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Efficient Inductance Extraction Using the Solenoidal Basis Method

This paper introduces the Solenoidal Basis Method for efficient inductance extraction, addressing the challenges of inductance computation between current-carrying filaments. By employing a linear system that satisfies Kirchhoff’s law, the methodology invokes preconditioned Krylov subspace methods. Key developments include robust preconditioners and hierarchical approximations, leading to rapid convergence rates. The approach is benchmarked against established methods like FASTHENRY, demonstrating significant improvements in time and memory efficiency across diverse frequency ranges (1 GHz - 1 THz) and geometries such as ground planes and spiral inductors.

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Efficient Inductance Extraction Using the Solenoidal Basis Method

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  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

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