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Fast Low-Frequency Impedance Extraction using a Volumetric 3D Integral Formulation

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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

Structure of the Talk

- Introduction
- Aim 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:

–Densematrices, with a singular kernel heavy computation

Critical point:

Generation, storage and inversion of largedensematrices

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
- FFTbased methods
- (exploiting thetranslation invariance of the integral operator, leading to a convolution product on a regular grid)

- Fast Multipoles Method (FMM)
- Block SVD method
- Wavelets
- …

- Basic idea: Separation of long and short range interactions

Structure of the talk

- Introduction
- The numerical model
- Problem definition
- Integral formulation

S2

S1

Problem Definition

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

Integral formulation

- Impose Ohm’s law in weak form :

Integral formulation

dense matrix

sparse matrix

Structure of the talk

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

Solving large scale problems

is a real symmetric and sparse NN matrix

is a symmetric and full NN matrix

The solution ofby a direct method

requires O(N3) operations

iterative methods

The productneeds N2 multiplications

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

rj

a

Fast Multipole Method (FMM)

“far” sources

Field points

rj

a

Fast Multipole Method (FMM)

Coarser level

already computed

rj

a

Fast Multipole Method (FMM)

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

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)

Fast Multipole Method (FMM)

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

Block SVD Method

Y=field domain

r-r’

X=source domain

Block SVD Method

is a low rank matrix

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

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 cases
- A microstrip line

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

A microstrip line

s=50 elements per box

A microstrip line

s=400 elements per box

N=11068, S=50, e=1.e-4

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.