This article presents a unified description of numerical methods for solving electromagnetic field problems. Traditionally, these methods are considered to be independent and alternative ways of solving Maxwell’s equations or equations derived therefrom. However, they all are projective approximations of the unknown solution by known expansion or basis functions with unknown amplitude coefficients that are determined using the so- called method of weighted residuals (MWR). This common feature forms the proposed unifying framework that may serve both as a systematic introduction to computational electromagnetics and as a way to provide insight into the nature, strengths, and limitations of the various algorithms used by present and future field solvers. For convenience, the fundamental equations and concepts of electromagnetics are summarized in order to facilitate the demonstration of the unified approach. Our aim is to provide students, designers, and researchers with a framework for developing new numerical algorithms, to guide users in the selection and application of electromagnetic design tools, and to foster informed engineering judgment. This article also serves as a review and summary of earlier theoretical works reported in different places and at different times.

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A Unified View of Computational Electromagnetics

A full-size airplane model (the EXPEDITE-RCS model) was developed as part of a benchmark suite for evaluating radar-cross-section (RCS) prediction methods. To generate accurate reference data for the benchmark problems formulated using the model, scale-model targets were additively manufactured, their material properties and RCS were measured, and the measurements were validated with a surface-integral-equation solver. To enable benchmarking of as many computational methods as possible, the following data are made available in a version-controlled online repository: (1) Exterior surface (outer mold line) of the CAD model in two standard file formats. (2) Triangular surface meshes. (3) Measured and predicted monostatic RCS data.

Adding a Reproducible Airplane Model to the Austin RCS Benchmark Suite

To solve scattering from multiple complex targets above the rough surface with a high root mean square, a hybrid method combining real coefficients fitting both Green's function and its gradient with the fast Fourier transform (RFGG-FG-FFT) and the iterative Kirchhoff approximation (IKA) is developed. Due to its great performance for solving complex fine structures, RFGG-FG-FFT is applied to compute scattering from complex targets, while IKA is used to solve scattering from rough surface. Interactions between the targets with rough surface are solved iteratively and these are the most time-consuming part of the entire computational process. To overcome this computational bottleneck, the combined multilevel fast multipole algorithm (MLFMA), ray-propagation fast multipole algorithm (RPFMA), and fast far-field approximation (FAFFA) are incorporated into interactions between the targets and rough surface. To effectively implement the MLFMA-RPFMA-FAFFA algorithm, the rough surface region is divided into near area, adjacent area, and far area according to distance, where the interactions between these areas with targets are solved by MLFMA, RPFMA, and FAFFA, respectively. The high efficiency and accuracy of the proposed method are verified by a comparison with the numerical method.

Fast RFGG-FG-FFT/IKA for Scattering From Multiple Targets Above a Rough Surface

This communication investigates high-performance computation schemes for local Lagrange interpolation and anterpolation operations in the parallel graphics processing unit (GPU)-accelerated distributed-memory multilevel fast multipole algorithm (MLFMA). Two ELLPACK format-based schemes, namely, block ELLPACK (ELL-B) and hybrid compressed sparse column (CSC)-block ELLPACK (CSC-ELL-B), are proposed for the evaluation of interpolation and anterpolation operations, respectively, which ensure high computational throughput for GPU calculation. Optimization using the GPU hierarchical memory architecture, the mechanism of the stream and the CPU/GPU asynchronous computation pattern are employed to further improve the overall performance. The proposed schemes are proven to be an order of magnitude faster than the conventional schemes for aggregation/disaggregation operations. For an aircraft model involving over 10 billion unknowns, the iteration time is reduced by over half, which is remarkable progress in the development of GPU-accelerated parallelization of MLFMA. 

High Performance Evaluation of the Interpolations and Anterpolations in the GPU-Accelerated Massively Parallel MLFMA

Evaluation of pair potentials is critical in a number of areas of physics The classicalN-body problem has its root in evaluating the Laplace potential, and has spawned tree-algorithms, the fast multipole method (FMM), as well as kernel independent approaches Over the years, FMM for Laplace potential has had a profound impact on a number of disciplines as it has been possible to develop highly scalable parallel algorithm for these potential evaluators This is in stark contrast to parallel algorithms for the Helmholtz (oscillatory) potentials The principal bottleneck to scalable parallelism are operations necessary to traverse up, across and down the tree, affecting both computation and communication In this paper, we describe techniques to overcome bottlenecks and achieve high performance evaluation of the Helmholtz potential for a wide spectrum of geometries We demonstrate that the resulting implementation has a load balancing effect that significantly reduces the time-to-solution and enhances the scale of problems that can be treated using full wave physics

High Performance Evaluation of Helmholtz Potentials usingthe Multi-Level Fast Multipole Algorithm.

Electromagnetic scattering from electrically large objects with multiscale features is an increasingly important problem in computational electromagnetics. A conventional approach is to use an integral equation-based solver that is then augmented with an accelerator, a popular choice being a parallel multilevel fast multipole algorithm (MLFMA). One consequence of multiscale features is locally dense discretization, which leads to low-frequency breakdown and requires nonuniform trees. To the authors’ knowledge, the literature on parallel MLFMA for such multiscale distributions capable of arbitrary accuracy is sparse; this paper aims to fill this niche. We prescribe an algorithm that overcomes this bottleneck. We demonstrate the accuracy (with respect to analytical data) and performance of the algorithm for both PEC scatterers and point clouds as large as $755{\lambda }$ with several hundred million unknowns and nonuniform trees as deep as 16 levels.

Parallel Wideband MLFMA for Analysis of Electrically Large, Nonuniform, Multiscale Structures

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Parallel Wideband MLFMA for Analysis of Electrically Large, Nonuniform, Multiscale Structures