THE UNIFED-FFT GRID TOTALIZING ALGORITHM FOR FAST O(N LOG N) METHOD OF MOMENTS ELECTROMAGNETIC ANALYSIS WITH ACCURACY TO MACHINE PRECISION (Invited Paper)

While considerable progress has been made in the realm of speed-enhanced electromagnetic (EM) solvers, these fast solvers generally achieve their results through methods that introduce additional error components by way of geometric type approximations, sparse-matrix type approximations, multilevel type decomposition of interactions, and assumptions regarding the stochastic nature of EM problems.This work introduces the O(N log N ) Unified-FFT grid totalizing (UFFT-GT) method, a derivative of method of moments (MoM), which achieves fast analysis with minimal to zero reduction in accuracy relative to direct MoM solution.The method uniquely combines FFT-enhanced Matrix Fill Operations (MFO) that are calculated to machine precision with FFT-enhanced Matrix Solve Operations (MSO) that are also calculated to machine precision, for an expedient solution that does not compromise accuracy. INTRODUCTIONThere are many speed-enhanced numerical methods of computational electromagnetics today that represent major achievements in terms of speed and efficiency compared to their predecessors.In the algorithms based on the Moment Method (MoM) solution of the integral equations (IEs), the discretization of Maxwell's equations produces a dense matrix equation.The traditional and direct methods for solving such equations require O(N 3 ) operations to arrive at a solution for a problem with N degrees of freedom.However, there are many developed methods which are able to solve the same systems with O(N log N ) operations by making certain approximations.In the class of the FFT-based algorithms such as the Pre-corrected FFT (PFFT) method [1] and CG-FFT algorithm [2], the translational invariance of the IE kernels is used for acceleration of MoM interactions.The methods based on pre-corrections, e.g., PFFT and Adaptive Integral Method (AIM), project an arbitrary geometry onto a uniform FFT grid.This allows for the majority of MoM interactions to be computed quickly using FFT.Only a sparse version of the MoM system pre-correcting the erroneous near field contributions is then directly computed.The other class of algorithms performing rapid MoM solution of the IEs is the Multi-Level-Fast-Multipole-Algorithm (MLFMA) [2].The latter relies on a hierarchical multipole decomposition of the MoM dense matrix interactions allowing for O(N log N ) solution of the dense matrix equations.While both the FFT and MLFMA techniques are highly scalable and able to make reliable workingapproximations, there are numerous applications, such as high-performance filters, inductors, and capacitors for RF integrated circuits (RFIC) making the FFT-class algorithms as the solution of choice.

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