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DiMES: Multilevel Fast Direct Solver based on Multipole Expansions for Parasitic Extraction of Massively Coupled 3D Microelectronic Structures PowerPoint PPT Presentation

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

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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


Class of problems l.jpg

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


Outline l.jpg

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


Increasing interconnect parasitics l.jpg

Deep Sub-Micron and Nano Fabrication Technology

- Gate delay reduces

Overall chip size does not decrease

- More functionalities added to the same chip

Switching Speed:

Function of Interconnect Parsitics

  • = 0.0103pF

  • = 0.0103pF

  • = 0.0153pF

  • = 0.0153pF

  • = 0.0069pF

  • = 0.0069pF

  • = 0.0153pF

  • = 0.0069pF

  • = 0.0103pF

  • = 0.0153pF

  • = 0.0103pF

  • = 0.0069pF

  • = 0.0153pF

  • = 0.0103pF

  • = 0.0069pF

  • = 0.0153pF

  • = 0.0103pF

  • = 0.0069pF

  • Spacing between traces reduced

  • Spacing between traces reduced

  • Spacing between traces reduced

  • Spacing between traces reduced

  • Spacing between traces reduced

  • Spacing between traces reduced

  • Aspect Ratio (H/W) Increases

  • Aspect Ratio (H/W) Increases

  • Aspect Ratio (H/W) Increases

  • Aspect Ratio (H/W) Increases

  • Aspect Ratio (H/W) Increases

  • Aspect Ratio (H/W) Increases

Size

Size

Size

250nm

250nm

250nm

70nm

70nm

70nm

ITRS

Data

ITRS

Data

ITRS

Data

Spacing

Spacing

Spacing

340nm

340nm

340nm

100nm

100nm

100nm

H/W

H/W

H/W

1.8:1

1.8:1

1.8:1

2.7:1

2.7:1

2.7:1

Courtesy: VLSI Systems WPI web-course

Increasing Interconnect Parasitics


Mems electrical force computations l.jpg

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


Solution scheme l.jpg

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


Method of moments l.jpg

Surface is Discretized into Patches (Basis Functions)

Pulse

Method of Moments

  • Basis Functions Interact through the Green’s Function

  • Generates a Dense Method of Moments Matrix


Fast solvers significance l.jpg

Courtesy: Ansoft Corporation

Practical problems: N ~ 1 million

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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α=2; β=1

Fast Setup and Solve

p1=2xp

ILL-Conditioned Problem

α=2; β=2

Fast LU Setup

α=3; β=2

Direct / LU

Fast Direct

Fast Iterative

Fast Iterative

Fast Iterative

Fast Iterative

Fast Direct

Fast Direct

  • ILL-Conditioned Problems (More Prominent for EFIE)

  • Large Number of Excitations / Large Number of RHS Vectors

Motivn 1: Large Number of RHS Vectors

Direct Setup + Solve

Fast Iterative Setup + Solve

Setup

Solve

Solve

Setup

N=10,000

p=90


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Motivn 2: Fast Updates in Re-simulation

Critical Transition: Analysis to Solution

1. Schur Complement

2. SMW-Updates

=

B

+

A

Ax+By=z1;

Cx+Dy=z2;

(A+BD-1C)x=z1-BD-1z2

D

A

U

V

M

C

Repeated Simulation: Update vs. Re-Solve


Existing literature l.jpg

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

Existing Literature

  • 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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Number of degrees of freedom

<< Number of basis functions

(Well-separated groups)

=

=

Translation: Same Size

Translation

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


Problems in single matrix formation l.jpg

=

=

Q2P

=

=

Reconstruct with Multipoles

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


Modified lhs l.jpg

Step 1: Increase LHS Size

Step 2: Use Multipole Expansions

Step 1 On Its Own Will NOT Expedite; Step 1 is ONLY Required To Achieve Step2

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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Q2P

q

Q2M

ML

L2P

ML-1

M2M

LL

LL-1

M2L

L2L

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


Slide22 l.jpg

Hughes Test Chip ic_hrl_tc1

Validation Example

Multipole Order (p): 2

1.5GB RAM and 1.6GHz Processor Speed

Capacitance Matrix

Norm Difference < 1e-3


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Hughes Test Chip ic_hrl_tc1

Time and Memory

α=3

Memory

Time: LU Setup

β=2

α=1.8

β=1.2

β=2

Time: LU Solve

β=1.2


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Substrate Coupling Problem

2500 Metal Contacts; 6500 Charge Basis Functions


Slide25 l.jpg

Comparison with FastCap

Cutoff Point: 360 RHS Vectors

Below Cutoff: Fast Iterative Solver

Above Cutoff: Fast Direct Solver


Slide26 l.jpg

Highlight

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


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