Model Order Reduction of Model Order Reduction of Parameterized - - PowerPoint PPT Presentation
Model Order Reduction of Model Order Reduction of Parameterized Interconnect Networks Parameterized Interconnect Networks via a Two- -Directional Directional Arnoldi Arnoldi Process Process via a Two Yung-Ta Li joint work with Zhaojun
1
joint work with
2
3
Image source: OIIC
Complex Structure of Interconnects
spacing parameters
MNA (Modified Nodal Analysis)
4
Disk resonator PML region Silicon wafer Electrode
Cutaway schematic (left) and SEM picture (right) of a micromachined disk resonator. Through modulated electrostatic attraction between a disk and a surrounding ring of electrodes, the disk is driven into mechanical resonance. Because the disk is anchored to a silicon wafer, energy leaks from the disk to the substrate, where radiates away as elastic waves. To study this energy loss, David Bindel has constructed finite element models in which the substrate is modeled by a perfectly matched absorbing layer. Resonance poles are approximated by eigenvalues of a large, sparse complex-symmetric matrix pencil. For more details, see D. S. Bindel and S. Govindjee, "Anchor Loss Simulation in Resonators," International Journal for Numerical Methods in Engineering, vol 64, issue 6. Resonator micrograph courtesy of Emmanuel Quévy.
thickness
thickness
5
6
One geometric parameter for clarity of presentation
7
8
9
10
11
for (1) (2)
12
13
(1,i)th Krylov subspace: Use Arnoldi process to compute an orthonormal basis
] 1 [ ] 1 [
] 1 [ ] 2 [
] 1 [ ] 3 [
] 1 [ ] 4 [
] 1 [ ] 5 [
] 1 [ ] 6 [
14
(2,i)th Krylov subspace: : How to efficiently compute an orthonormal basis ?
] 2 [ ] 1 [
] 2 [ ] 2 [
] 2 [ ] 3 [
] 2 [ ] 4 [
] 2 [ ] 5 [
] 2 [ ] 6 [
15
] 2 [ ] 1 [
] 2 [ ] 2 [
] 2 [ ] 3 [
] 2 [ ] 4 [
] 2 [ ] 5 [
] 2 [ ] 6 [
Computed by 2D Arnoldi process
16
Define We have the linear recurrence where
17
where
Computed by 2D Arnoldi process vectors in the projection subspace be used to construct projection matrix
18
Use V to construct reduced-order models by orthogonal projection PIMTAP (Parameterized Interconnect Macromodeling via a Two-directional Arnoldi Process)
19
20
RLC network: 8-bit bus with 2 shield lines variations on and
21
1 2 3 4 5 6 7 8 9 10 10
−16
10
−14
10
−12
10
−10
10
−8
10
−6
10
−4
10
−2
10 Frequency (GHz) Relative error: | h − h
^ | / | h |
CORE (p,q)=(40,1) CORE (p,q)=(80,1) PIMTAP (p,q)=(40,1)
CORE (p,q)=(40,1) CORE (p,q)=(80,1) PIMTAP (p,q)=(40,1)
22
1 2 3 4 5 6 7 8 9 10 5 6 7 8 9 10 11 Frequency (GHz) |h(s)|
CORE (p,q)=(80,2) PIMTAP (p,q)=(40,2)
CORE (p,q)=(80,2) PIMTAP (p,q)=(40,2)
23
2 4 6 8 10 10
−16
10
−14
10
−12
10
−10
10
−8
10
−6
10
−4
10
−2
Frequency (GHz) Relative error: abs( Horiginal − Hreduced )/ abs( Horiginal ) bus8b: λ = 0.06 PIMTAP (p,q)=(40,1) PIMTAP (p,q)=(40,2) PIMTAP (p,q)=(40,3) PIMTAP (p,q)=(40,4)
4
2
24
Thermal model with parameters and Power series expansion of the transfer function on satisfies the recursion:
25
The sequence :
26
27
Define We have the linear recurrence where
28
29
30
31
32
33