Fast Low-FrequencyImpedance Extraction using a

Volumetric 3D Integral Formulation

A.MAFFUCCI, A. TAMBURRINO, S. VENTRE, F. VILLONEEURATOM/ENEA/CREATE Ass., Università di Cassino, Italy

G. RUBINACCIEURATOM/ENEA/CREATE Association,

Università di Napoli “Federico II”, Italy

Structure of the Talk

•IntroductionAim of the work

“Fast” methods

Aim of the work

Big interconnect delay and coupling increases the importance of interconnect parasitic parameter extraction.

In particular, on-chip inductance effect becomes more and more critical, for the huge element number and high clock speed

Precise simulation of the current distribution is a key issue in the extraction of equivalent frequency dependent R an L for a large scale integration circuit.

Difficulties arise because of the skin-effect and the related proximity effect

Aim of the work

Eddy current volume integral formulations:

Advantages:– Only the conducting domain meshed

no problems with open boundaries –“Easy” to treat electrodes and to include electric non linearity.

Disadvantages:– Dense matrices, with a singular kernel heavy computation

Critical point: Generation, storage and inversion of large dense matrices

Aim of the work

• Direct methods: O(N3) operations (inversion)• Iterative methods: O(N2) operations per solution

Fast methods: O(N log(N) ) or O(N) scaling

required to solve large-scale problems

“Fast” methods

Two families of approaches:For regular meshes

FFT based methods (exploiting the translation invariance of the integral

operator, leading to a convolution product on a regular grid) •

For arbitrary shapes Fast Multipoles Method (FMM)Block SVD methodWavelets…

Basic idea: Separation of long and short range interactions

(Compute large distance field by neglecting source details)

Structure of the talk

•Introduction

•The numerical modelProblem definition

Integral formulation

Problem Definition

S1S2

AE j

sc AJAA

0dˆ SnJ

'd')'(

4ˆ)( 0 r

rrrJrJA

c

kS on 0n̂J

tt ,, rJrrE

kk SonV

•Set of admissible current densities :

•Integral formulation in terms of the electric vector potential T:

J = T “two components” gauge condition

•Edge element basis functions:

“tree-cotree” decomposition

Integral formulation

N

kkkI

1

)()( xNxJ

kkdiv SS on0ˆ,in0),(2 nJJLJ

•Impose Ohm’s law in weak form :

Integral formulation

e

k

N

k Skks dsVdvj

dvdvj

1

0

ˆ)()(

'')'(

4)()(

nrWArW

rrrJrJrW

SS JW ,

Integral formulation

sjω VVFILR

'

')'()(

40 dvdvL ji

ij rrrNrN

dvR jiij )()( rNrN

dvjV siis )()( rArN kS

kiik dSF nrN ˆ)(

dense matrix

sparse matrix

Structure of the talk

•Introduction

•The numerical model

•Solving Large Scale ProblemsThe Fast Multipoles Method

The block SVD Method

is a real symmetric and sparse NN matrix is a symmetric and full NN matrix

The solution of by a direct method requires O(N3) operations

iterative methods

The product needs N2 multiplications

Solving large scale problems

LjωRZ RL

sVVFIZ

IZ

Fast Multipole Method (FMM)

• Goal: computation of the potential due to N charges in the locations of the N charges themselves with O(N) complexity

• Idea: the potential due to a charge far from its source can be accurately approximated by only a few terms of its multipole expansion

p

n

n

nmjj

mnn

j

mn

jini ij

ij Y

rM

xxqx

01

,1

),()(

Fast Multipole Method (FMM)

a

rj

Field points

“far” sources

1

01 ),()(

p

jj

ip

n

n

nmjj

mnn

j

mn

j ra

ar

qY

rM

x

Fast Multipole Method (FMM)

a

rjCoarser level

already computed

1

01 ),()(

p

jj

ip

n

n

nmjj

mnn

j

mn

j ra

ar

qY

rM

x

Fast Multipole Method (FMM)

1

01 ),()(

p

jj

ip

n

n

nmjj

mnn

j

mn

j ra

ar

qY

rM

x

arj N log(N) algorithm!

Fast Multipole Method (FMM)

• To get a O(N) algorithm: local expansion (potential due to all sources outside a given sphere) inside a target box, rather than evaluation of the far field expansion at target positions

p

n

n

nmjj

mn

nj

mnj YrLx

0

),()(

Fast Multipole Method (FMM)

1. Multipole Expansion (ME) for sources at the finest level

2. ME of coarser levels from ME of finer levels (translation and combination)

3. Local Expansion (LE) at a given level from ME at the same level

4. LE of finer levels from LE of coarser levels

Additional technicalities needed for adaptive algorithm (non-uniform meshes)

Fast Multipole Method (FMM)

• Key point: fast calculation of i-th component of the matrix-vector product

• Compute cartesian components separately: three scalar computations

dvii NrAIL

farnear AAA

p

n

n

nm

pmn

nmn

far lOYr0

2/)1(10 3),(4

)(

LrA

p

n

n

nm

pmni

mni

neari lO

0

2/)1(1,

0 3),(4

MLILLI

Block SVD Method

r-r’X=source domain

Y=field domain

XinYin

dvdvLj

i

Y X

jiXYij 0

0,'

')'()(

40

NN

rrrNrN

Block SVD Method

nr

rm

nm

Y

X

XY

YXXY

)Rdim(

)Qdim(

)Ldim(

RQL

nmr

operationsmnr

operationsnmYYX

YXY

,

)(IRQ

IL

is a low rank matrix

rank r decreases as the separation between X and Y is increased

XYL

Block SVD Method

•The computation of the LI product follows the same lines of the FMM adaptive approach

•Each QR decomposition is obtained by using the modified GRAM-SCHMIDT procedure

•An error threshold is used to stop the procedure for having the smallest rank r for a given approximation

The iterative solver

• The solution of the linear system has been obtained in both cases by using the preconditioned GMRES.

• Preconditioner: sparse matrix Rnear + jLnear,

or with the same sparsity as R, or diagonal

• Incomplete LU factorisation of the preconditioner: dual-dropping strategy (ILUT)

Structure of the talk

•Introduction

•The numerical model

•Solving Large Scale Problems

•Test casesA microstrip line

Critical point: the rather different dimensions of the finite elements in the three dimensions, since the error scales as a/R

A microstrip line

a

R

s=50 elements per box

A microstrip line

s=400 elements per box

A microstrip line

N=11068, S=50, e

The relative error in the LfarI product as a function of the compression rate

N=11068, S=50

Conclusions

• The magnetoquasistationary integral formulation here presented is a flexible tool for the extraction of resistance and inductance of arbitrary 3D conducting structures.

• The related geometrical constraints due to multiply connected domain and to field-circuit coupling are automatically treated.

• FMM and BLOCK SVD are useful methods to reduce the computational cost.

• BLOCK SVD shows superior performances in this case, due to high deviation from regular mesh.