Iterative near-field preconditioner for the multilevel fast multipole algorithm

For iterative solutions of large and difficult integral-equation problems in computational electromagnetics using the multilevel fast multipole algorithm (MLFMA), preconditioners are usually built from the available sparse near-field matrix. The exact solution of the near-field system for the preconditioning operation is infeasible because the LU factors lose their sparsity during the factorization. To prevent this, incomplete factors or approximate inverses can be generated so that the sparsity is preserved, but at the expense of losing some information stored in the near-field matrix. As an alternative strategy, the entire near-field matrix can be used in an iterative solver for preconditioning purposes. This can be accomplished with low cost and complexity since Krylov subspace solvers merely require matrix-vector multiplications and the near-field matrix is sparse. Therefore, the preconditioning solution can be obtained by another iterative process, nested in the outer solver, provided that the outer Krylov subspace solver is flexible. With this strategy, we propose using the iterative solution of the near-field system as a preconditioner for the original system, which is also solved iteratively. Furthermore, we use a fixed preconditioner obtained from the near-field matrix as a preconditioner to the inner iterative solver. MLFMA solutions of several model problems establish the effectiveness of the proposed nested iterative near-field preconditioner, allowing us to report the efficient solution of electric-field and combined-field integral-equation problems involving difficult geometries and millions of unknowns. © 2010 Societ y for Industrial and Applied Mathematics

Schur complement preconditioners for surface integral-equation formulations of dielectric problems solved with the multilevel fast multipole algorithm

Cataloged from PDF version of article.Surface integral-equation methods accelerated with the multilevel fast multipole algorithm (MLFMA) provide a suitable mechanism for electromagnetic analysis of real-life dielectric problems. Unlike the perfect-electric-conductor case, discretizations of surface formulations of dielectric problems yield 2 x 2 partitioned linear systems. Among various surface formulations, the combined tangential formulation (CTF) is the closest to the category of first-kind integral equations, and hence it yields the most accurate results, particularly when the dielectric constant is high and/or the dielectric problem involves sharp edges and corners. However, matrix equations of CTF are highly ill-conditioned, and their iterative solutions require powerful preconditioners for convergence. Second-kind surface integral-equation formulations yield better conditioned systems, but their conditionings significantly degrade when real-life problems include high dielectric constants. In this paper, for the first time in the context of surface integral-equation methods of dielectric objects, we propose Schur complement preconditioners to increase their robustness and efficiency. First, we approximate the dense system matrix by a sparse near-field matrix, which is formed naturally by MLFMA. The Schur complement preconditioning requires approximate solutions of systems involving the (1,1) partition and the Schur complement. We approximate the inverse of the (1,1) partition with a sparse approximate inverse (SAI) based on the Frobenius norm minimization. For the Schur complement, we first approximate it via incomplete sparse matrix-matrix multiplications, and then we generate its approximate inverse with the same SAI technique. Numerical experiments on sphere, lens, and photonic crystal problems demonstrate the effectiveness of the proposed preconditioners. In particular, the results for the photonic crystal problem, which has both surface singularity and a high dielectric constant, shows that accurate CTF solutions for such problems can be obtained even faster than with second-kind integral equation formulations, with the acceleration provided by the proposed Schur complement preconditioners

Envelope tracking integral equation based hybrid electromagnetic circuit simulators

This dissertation presents envelope-tracking hybrid field-circuit simulator for efficiently analyzing narrowband scattering from distributed structures loaded with nonlinear devices. The simulator models the interactions of fields with distributed structures and lumped elements by coupling and simultaneously solving the electric field integral equation and Kirchhoff’s equations, respectively. The coupled nonlinear system of equations is iteratively solved by a time marching scheme that represents the fields, voltages, and currents of interest (signals) as a truncated series of harmonic sinusoids (carriers) multiplied with complex-valued time-varying coefficients (envelopes). Unlike time-domain simulators, which sample the signals at a rate proportional to their maximum frequency content, the proposed envelope-tracking simulator samples the envelopes at a rate proportional to their maximum bandwidth; thus, it requires significantly fewer time steps when solving narrowband problems. Moreover, the envelope-tracking simulator is generally more accurate than its time-domain counterpart because of smaller integration and interpolation errors. Numerical results demonstrate that the proposed simulator improves the tradeoff between accuracy and computational cost, especially when analyzing distributed structures excited by narrowband signals or/and loaded with weakly nonlinear devices. Although the Fourier envelope simulator uses smaller number of time steps, there are other issues relating to the Fourier envelope simulator which are addressed in this thesis: (i) lumped element models that relate voltage envelopes and current envelopes for nonlinear elements are generally unavailable and the approximations used in the simulator to find them are inaccurate for broader band excitations. Higher order interpolation schemes were used in this dissertation to improve the accuracy of these approximations. Numerical results that demonstrate the ability to solve for problems with broader bandwidth of excitation are presented. (ii) As in its timedomain counterpart, adaptive integral method is used to reduce the computational cost of the simulator thus enabling the simulation of larger problems and (iii) Sparse preconditioners are used to improve the convergence of the solution algorithms. Finally, the Fourier envelope method is extended to the analysis of infinitely periodic arrays containing lumped nonlinear loads. Numerical results are presented to highlight the .features of this algorithm.Electrical and Computer Engineerin