1. A Solenoidal Basis Method For Efficient Inductance Extraction H emant Mahawar Vivek Sarin Weiping Shi Texas A&M University College Station, TX

2. Introduction

3. Background Inductance between current carrying filaments Kirchoff’s law enforced at each node

4. Background … Current density at a point Linear system for current and potential Inductance matrix Kirchoff’s Law

5. Linear System of Equations Characteristics  Extremely large; R, B: sparse; L: dense  Matrix-vector products with L use hierarchical approximations Solution methodology  Solved by preconditioned Krylov subspace methods  Robust and effective preconditioners are critical Developing good preconditioners is a challenge because system is never computed explicitly!

6. First Key Idea Current Components  Fixed current satisfying external condition I d (left)  Linear combination of cell currents (right)

7. Solenoidal Basis Method Linear system Solenoidal basis  Basis for current that satisfies Kirchoff’s law  Solenoidal basis matrix P:  Current obeying Kirchoff’s law: Reduced system  Solve via preconditioned Krylov subspace method

8. Local Solenoidal Basis Cell current k consists of unit current assigned to the four filaments of the kth cell There are four nonzeros in the kth column of P: 1, 1, -1, -1

9. Second Key Idea Observe: where Approximate reduced system Approximate by

10. Preconditioning Preconditioning involves multiplication with

11. Hierarchical Approximations Components of system matrix and preconditioner are dense and large Hierarchical approximations used to compute matrix-vector products with both L and  Used for fast decaying Greens functions, such as 1/r (r : distance from origin)  Reduced accuracy at lower cost Examples  Fast Multipole Method: O(n)  Barnes-Hut: O(nlogn)

12. FASTHENRY Uses mesh currents to generate a reduced system Approximation to reduced system computed by sparsification of inductance matrix Preconditioner derived from Sparsification strategies  DIAG: self inductance of filaments only  CUBE: filaments in the same oct-tree cube of FMM hierarchy  SHELL: filaments within specified radius (expensive)

13. Experiments Benchmark problems  Ground plane  Wire over plane  Spiral inductor Operating frequencies: 1GHz-1THz Strategy  Uniform two-dimensional mesh  Solenoidal function method  Preconditioned GMRES for reduced system Comparison  FASTHENRY with CUBE & DIAG preconditioners

14. Ground Plane

15. Problem Sizes Mesh Potential Nodes Current Filaments Linear System Solenoidal functions 33x331,0892,1123,2011,024 65x654,2258,32012,5454,096 129x12916,64133,02449,66516,384 257x25766,049131,584197,63365,536

16. Comparison with FastHenry Preconditioned GMRES Iterations (10GHz) Mesh FASTHENRY DIAG FASTHENRY CUBE Solenoidal Method 33x3313 5 65x6516176 129x12921197 257x25726289 513x513--14

17. Comparison … Time and Memory (10GHz) Mesh FASTHENRY DIAG FASTHENRY CUBE Solenoidal Method Time (sec) Mem (MB) Time (sec) Mem (MB) Time (sec) Mem (MB) 33x332102 21 65x6513421742125 129x129951771421776817 257x257835734136473440969 513x513----2925298

18. Preconditioner Effectiveness Preconditioned GMRES iterations Mesh Filament Length Frequency (GHz) 1101001000 33x331/326555 65x651/646655 129x1291/1288776 256x2561/25611988

19. Wire Over Ground Plane

20. Comparison with FastHenry Preconditioned GMRES Iterations (10GHz) Mesh FASTHENRY DIAG FASTHENRY CUBE Solenoidal Method 33x3313114 65x6513145 129x12913126 257x257338 513x513--12

21. Comparison … Time and Memory (10GHz) Mesh FASTHENRY DIAG FASTHENRY CUBE Solenoidal Method Time (sec) Mem (MB) Time (sec) Mem (MB) Time (sec) Mem (MB) 33x332102 11 65x651242164294 129x129791781241785515 257x257719735273273535161 513x513----2427260

22. Preconditioner Effectiveness Preconditioned GMRES iterations Mesh Filament Length Frequency (GHz) 1101001000 33x331/325444 65x651/646555 129x1291/1288666 257x2571/25612887

23. Spiral Inductor

24. Preconditioner Effectiveness Preconditioned GMRES iterations Mesh Filament Length Frequency (GHz) 1101001000 33x331/327666 65x651/648777 129x1291/12810999 257x2571/256161211

25. Concluding Remarks Preconditioned solenoidal method is very effective for linear systems in inductance extraction Near-optimal preconditioning assures fast convergence rates that are nearly independent of frequency and mesh width Significant improvement over FASTHENRY w.r.t. time and memory Acknowledgements: National Science Foundation