mesh to resolve quality of resonant peaks Reduced Order Models in Microsystems and RF MEMS – p.10/14
PML ROM: Preserving structure Weak form of the PML equation (frequency domain) is Z Z Z s2 ρw · u JdΩ + ˜(w) : C˜(u) JdΩ = w · p˜ n JdΓ Ω
Ω
Γ
Finite element discretization (Bubnov-Galerkin) yields complex-symmetric mass and stiffness. Reduction procedure: ˆn Run Arnoldi on (s20 M + K)−1 to get basis Q ˆ n ), Im(Q ˆ n )]) Qn = orth([Re(Q Mn = QTn M Qn and Kn = QTn KQn are symmetric
ROM is a Bubnov-Galerkin projection of original equation. Reduced Order Models in Microsystems and RF MEMS – p.11/14
Anchor loss N = 57475 → n = 3 Magnitude (dB)
−10 −20 −30 −40 −50 4.53
4.531
4.532
4.533
4.534 4.535 4.536 Frequency (Hz)
4.537
4.538
4.539
4.54 x 108
4.531
4.532
4.533
4.534 4.535 4.536 Frequency (Hz)
4.537
4.538
4.539
4.54 x 108
Phase(degree)
200 150 100 50 0 4.53
Reduced Order Models in Microsystems and RF MEMS – p.12/14
Why n = 3? 0
x 10 5
−2 −4 −6 −8 −10 −12 −14 −16 4.524 4.526 4.528
4.53
4.532 4.534 4.536 4.538
4.54 x 10 8
Behavior in this region is dominated by two poles Mode “mixing” dramatically affects predicted resonance! Impact: Eliminate most anchor loss for disk resonator? Reduced Order Models in Microsystems and RF MEMS – p.13/14
Ongoing and future work Continued development of predictive CAD for RF MEMS Parameter-dependent model reduction Structure-preserving model reduction for thermoelasticity: # " #00 " " # " #0 " #" # " # Muu 0 u u P 0 0 Kuu Kut u = φ(t) + + 0 0 θ 0 Ktt θ 0 Dtu Dtt θ Integration into the SUGAR framework for MEMS simulation
Reduced Order Models in Microsystems and RF MEMS – p.14/14
