Introduction

1. A Solenoidal Basis Method For Efficient Inductance ExtractionHemant MahawarVivek SarinWeiping ShiTexas A&M UniversityCollege 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 Id (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

16. Comparison with FastHenry Preconditioned GMRES Iterations (10GHz)

17. Comparison … Time and Memory (10GHz)

18. Preconditioner Effectiveness Preconditioned GMRES iterations

19. Wire Over Ground Plane

20. Comparison with FastHenry Preconditioned GMRES Iterations (10GHz)

21. Comparison … Time and Memory (10GHz)

22. Preconditioner Effectiveness Preconditioned GMRES iterations

23. Spiral Inductor

24. Preconditioner Effectiveness Preconditioned GMRES iterations

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