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Fast Low-Frequency Impedance Extraction using a Volumetric 3D Integral Formulation. A.MAFFUCCI, A. TAMBURRINO, S. VENTRE, F. VILLONE EURATOM/ENEA/CREATE Ass., Università di Cassino, Italy G. RUBINACCI EURATOM/ENEA/CREATE Association , Università di Napoli “Federico II”, Italy.

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slide1

Fast Low-Frequency

Impedance Extraction using a Volumetric 3D Integral Formulation

A.MAFFUCCI, A. TAMBURRINO, S. VENTRE, F. VILLONE

EURATOM/ENEA/CREATE Ass., Università di Cassino, Italy

G. RUBINACCI

EURATOM/ENEA/CREATE Association,

Università di Napoli “Federico II”, Italy

slide2

Structure of the Talk

  • Introduction
  • Aim of the work
  • “Fast” methods
slide3

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

slide4

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:

–Densematrices, with a singular kernel  heavy computation

Critical point:

Generation, storage and inversion of largedensematrices

slide5

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

slide6

“Fast” methods

  • Two families of approaches:
  • For regular meshes
      • FFTbased methods
      • (exploiting thetranslation invariance of the integral operator, leading to a convolution product on a regular grid)
  • For arbitrary shapes
      • Fast Multipoles Method (FMM)
      • Block SVD method
      • Wavelets
      • Basic idea: Separation of long and short range interactions
  • (Computelarge distance field by neglecting source details)
slide7

Structure of the talk

  • Introduction
  • The numerical model
  • Problem definition
  • Integral formulation
slide8

S2

S1

Problem Definition

slide9

Integral formulation

  • Set of admissible current densities :
  • Integral formulation in terms of the electric vector potential T:
  • J = T“two components” gaugecondition
  • Edge element basis functions:
  • “tree-cotree” decomposition
slide10

Integral formulation

  • Impose Ohm’s law in weak form :
slide11

Integral formulation

dense matrix

sparse matrix

slide12

Structure of the talk

  • Introduction
  • The numerical model
  • Solving Large Scale Problems
  • The Fast Multipoles Method
  • The block SVD Method
slide13

Solving large scale problems

is a real symmetric and sparse NN matrix

is a symmetric and full NN matrix

The solution ofby a direct method

requires O(N3) operations

iterative methods

The product needs N2 multiplications

slide14

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
slide15

rj

a

Fast Multipole Method (FMM)

“far” sources

Field points

slide16

rj

a

Fast Multipole Method (FMM)

Coarser level

already computed

slide17

rj

a

Fast Multipole Method (FMM)

N log(N) algorithm!

slide18

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
slide19

Fast Multipole Method (FMM)

  • Multipole Expansion (ME) for sources at the finest level
  • ME of coarser levels from ME of finer levels (translation and combination)
  • Local Expansion (LE) at a given level from ME at the same level
  • LE of finer levels from LE of coarser levels

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

slide20

Fast Multipole Method (FMM)

  • Key point: fast calculation of i-th component of the matrix-vector product
  • Compute cartesian components separately:
  •  three scalar computations
slide21

Block SVD Method

Y=field domain

r-r’

X=source domain

slide22

Block SVD Method

is a low rank matrix

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

slide23

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
slide24

The iterative solver

  • The solution of the linear system has been obtained in both cases by using the preconditioned GMRES.
  • Preconditioner: sparse matrix Rnear + jLnear, or with the same sparsity as R, or diagonal
  • Incomplete LU factorisation of the preconditioner: dual-dropping strategy (ILUT)
slide25

Structure of the talk

  • Introduction
  • The numerical model
  • Solving Large Scale Problems
  • Test cases
  • A microstrip line
slide26

a

R

A microstrip line

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

slide27

A microstrip line

s=50 elements per box

slide28

A microstrip line

s=400 elements per box

slide33

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.
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