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

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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

Magnetostatic

Problems

Electrostatic

Problems

DiMES: FAST DIRECT

SOLVER ALGORITHM

Electric Field

Integral Equations

Magnetic Field

Integral Equations

PMCHW: Multi-Region Dielectric Problems

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

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 ParasiticsMEMS: 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

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

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

Practical problems: N ~ 1 million

Fast Solvers: SignificanceN = 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?

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

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

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

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

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

<< Number of basis functions

(Well-separated groups)

=

=

Translation: Same Size

Translation

Fast Multipole Basics

1D Geometry

MoM

Matrix

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

=

Q2P

=

=

Reconstruct with Multipoles

Problems in Single Matrix FormationM2L

M2Ms

Q2M

L2Ls

L2P

Fast Matrix Vector Products

Fast Multipole Iterative Method Does Not Inherently

Lend Itself to Fast Direct Solution

Step 2: Use Multipole Expansions

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

Modified LHSZ

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

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

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)

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

Validation Example

Multipole Order (p): 2

1.5GB RAM and 1.6GHz Processor Speed

Capacitance Matrix

Norm Difference < 1e-3

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

2500 Metal Contacts; 6500 Charge Basis Functions

Cutoff Point: 360 RHS Vectors

Below Cutoff: Fast Iterative Solver

Above Cutoff: Fast Direct Solver

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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