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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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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

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


Fast low frequency impedance extraction using a volumetric 3d integral formulation

Structure of the Talk

  • Introduction

  • Aim of the work

  • “Fast” methods


Fast low frequency impedance extraction using a volumetric 3d integral formulation

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


Fast low frequency impedance extraction using a volumetric 3d integral formulation

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


Fast low frequency impedance extraction using a volumetric 3d integral formulation

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 low frequency impedance extraction using a volumetric 3d integral formulation

“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)


  • Fast low frequency impedance extraction using a volumetric 3d integral formulation

    Structure of the talk

    • Introduction

    • The numerical model

    • Problem definition

    • Integral formulation


    Fast low frequency impedance extraction using a volumetric 3d integral formulation

    S2

    S1

    Problem Definition


    Fast low frequency impedance extraction using a volumetric 3d integral formulation

    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


    Fast low frequency impedance extraction using a volumetric 3d integral formulation

    Integral formulation

    • Impose Ohm’s law in weak form :


    Fast low frequency impedance extraction using a volumetric 3d integral formulation

    Integral formulation

    dense matrix

    sparse matrix


    Fast low frequency impedance extraction using a volumetric 3d integral formulation

    Structure of the talk

    • Introduction

    • The numerical model

    • Solving Large Scale Problems

    • The Fast Multipoles Method

    • The block SVD Method


    Fast low frequency impedance extraction using a volumetric 3d integral formulation

    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 productneeds N2 multiplications


    Fast low frequency impedance extraction using a volumetric 3d integral formulation

    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


    Fast low frequency impedance extraction using a volumetric 3d integral formulation

    rj

    a

    Fast Multipole Method (FMM)

    “far” sources

    Field points


    Fast low frequency impedance extraction using a volumetric 3d integral formulation

    rj

    a

    Fast Multipole Method (FMM)

    Coarser level

    already computed


    Fast low frequency impedance extraction using a volumetric 3d integral formulation

    rj

    a

    Fast Multipole Method (FMM)

    N log(N) algorithm!


    Fast low frequency impedance extraction using a volumetric 3d integral formulation

    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 low frequency impedance extraction using a volumetric 3d integral formulation

    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 low frequency impedance extraction using a volumetric 3d integral formulation

    Fast Multipole Method (FMM)

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

    • Compute cartesian components separately:

    •  three scalar computations


    Fast low frequency impedance extraction using a volumetric 3d integral formulation

    Block SVD Method

    Y=field domain

    r-r’

    X=source domain


    Fast low frequency impedance extraction using a volumetric 3d integral formulation

    Block SVD Method

    is a low rank matrix

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


    Fast low frequency impedance extraction using a volumetric 3d integral formulation

    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


    Fast low frequency impedance extraction using a volumetric 3d integral formulation

    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)


    Fast low frequency impedance extraction using a volumetric 3d integral formulation

    Structure of the talk

    • Introduction

    • The numerical model

    • Solving Large Scale Problems

    • Test cases

    • A microstrip line


    Fast low frequency impedance extraction using a volumetric 3d integral formulation

    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


    Fast low frequency impedance extraction using a volumetric 3d integral formulation

    A microstrip line

    s=50 elements per box


    Fast low frequency impedance extraction using a volumetric 3d integral formulation

    A microstrip line

    s=400 elements per box


    Fast low frequency impedance extraction using a volumetric 3d integral formulation

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


    Fast low frequency impedance extraction using a volumetric 3d integral formulation

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

    N=11068, S=50


    Fast low frequency impedance extraction using a volumetric 3d integral formulation

    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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