Scattering by large, complex, composite targets using hybrid FE/BI/MLFMA

Many problems in Computational Electromagnetics (CEM) and Computational Fluid Dynamics (CFD) require millions of grid cells to properly represent the physics. Advances in parallel computing, both in hardware and software, make it feasible to solve such large scale simulations in a timely manner. The objective of this paper is to describe some of the recent work at Rockwell in performing parallel computing for providing quick-turnaround solutions to problems in CFD, CEM and other computational disciplines.

In a recent invited paper in the IEEE Antennas and Propagation Magazine, some of the challenging problems in computational electromagnetics were presented. One of the objectives of this note is to simply point out that challenging to one may be simple to another. This is demonstrated through an example cited in that article. The example chosen is a Vivaldi antenna array. What we discuss here also applies to the other examples presented in that article, but we have chosen the Vivaldi antenna array to help us make our point. It is shown in this short article that a higher-order basis using a surface integral equation a la a PMCHWT (Poggio-Miller-Chu-Harrington-Wu-Tsai) method-of-moments formulation may still be the best weapon that one have in today's arsenal to deal with challenging complex electromagnetic analysis problems. Here, we have used the commercially available code WIPL-D to carry out all the computations using laptop/desktop systems. The second objective of this paper is to present an out-of-core solver. The goal is to demonstrate that an out-of-core 32-bit-system-based solver can be as efficient as a 64-bit in-core solver. This is quite contrary to the popular belief that an out-of-core solver is generally much slower than an in-core solver. This can be significant, as the difference in the cost of a 32-bit system can be 1/30 of a 64-bit system of similar capabilities using current computer architectures. For the 32-bit system, we consider a Pentium 4 system, whereas for the 64-bit system, we consider an Itanium 2 system for comparison. The out-of-core solver can go beyond the 2 GB limitation for a 32-bit system and can be run on ordinary laptop/desktop; hence, we can simultaneously have a much lower hardware investment while better performance for a sophisticated and powerful electromagnetic solver. The system resources and the CPU times are also outlined.

Maxwell's equations are valid from the atomic to intergalactic length scales. Hence, features at various spatial or temporal scales may all be significant in electromagnetic (EM) problems of interest and may affect the results of multiscale modeling and simulation in computational electromagnetics (CEM). Solving electromagnetic problems that involve multiscale structures efficiently is challenging and is of growing interest in CEM. In recent decades, various approaches have been proposed for addressing the multiscale issue of electrically large problems. These methods can be roughly categorized into three types: the “geometry‐based,” “algebra‐based,” and “physics‐based” approaches. In this chapter, we introduce these three representative approaches for modeling multiscale problems in CEM: (1) the discontinuous Galerkin integral equation method (DGIE), which is a general domain decomposition framework for dealing with structures with mesh nonconformity; (2) the nested dissection accelerated H‐LU‐based fast direct solver for sparse matrices, which is used to construct effective preconditioners; and (3) the resistive sheet boundary condition‐based hybrid finite element‐boundary integral‐multilevel fast multipole algorithm (FE–BI–MLFMA), which can simulate electromagnetic field interactions with microwave‐absorbing honeycomb structures efficiently. After we present the formulation for each approach, we present examples to demonstrate its accuracy, capability, and performance.

Surface-integral-equation (SIE) approaches are presented for the accurate solution of different problems in computational electromagnetics. First, an efficient message passing interface (MPI)/OpenMP parallel implementation of the multilevel fast multipole algorithm-fast Fourier transform (MLFMA-FFT) is presented for the solution of large-scale conducting bodies. Second, looking for the extension of these rigorous approaches to new demanded areas, the SIE method is successfully applied to the solution of left-handed metamaterials and plasmonic nanostructures.

Electromagnetic wave scattering from large and complex bodies is currently the most challenging problem in computational electromagnetics. There is an increasing need for more efficient algorithms with reduced computational complexity and memory requirements. In this work we solve the problem of electromagnetic wave scattering involving three-dimensional, homogeneous, arbitrarily shaped dielectric objects. The fast multipole method (FMM) is used along with the algebraic multigrid (AMG) method, that is employed as a preconditioner, in order to accelerate the convergence rate of the Krylov iterations. Our experimental results suggest much faster convergence compared to the non preconditioned FMM, and hence significant reduction to the overall computation time.

Recent years have seen a spectacular increase in our capability to model, simulate the performance of, and design complex electromagnetic systems. Much progress has been made in enhancing the available numerical techniques, viz., the method of moments (MoM), the finite-element method (FEM), and the finite-difference time-domain (FDTD) or its variants. Great strides have recently been made in enlarging the scope of MoM via the use of the fast multipole method (FMM), which has made it feasible for us to solve problems that require the handling of 106 degrees of freedom, or even higher, and distributed processing has enabled the FDTD to handle upward of 109 degrees of freedom on a moderate-size computing platform. Despite this recent progress, many practical computational electromagnetic (CEM) modeling problems of interest present formidable challenges, and the search for numerically efficient techniques to solve large problems involving complex structures continues unabated. The objectives of this paper are to identify some of these challenging problems encountered by the author during the last five years, and to present the results of application of a technique called CBFM - developed at the EMC Laboratory at Penn State - that has been found useful for addressing them.

The analysis of electromagnetic diffraction phenomena in arbitrarily linear media is a challenging problem in computational electromagnetics either with numerical, analytical, semi-analytical and asymptotic techniques.

This publication documents the proceedings of the first Workshop on Computational Electromagnetics (CEM) and Applications, hosted by the Institute for Computer Applications in Science and Engineering (ICASE) and the NASA Langley Research Center, Hampton, Virginia, 29-31 May, 1996, and attended by approximately 70 people from academia, government laboratories, and industry. ICASE's charter mission in 1972 remains today - to explore novel computer environments (vector in the 1970s; parallel in the 1990s) for scientific computing. These proceedings provide a necessary foundation for symposia in computational electromagnetics for future aerospace applications. The objectives of this CEM Workshop were to provide a forum for many of the leaders of the community to assess the state of CEM technology and to discuss areas of research for future programmatic planning activities. Workshop sessions included topics on optimization, industrial applications, algorithms, and a special panel session was provided during which issues were discussed and future research areas were identified. Hopefully, this publication will stimulate and improve communication among multidisciplinary researchers as well as highlighting several CEM areas that need improvement - especially for highly challenging problems. The two most important criteria in the selection of speakers for the workshop were their substantial contribution to large-scale CEM problems and their ability to articulate the issues confronting the CEM research community. Based on the results obtained, it is anticipated that this publication will be useful to government, industry, and university researchers to plan future research tasks in CEM analytical methods and applications.

This study investigates and evaluates applications of the adjoint problem and its solution in frequency-domain computational electromagnetics (CEM). The study establishes and validates adjoint-based applications including higher order parameter sampling, a posteriori error estimate evaluation, and p- and h-refinements. These applications can improve the efficiency, automation, and robustness of CEM methods. We employ a 1-D finite-element-method scattering solver, simplifying the implementation, replicability, clarity, and intuitiveness of analysis results and conclusions, which then extend naturally to higher dimensional solvers and more complicated CEM problems. While demonstrated with a higher order solver, the derived techniques apply to low-order methodology as well. This is the first demonstration of the applicability of adjoint-based a posteriori error estimation techniques to adaptive discretization refinement in frequency-domain CEM with arbitrary-order basis functions. This work introduces the application of dual-weighted residual error estimation and selective adaptivity based on error cancellation. The proposed targeted, adaptive mesh/model p- and/or h-refinement heuristics informed by adjoint element-wise error contribution estimates show near-monotonic reduction of the quantity-of-interest error with increased numbers of refined elements. In general, adjoint techniques are underutilized in CEM, and another goal of this work is to promote their future use for refinement, optimization, and uncertainty quantification.

We study the threshold conditions for the natural modes of the microsize plasmonic laser shaped as an infinite flat graphene strip grating symmetrically embedded into the gain-material layer, in the H-polarization case. For this purpose, we solve the lasing eigenvalue problem (LEP), which is a classical source-free electromagnetic field boundary-value problem, adapted to the presence of the active region by the corresponding sign of the imaginary part of the refractive index. In such a way we look for the eigenpairs, i.e. the stimulated emission real-valued frequency and the threshold gain index, specific to each mode. We transform LEP to a hypersingular integral equation for the on-strip current density and discretize it by the regularizing Galerkin technique. This procedure leads to a determinantal equation with guaranteed convergence to the exact LEP eigenpairs and controlled accuracy of their computation. The numerical analysis allows us to study the threshold conditions for various lasing modes of the microsize laser, identify them and trace their change when varying the parameters of the lasing structure.This article is part of the theme issue 'Analytically grounded full-wave methods for advances in computational electromagnetics'.

Computational Electromagnetics (CEM) is a technique used to simulate the interaction of electromagnetic fields with structures in a particular environment. The geometries to analyze are typically very complexes and they could be simulated with a CAD model, that can include a fine level of details. This is reflected in a higher computational effort that must be faced by high performances infrastructures. The paper addresses the potential benefit of adopting a grid computing infrastructure to solve CEM problems. In particular here we focus on a Domain Decomposition (DD) technique to reduce the complexity of a MoM analysis of large and complex structures. It was build a low-cost grid infrastructure, composed of heterogeneous recovery machines, both physical and virtual, on which to run scientific applications without the need to modify the source codes of the used algorithms. Virtualization enables the optimization of resources and simplify infrastructure management. Were carried out some performance tests to verify the time gain and the overhead introduced by the grid infrastructure.

The standard Yee FDTD algorithm is widely used in computational electromagnetics because of its simplicity and divergence free nature. A generalization of this classical scheme to 3D unstructured co-volume meshes is adopted, based on the use of a Delaunay primal mesh and its high quality Voronoi dual. This circumvents the problem of accuracy losses, which are normally associated with the use of a staircased representation of curved material interfaces in the standard Yee scheme. The procedure has been successfully employed for modelling problems involving both isotropic and anisotropic lossy materials. Here, we consider the novel extension of this approach to allow for the challenging modelling of chiral materials, where the material parameters are frequency dependent. To adequately model the dispersive behaviour, the Z-transform is employed, using second order Padé approximations to maintain the accuracy of the basic scheme. To validate the implementation, the numerical results produced are compared with available analytical solutions. The stability of the chiral algorithm is also studied.

An a posteriori-driven adaptive Mixed High-Order method with application to electrostatics

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.

Multi-objective design is very common in electrical engineering, and the development of new multi-objective optimizers becomes recently a topical issue in computational electromagnetics because of the unavailability of a universal multi-objective optimizer. In this regard, an improved physical programming (PP) method is presented to solve multi-objective electromagnetic inverse problems. The improvements include: (1) the multi-pseudo preference concept is introduced to overcome the deficiency of existing PP method in finding only one Pareto solution in a simulation and enhance the usage rate of sampling points; (2) a novel dynamically guiding mechanism is proposed to improve the computational efficiency and to provide the evolutionary direction; (3) some new rules for algorithm parameter updating are proposed to realize the two ultimate goals of an idea multi-objective optimizer of finding as many as possible Pareto optimal solutions and having them uniformly distributed on the Pareto Front. The results of typical mathematic test functions and a practical optimization design demonstrate the validity and advantages of the proposed method.