Multiple scattering among vias in planar waveguides using preconditioned SMCG method

Full-wave modeling for cylindrical vias in planar waveguides is formulated using Foldy-Lax multiple scattering equations. Recently, a sparse-matrix canonical-grid method based on fast Fourier transform and an iterative algorithm was proposed to solve a large-scale via problem. In this paper, we further improve computational efficiency by a preconditioning scheme based on the dominant information contained in the near field. We also discuss two methods-the sparse-matrix LU decomposition and sparse matrix iterative methods-for constructing the preconditioner, providing the tradeoffs between CPU time and memory. Results show an order of magnitude improvement over the nonpreconditioned case on the convergence rate. As an example, for 20 000 vias simulation, the solution time (per excitation and per frequency) is approximately 13 s per iteration, and 75 min to reach convergence in 320 iterations with memory requirements of 560 MB on a single Pentium 2.4-GHz processor machine. Numerical simulations are illustrated for physical problems such as ball-grid array and large-scale randomly distributed vias, where we have observed the shielding effects of the terminated vias that reduce the coupling between vias.