Loading in 2 Seconds...

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

Loading in 2 Seconds...

- By
**abie** - Follow User

- 64 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about ' Fast Low-Frequency Impedance Extraction using a Volumetric 3D Integral Formulation' - abie

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

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

- Introduction
- Aim of the work
- “Fast” methods

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

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

- 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

- 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)

- Introduction
- The numerical model
- 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

- Impose Ohm’s law in weak form :

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

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 product needs N2 multiplications

- 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

- 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

- 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)

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

- 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 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)

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

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

s=50 elements per box

s=400 elements per box

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

N=11068, S=50

- 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.

Download Presentation

Connecting to Server..