DiMES: Multilevel Fast Direct Solver based on Multipole Expansions for Parasitic Extraction of Massi

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DiMES Multilevel Fast Direct Solver based on
Multipole Expansions for Parasitic Extraction of
Massively Coupled 3D Microelectronic Structures
Indranil Chowdhury, Vikram Jandhyala
Dipanjan Gope
ACE Research Department of Electrical
Engineering University of Washington
Design and Technology Solutions INTEL Corporation
Supported by NSF, SRC and DARPA
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Class of Problems
Magnetostatic Problems
Electrostatic Problems
DiMES FAST DIRECT SOLVER ALGORITHM
Electric Field Integral Equations
Magnetic Field Integral Equations
PMCHW Multi-Region Dielectric Problems
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Outline

• Focus Application Accurate Charge Distribution
• - Circuit Parasitic Estimation
• - MEMS Charge Distribution
• Motivation behind Fast Direct Solution
• - Large Number of RHS Vectors
• - Re-simulation Advantages
• DiMES Fast Direct FMM based Solver
• - Sparsification of MoM Using FMM
• - Sparse 1.3 Solution
• Numerical Results

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Increasing Interconnect Parasitics

• Deep Sub-Micron and Nano Fabrication Technology
• - Gate delay reduces
• Overall chip size does not decrease
• - More functionalities added to the same chip

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MEMS Electrical Force Computations
MEMS Electrical Force Computation Requires
Accurate Simulation of Charge Distribution

• Approximate Solutions Inaccurate Charge
  Distribution
• Inaccurate Charge Distribution Inaccurate Force
  Computation

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Solution Scheme
Solution Scheme
Analytic
Numerical
Inexpensive but Inaccurate
Accurate for 3D Arbitrary Shaped Objects
Accurate Prediction of Charge Distribution

• Method of Moments (MoM)
• Well-Conditioned System
• Smaller Sized Matrix
• Dense Matrix

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Method of Moments

• Surface is Discretized into Patches (Basis
  Functions)

• Basis Functions Interact through the Greens
  Function

• Generates a Dense Method of Moments Matrix

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Fast Solvers Significance
N Number of basis functions (50,000)
p Number of iterations per RHS
r Number of RHS

• Fast Iterative Methods Mature Field
• - Fast Multipole Method (FastCap) Nabors and
  White 1992
• - Pre-Corrected FFT Method Phillips and White
  1997
• QR Based Method (IES3) Kapur and Long 1997
• QR Based Method (PILOT) Gope and Jandhyala
  2003
• O(N)-O(NlogN) Matrix Vector Products
• Why Look Any Further?

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Outline

• Focus Application Accurate Charge Distribution
• - Circuit Parasitic Estimation
• - MEMS Charge Distribution
• Motivation behind Fast Direct Solution
• - Large Number of RHS Vectors
• - Re-simulation Advantages
• DiMES Direct Multipole Expansion Solver
• - Sparsification of MoM Using FMM
• - Sparse 1.3 Solution
• Numerical Results

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Motivn 1 Large Number of RHS Vectors
Direct Setup Solve
Fast Iterative Setup Solve
Setup
Solve
Solve
Setup
N10,000 p90
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Motivn 2 Fast Updates in Re-simulation
Critical Transition Analysis to Solution
1. Schur Complement
2. SMW-Updates

B

A
AxByz1 CxDyz2 (ABD-1C)xz1-BD-1z2
D
M
A
U
V
C
Repeated Simulation Update vs. Re-Solve
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Existing Literature

• Advances In Fast Direct Solvers NOT Comparable To
  Advances In Fast Iterative Solvers

• Michielssen, Boag and Chew (1996)
• - Reduced Source Field Representation
• Canning and Rogovin (1999)
• - SMW Method
• - LUSIFER
• Hackbusch (2000)
• - H-Matrices
• Gope and Jandhyala (2001)
• - Compressed LU Method
• Yan, Sarin and Shi (2004)
• - Inexact Factorization

• Forced Matrix Structure Unsuitable for Arbitrary
  3D Shapes
• Fillins Chief Cost Factor / Neglected

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Outline

• Focus Application Accurate Charge Distribution
• - Circuit Parasitic Estimation
• - MEMS Charge Distribution
• Motivation behind Fast Direct Solution
• - Large Number of RHS Vectors
• - Re-simulation Advantages
• DiMES Direct Multipole Expansion Solver
• - Sparsification of MoM Using FMM
• - Sparse 1.3 Solution
• Numerical Results

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Fast Multipole Basics
1D Geometry
MoM
Matrix
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Multilevel Multipole Operators
Q Q2M M2M M2L L2L L2P P
M2L
Finest - 1 Level
M2M
L2L
L2L
M2M
M2L
Finest Level
M2L
Q2M
L2P
Down Tree
Up Tree
Across Tree
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Problems in Single Matrix Formation
M2L
M2Ms
Q2M
L2Ls
L2P
Fast Matrix Vector Products
Fast Multipole Iterative Method Does Not
Inherently Lend Itself to Fast Direct Solution
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Modified LHS
Z
q
V
Are We Simply Increasing the Size of the Matrix
to Make it Sparse?

No

• Size of the Matrix Increases
• Non-Zero Entries O(No)
• Non-Zero Entries NOT No2

q
ML
Nn
Multipole Expansions
ML-1
LL
Local Expansions
LL-1
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Modified Set of Equations
LHS

• 1st Set of Equations Formation of V
• - Contribution from q via Q2P (Finest Level)
• - Contribution from L via L2P (Finest Level)

• 2nd Set of Equations Formation of M
• - Contribution from q via Q2M (Finest Level)
• - Contribution from M (From Level Below) via
  M2M

• 3rd Set of Equations Formation of L
• - Contribution from M via M2L (Same Level)
• - Contribution from L (From Level Above) via
  L2L

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4 Level Sparse Matrix
Set 1
Set 2
Set 3

• Total Number of Non-zero Entries is O(N)

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Optimization Number of Levels

• Increase Levels More Sparsity
• Increase Levels Larger Size of the Matrix

Dry Run Pre-Estimation of Number of Levels

• Re-Order The Unknowns Based on Geometry
• Dry-Run Cost is a Function of Fillin-Factor (w)

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Outline

• Focus Application Accurate Charge Distribution
• - Circuit Parasitic Estimation
• - MEMS Charge Distribution
• Motivation behind Fast Direct Solution
• - Large Number of RHS Vectors
• - Re-simulation Advantages
• DiMES Direct Multipole Expansion Solver
• - Sparsification of MoM Using FMM
• - Sparse 1.3 Solution
• Numerical Results

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Validation Example
Multipole Order (p) 2 1.5GB RAM and 1.6GHz
Processor Speed
Capacitance Matrix Norm Difference lt 1e-3
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Time and Memory
a3
Memory
Time LU Setup
ß2
a1.8
ß1.2
ß2
Time LU Solve
ß1.2
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Substrate Coupling Problem
2500 Metal Contacts 6500 Charge Basis Functions
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Comparison with FastCap
Cutoff Point 360 RHS Vectors
Below Cutoff Fast Iterative Solver Above Cutoff
Fast Direct Solver
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Conclusions and Future Work

• Conclusions
• First of Its Kind Multilevel Multipole-based
  Direct Solver
• Matrix Structure is Not Forced
• - Valid for Arbitrary 3D Structures
• Fillins are Not Neglected
• - Guaranteed High Accuracy

• Future Work
• Reduce Setup Time
• - Increasing N will Increase Cut-off Point
  More than Linearly