unc seminar numerical issues in fast maxwell’s equation ...vv vv vvvv. unc seminar mathematical...

UNC Seminar

Numerical IssuesNumerical IssuesIn Fast Maxwell’s Equation In Fast Maxwell’s Equation

Solvers for Integrated Circuit Solvers for Integrated Circuit InterconnectInterconnect

Zhenhai Zhu, Ben Song and Jacob White

RLE Computational prototyping group, MITrleweb.mit.edu/vlsi

UNC Seminar

•• BackgroundBackground•• PrePre--corrected FFT algorithmcorrected FFT algorithm•• Unit testing resultsUnit testing results•• Surface integral formulationSurface integral formulation•• Numerical ResultsNumerical Results

OutlineOutline

UNC SeminarMathematical PreliminariesMathematical Preliminaries

SrrfrrrKSdS

∈=′′′∫vvvvv ),()(),( ρ

A simple integral equation:A simple integral equation:

)(span ,)()(1

rbBrbr jn

n

jjjn ′=′=′ ∑

=

vvv αρ

Project the solution on a functional space:Project the solution on a functional space:

1( , ) ,

ik r reK r r

r r r r

′−

′ =′ ′− −

v vv v v v v v

UNC SeminarMathematical PreliminariesMathematical Preliminaries

)(span ,0)(e),( n rtTrrt inivvv ==

Enforce the residual to be orthogonal toEnforce the residual to be orthogonal toanother functional space:another functional space:

fA =αA dense linear system:A dense linear system:

)()(),()(en rfrrrKSdrS

nvvvvv −′′′= ∫ ρ

Residual:Residual:

UNC SeminarSome very useful applicationsSome very useful applications

Figures thank to Figures thank to CoventorCoventor

Electrostatic analysisto compute the capacitance

Magneto-quasi-static analysisto compute impedance

UNC Seminar

© Carleton University Hammerhead UAV Project, 2000, David Willis

Picture thanks to David Joe Willis

Some very useful applicationsSome very useful applications

Computational Aerodynamics

MirrorGimbal

Rotate

Picture thanks to Xin Wang

Stokes Flow SolverViscous drag

UNC SeminarIntroduction to Iterative SolversIntroduction to Iterative Solvers

A simple iterative solver:A simple iterative solver:

0

1

Solve step 1: guess an initial solution , let 0

step 2: compute the residual -

step 3: find an update from step 4: update the solution

step 5: 1, go to step 2

k

k k

Ax bx k

r b Ax

x rx x x

k k+

==

=

= += +

VV

UNC SeminarFast MatrixFast Matrix--Vector ProductVector Product

The most expensive step:The most expensive step:

AxGoal:

2( ) ( ) or ( log( ))O N O N O N N⇒

UNC SeminarWellWell--known Fast Algorithmsknown Fast Algorithms

•• Fast Multiple Method Fast Multiple Method •• Hierarchical SVDHierarchical SVD•• Panel Clustering MethodPanel Clustering Method

Key idea:interaction matrix is low rank

UNC SeminarKernel “Independent” TechniqueKernel “Independent” Technique

Basic requirements:

Reciprocity: ( , ) ( , )G r r G r r′ ′=v v v v

Commonly used Green’s function all satisfythese requirements

1 1, , ( ), ( )

n n

ik r r ik r re er r r r r r r r

′ ′− −∂ ∂′ ′ ′ ′− − ∂ − ∂ −

v v v v

v v v v v v v v

Shift invariance: ( , ) ( , )G r r r r G r r′ ′+ + =v v v v v vV V

UNC Seminar

•• BackgroundBackground•• PrePre--corrected FFT Algorithmcorrected FFT Algorithm•• Unit testing resultsUnit testing results•• Surface Integral FormulationSurface Integral Formulation•• Numerical ResultsNumerical Results

OutlineOutline

UNC SeminarFFTFFT--based Methodbased Method

( , ) ( ) ( ), S

dS G r r r f r r Sρ′ ′ ′ = ∈∫v v v v v

( , ) ( ,0) ( )G r r G r r G r r′ ′ ′= − = −v v v v v v%Key idea: kernel is shift-invariant

A simple example:

H fα =

UNC SeminarFFTFFT--based Methodbased Method

, ( )j

i j i jpanel

H dS G r r′ ′= −∫v v%

1,Only ( 1,2,..., ) are unique. H is a Toeplitz

matrix. Matrix vector product could be computed using FFT in O( log( )) time.

jH j N

N N

=

Operations: Operations: O(O(NNlog(log(NN)))) Memory: O(Memory: O(NN))

If collocation method with constant basis is used

UNC Seminar

Separation of Regular Grid From Separation of Regular Grid From DiscretizationDiscretization PanelsPanels

UNC Seminar

pFFTpFFT Algorithm:Algorithm:Basic stepsBasic steps

(1)

(2)

(3)

(4)

[ ]αPQg = :Project (1)

[ ]αDd =Ψ :Direct (4)

[ ] gg I φ=Ψ :eInterpolat (3)

[ ] gg QH=φ :Convolve (2)

[ ] [ ][ ][ ] α)( PHIDdg +=Ψ+Ψ=Ψ

UNC Seminar

pFFTpFFT Algorithm:Algorithm:Basic IdeaBasic Idea

bggggbbbbb NNNNNNNNNN PHIDA ××××× += ][][][][][

A sparse representation A sparse representation of the system matrixof the system matrix

)( ))log(( )( )( )( 2bggbbb NONNONONONO

2( ) ( ) ( ) ( ) ( )b b b g bO N O N O N O N O N

UNC Seminar

pFFTpFFT Algorithm:Algorithm:Interpolation MatrixInterpolation Matrix

),( Compute

Given

yxg

φ

φ

UNC Seminar


( , ) ( , ) ( , )tk k

k

x y c f x y f x y cφ = =∑

2 2 2 2 2 2

An example of ( , ) :

1, , , , , , , ,kf x y

x x y xy x y y xy x y

UNC Seminar

[ ]cF

c

cc

yxfyxfyxf

yxfyxfyxfyxfyxfyxf

g

g

g

g =

=

=

9

2

1

999992991

229222221

119112111

9,

2,

1,

),(),(),(

),(),(),(),(),(),(

ML

MOMMLL

Mφ

φφ

φ

[ ] 1( , ) ( , )tgx y f x y Fφ φ−=


[ ] cyxfc

cyxfyxfyxfyx t ),(),(),(),(),(

9

1

921 =

= MLφ

UNC Seminar


[ ] gt

gt

iii WFrfrdStrrtti

φφφ ===Ψ ∫∆

−1)()()(),( vvvv

Operations: 9Operations: 9NNbb Memory: 9Memory: 9NNbb

[ ] g

g

g

i IWW φφ

φ=

=

Ψ=Ψ

M

M

M

LLLM

M

9,

1,

91

UNC Seminar

pFFTpFFT Algorithm:Algorithm:Outer Differential OperatorOuter Differential Operator

[ ] 1( ) ( )( )t

i

t tn iW dSt r f r F

n r−

∆

∂=

∂∫v vv

1( ) ( )( ) ( )

tgr f r F

n r n rφ φ−∂ ∂

=∂ ∂

v vv v

( , ) ( )( ) S

dS G r r rn r

ρ∂ ′ ′ ′

∂ ∫v v vv

If the kernel has a differential operator outside:

The operator works on the interpolation

UNC Seminar

pFFTpFFT Algorithm:Algorithm:Projection MatrixProjection Matrix

),()1(EsE rrG vv=φ

gt

gi

EiigE rrG φρρφ )(),(,)2( == ∑ vv

gρ charge grid find

Assume a unit charge at point SE

s

)2()1(such that EE φφ =

UNC Seminar


gt

gi

EiigE rrG φρρφ )(),(,)2( == ∑ vv

kk

kE crfrrG ∑= )(),( vvv

match both sides at grid point irv

Expand the Green’s functionE

s gFc φ1−=

gst

EsE FrfrrG φφ 1)1( )(),( −== vvv

[ ] 1)()( −= Frf stt

gvρ

UNC Seminar


)r(b jvon distributi a

is charge theIf

[ ] 1)( )()()( −

∆∫= FrfrdSb t

jtj

gbj

vvρ

[ ] 1)()( −= Frf stt

gvρ

For unit point charge

UNC Seminar


∑=

=bN

j

tjgjgQ

1

)( )(ρα

For multiple panels:

( ) ( )j jj

r b rρ α= ∑v v

[ ] 1( )( ) ( ) ( )bj

j t tg jdSb r f r Fρ −

∆

= ∫v v

UNC Seminar


[ ],1 ,1

,9 ,9

( ) ( )

,1

( ) ( ),9

0 0

0

g g

g g

j k

g j

gj k

g k

QQ P

Q

ρ ρ αα

ρ ρ α

= = =

O M MM MM M M

M M M M M M MM M M

M MM M M M

Operations: 9Operations: 9NNbb Memory: 9Memory: 9NNbb

∑=

=bN

j

tjgjgQ

1

)( )(ρα

UNC Seminar

pFFTpFFT Algorithm:Algorithm:Inner Differential OperatorInner Differential Operator

[ ]∫∆

−=bj

Frfrdn

drdSb t

jtg

1)()(

)( vvvρ

gst

s Frfrdn

dr

rdnd

φφ 1)()(

)()(

−= vvvv

)(),()(

rrrGrdn

dSd

S

′′′

′∫vvvv ρ

If the kernel has a differential operator inside:

The operator works on the projection

UNC Seminar

pFFTpFFT Algorithm:Algorithm:Duality of [Duality of [II] and [] and [PP]]

[ ] 1)()( −

∆∫ FrfrdSb t

jbj

vv

[ ]∫∆

−

ti

FrfrdSt ti

1)()( vv

then,or ),()( If nnji BTrbrt == vv

ith row of [I]:

jth column of [P]:

tIP =

UNC Seminar

pFFTpFFT Algorithm:Algorithm:Convolution MatrixConvolution Matrix

(1)

(2)

(3)

(4) [ ] gg

igi

jijg

QH

QrrG

=

′= ∑φ

φ ,, ),( vv

UNC Seminar

pFFTpFFT Algorithm:Algorithm:Convolution MatrixConvolution Matrix

, ( )i j i jH G r r′= −v v%1,Only ( 1, 2,..., ) are uniquej gH j N=

Regular grid and position invariance

Operations: Operations: O(O(NNgglog(log(NNgg)))) Memory: O(Memory: O(NNgg))

, ,( )g j i j g ii

G r r Qφ ′= −∑ v v%

Fast convolution by FFT in O(Nlog(N)) time

UNC Seminar

pFFTpFFT Algorithm:Algorithm:Direct MatrixDirect Matrix

[ ][ ][ ]

=Ψ==

g

gg

g

IQH

PQ

φφ

α

[ ][ ][ ]αPHI=Ψ

Summary of the first three steps:

UNC Seminar

pFFTpFFT Algorithm:Algorithm:Direct MatrixDirect Matrix

[ ])(

)( )(),(,, )(

i

g

i

j

HWAD jjitjiji

Ν∈

−= ρ

Operations: Operations: O(O(NNbb)) Memory: Memory: O(O(NNbb))

UNC Seminar

pFFTpFFT Algorithm:Algorithm:Four sparse matricesFour sparse matrices

[ ] [ ] [ ][ ][ ] αα )( PHIDA +≈=Ψ•• Projection:Projection:

Operations: Operations: O(O(NNbb)) Memory: Memory: O(O(NNbb))•• Convolution:Convolution:

Operations: Operations: O(O(NNgglog(log(NNgg)))) Memory: O(Memory: O(NNgg))•• Interpolation:Interpolation:

Operations: Operations: O(O(NNbb)) Memory: Memory: O(O(NNbb))•• Nearby interaction:Nearby interaction:

Operations: Operations: O(O(NNbb)) Memory: Memory: O(O(NNbb))

UNC Seminar


OutlineOutline

UNC Seminar

Unit testingUnit testing

Let x be a random vector

1

2

1 2 2

1 2

( )

y Ax

y pfft x

y yerror

y

==

−=

( , ) ( )S

dS K r r r Axρ′ ′ ′ ⇒∫v v v

On the surface of a sphere with radius R

UNC Seminar

SingleSingle--Layer KernelLayer KernelAccuracy (4Accuracy (4--5 digits)5 digits)

Number of Panel

1r

UNC Seminar

SingleSingle--Layer Kernel Layer Kernel Accuracy (4Accuracy (4--5 digits), 5 digits), R/R/λ λ = 1e= 1e--66

Number of Panel

ikrer

UNC Seminar

SingleSingle--Layer Kernel Layer Kernel Accuracy (4Accuracy (4--5 digits), 5 digits), R/R/λ λ = 1e2= 1e2

Number of Panel

ikrer

UNC Seminar

DoubleDouble--Layer Kernel Layer Kernel Accuracy (2Accuracy (2--3 digits)3 digits)

Number of Panel

1r

UNC Seminar

DoubleDouble--Layer Kernel Layer Kernel Accuracy (2Accuracy (2--3 digits ), 3 digits ), R/R/λ λ = 1e= 1e--66

ikrer

Number of Panel

UNC Seminar

DoubleDouble--Layer Kernel Layer Kernel Accuracy (2Accuracy (2--3 digits ), 3 digits ), R/R/λ λ = 1e2= 1e2

Number of Panel

ikrer

UNC SeminarMatrix vector product time O(Matrix vector product time O(NN))

Break even point

Number of Panel

UNC SeminarMemory O(Memory O(NN))

Number of Panel

UNC Seminar


OutlineOutline

UNC Seminar

Summary of the Summary of the Surface Integral FormulationSurface Integral Formulation

iSSrrE

rnrrG

rnrE

rrGSdrEi

∈′′∂

′∂−

′∂′∂′′= ∫

vvvvv

vvvvvvv

))()(

),()()(

),(()(21 1

1

SrrrrrGrd sS∈=′′′∫

vvvvvv )()(),(0 εφρ

SrrrErn

rrGrnrE

rrGSdrES

∈∇+′′∂

′∂−

′∂′∂′′=− ∫

vvvvvv

vvv

vvvv )())(

)(),(

)()(

),(()(21 0

0 φ

contact ,)( ∈= rcrvvφ

Current Conservation0)( =•∇ rE vv

UNC SeminarSystem Matrix StructureSystem Matrix Structure

1 1

1 1

1 1

1 0 1 0 1 0 1 0 1 0 1 0 1

2 0 2 0 2 0 2 0 2 0 2 0 2

0

ˆˆ

x

y

z

x y z x y xx

x y z x y zy

zx y z

x y z

P D dE dnP D dE dn

P D dE dnt P t P t P t D t D t D t Et P t P t P t D t D t D t E

j En n n

a a a c c cA P

ωσ

φρ

∇ ∇

ii

UNC SeminarPrePre--conditionerconditioner

1

1

1

1 0 1 0 1 0 1 1 1 1

2 0 2 0 2 0 2 2 2 2

0

2

2

2ˆ2 2 2ˆ2 2 2

d

xd

yd

zd d d

x y z x y x xd d d

x y z x y z y

zx y z

x y zd

P I dE dnP I dE dn

P I dE dnt P t P t P t I t I t I t Et P t P t P t I t I t I t E

j En n n

a a a c c cA P

π

π

π

π π ππ π π

ωσ φ

ρ

− − − ∇ ∇

ii

UNC Seminar


OutlineOutline

UNC SeminarA Ring ExampleA Ring Example

Cross-section: 0.5x0.5 mm2

Radius: 10 mm

Picture thanks to Junfeng Wang

UNC Seminar

Frequency(Hz)

Res

ista

nce

(O

hm

)

A Ring ExampleA Ring Example

+ FastHenry 960o FastHenry 3840* FastHenry 15360-- Surface 992-- DC Analytical formulae

UNC Seminar

Frequency(Hz)

Ind

uct

ance

(n

H)

A Ring ExampleA Ring Example

+ FastHenry 960o FastHenry 3840* FastHenry 15360-- Surface 992-- DC Analytical formulae

UNC SeminarA Spiral ExampleA Spiral Example

Cross-section: 0.5x0.5 mm2

Inner Radius: 10 mm Spacing: 0.5 mmNumber of turns: 2

Picture thanks to Junfeng Wang

UNC Seminar

Frequency(Hz)

Res

ista

nce

(O

hm

)A Spiral ExampleA Spiral Example

o FastHenry 7680+ FastHenry 1920- Surface 488

UNC Seminar

Frequency(Hz)

Ind

uct

ance

(n

H)

A Spiral ExampleA Spiral Example

o FastHenry 7680+ FastHenry 1920- Surface 488

UNC Seminar

Frequency(Hz)

Imp

edan

ce (

oh

m)

A Shorted Transmission LineA Shorted Transmission Line

x 1011

+ Without Ground- With Ground

UNC SeminarA Spiral Over GroundA Spiral Over Ground

4-turn spiral over substrate 15162 panelsMQS: 106k unknowns, 69 minutes, 348 MbEMQS: 121k unknowns, 93minutes, 379 Mb

UNC SeminarMultiMulti--conductor busconductor bus

3 layer, 10 conductors each layer 12540 panelsMQS: 87.5k unknowns, 41 minutes, 165 MbEMQS: 100k unknowns, 61 minutes, 218 Mb

UNC Seminar

•• PrePre--corrected FFT algorithmcorrected FFT algorithm•• Performance of the algorithmPerformance of the algorithm•• A surface integral formulationA surface integral formulation•• Numerical resultsNumerical results

ConclusionsConclusions

unc seminar numerical issues in fast maxwell’s equation ...vv vv vvvv. unc seminar mathematical...

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