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

1. 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

2. Structure of the Talk • Introduction • Aim of the work • “Fast” methods

3. 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

4. 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

5. 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

6. “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)

7. Structure of the talk • Introduction • The numerical model • Problem definition • Integral formulation

8. S2 S1 Problem Definition

9. 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

10. Integral formulation • Impose Ohm’s law in weak form :

11. Integral formulation dense matrix sparse matrix

12. Structure of the talk • Introduction • The numerical model • Solving Large Scale Problems • The Fast Multipoles Method • The block SVD Method

13. 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

14. 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

15. rj a Fast Multipole Method (FMM) “far” sources Field points

16. rj a Fast Multipole Method (FMM) Coarser level already computed

17. rj a Fast Multipole Method (FMM) N log(N) algorithm!

18. 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

19. 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)

20. Fast Multipole Method (FMM) • Key point: fast calculation of i-th component of the matrix-vector product • Compute cartesian components separately: •  three scalar computations

21. Block SVD Method Y=field domain r-r’ X=source domain

22. Block SVD Method is a low rank matrix rank r decreases as the separation between X and Y is increased

23. 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

24. 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)

25. Structure of the talk • Introduction • The numerical model • Solving Large Scale Problems • Test cases • A microstrip line

26. 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

27. A microstrip line s=50 elements per box

28. A microstrip line s=400 elements per box

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

32. The relative error in the LfarI product as a function of the compression rate N=11068, S=50

33. 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.