Problems In Computational Electromagnetics Research Articles

Predicting Macro Basis Functions for Method of Moments Scattering Problems Using Deep Neural Networks

In this letter, we present research on the application of deep neural networks to predicting macro basis functions for complicated computational electromagnetics problems. We provide error statistics and representative examples for networks trained on simple and complicated datasets of method of moments scattering problems. Notably, we demonstrate that the networks learn generalizable knowledge applicable to problem types on which they were not trained. We conclude that the networks produce encouraging results, especially for cross validation, and larger training datasets will improve reliability for general scattering problems.

Haar Wavelet Solution Analysis of Compound Pendulum-based Computational Electromagnetic Damping Oscillation Problem

This paper reports the application of Haar wavelet algorithm in achieving the numerical solution of 2nd order computational electromagnetic problems related to dynamics of small, damped oscillations for compound pendulum based system. Flow chart of problem approximation using Haar wavelet has also been incorporated for the solution analysis. In this paper, a comparative analysis has been presented in the terms of efficiency and average computation cost of CPU for the numerical approximations obtained. The analysis shows that the Haar wavelet based techniques give results with less computation cost and more accurate approximations when compared to other numerical methods such as Taylor Series and Runge-Kutta available in literature.

Space-time computation and visualization of the electromagnetic fields and potentials generated by moving point charges

We present a computational method to directly calculate and visualize the directional components of the Coulomb, radiation, and total electromagnetic fields, as well as the scalar and vector potentials generated by moving point charges in arbitrary motion with varying speeds. Our method explicitly calculates the retarded time of the point charge along a discretized grid, which is then used to determine the fields and potentials. The computational approach, implemented in python, provides an intuitive understanding of the electromagnetic waves generated by moving point charges and can be used as a pedagogical tool for undergraduate and graduate-level electromagnetic theory courses. Our computer code, freely available for download, can also approximate complicated time-varying continuous charge and current densities, and can be used in conjunction with grid-based numerical modeling methods to solve real-world computational electromagnetics problems such as experiments with high-energy electron sources. We simulate and discuss several interesting example applications and lab experiments including electric and magnetic dipoles, oscillating and linear accelerating point charges, synchrotron radiation, and Bremsstrahlung.

Constraint-Preserving Hybrid Finite Element Methods for Maxwell’s Equations

Maxwell’s equations describe the evolution of electromagnetic fields, together with constraints on the divergence of the magnetic and electric flux densities. These constraints correspond to fundamental physical laws: the nonexistence of magnetic monopoles and the conservation of charge, respectively. However, one or both of these constraints may be violated when one applies a finite element method to discretize in space. This is a well-known and long-standing problem in computational electromagnetics. We use domain decomposition to construct a family of primal hybrid finite element methods for Maxwell’s equations, where the Lagrange multipliers are shown to correspond to a numerical trace of the magnetic field and a numerical flux of the electric flux density. Expressing the charge conservation constraint in terms of this numerical flux, we show that both constraints are strongly preserved. As a special case, these methods include a hybridized version of Nédélec’s method, implying that it preserves the constraints more strongly than previously recognized. These constraint-preserving properties are illustrated using numerical experiments in both the time domain and frequency domain. In 2-D, we also observe a superconvergence phenomenon, where hybrid post-processing yields an improved estimate of the magnetic field.

Study on a Fast Solver for Poisson’s Equation Based on Deep Learning Technique

Fast and efficient computational electromagnetic simulation is a long-standing challenge. In this article, we propose a data-driven model to solve Poisson’s equation that leverages the learning capacity of deep learning techniques. A deep convolutional neural network (ConvNet) is trained to predict the electric potential with different excitations and permittivity distribution in 2-D and 3-D models. With a careful design of cost function and proper training data generated from finite-difference solvers, the proposed network enables a reliable simulation with significant speedup and fairly good accuracy. Numerical experiments show that the same ConvNet architecture is effective for both 2-D and 3-D models, and the average relative prediction error of the proposed ConvNet model is less than 3% in both 2-D and 3-D simulations with a significant reduction in computation time compared to the finite-difference solver. This article shows that deep neural networks have a good learning capacity for numerical simulations. This could help us to build some fast solvers for some computational electromagnetic problems.

A Posteriori Error Estimation and Adaptive Discretization Refinement Using Adjoint Methods in CEM: A Study With a 1-D Higher Order FEM Scattering Example

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.

Novel techniques for numerically efficient solution of multiscale problems in computational electromagnetics

AbstractMultiscale problems in numerical electromagnetics are becoming increasingly common with the advent and widespread usage of compact mobile phones, body area networks, small and nano antennas, sensors, high‐speed interconnects, integrated circuits, and complex electronic packaging structures, to name just a few commercial applications. Numerical electromagnetic modeling and simulation of structures with multiscale features are highly challenging due to the fact that electrically small as well as large features are simultaneously present in the model that demands for discretization of the computational domain such that the number of degrees of freedom is very large, thus levying a heavy burden on the computational resources. The multiscale nature of a given problem also exacerbates the challenge of generating very fine meshes that do not introduce instabilities or ill‐conditioned behaviors.In this paper, we present novel techniques for an efficient solution of multiscale problems in the time‐domain. The proposed techniques handle arbitrarily shaped objects with fine features by using the dipole moment–based method of moments (MoM) type formulation in the MoM domain and electrically large objects in the finite‐difference time‐domain (FDTD) domain. Scattered fields in the MoM domain are obtained in closed forms and directly in the time‐domain at the desired observation points on a planar interface, which are combined with the conventional FDTD update equations. The time‐domain scattered fields computed by using the MoM formulations presented herein are stable in their implementation.

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

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 755λ with several hundred million unknowns and nonuniform trees as deep as 16 levels.

An Impedance Transfer Function Formulation for Reduced-Order Macromodels of Subgridded Regions in FDTD

Recently, model order reduction was applied to the macromodel of a subgridded region in the method of finite-difference time domain (FDTD), where an admittance transfer function formulation was used to model materials with electric conductivity. In this communication, we develop an impedance transfer function formulation for the reduced-order macromodel of a subgridded region; this formulation may be used to model materials exhibiting magnetic conductivity. The two macromodel types may be used together, in a complementary fashion, to model fine-featured subgridded regions in FDTD that contain materials exhibiting both electric and magnetic conductivity. Additionally, this communication greatly extends the range of both the deterministic and the stochastic macromodeling techniques recently developed in FDTD, thus enabling the solution of a much wider range of problems in computational electromagnetics.

The SCBiCG class of algorithms for complex symmetric linear systems with applications in several electromagnetic model problems

Frequency domain formulations of computational electromagnetic problems often require the solutions of complex-valued non-Hermitian systems of equations, which are still symmetric. For this kind of problems a whole class of sub-variant solver methods derived from the complex-valued Bi-Conjugate Gradient method is available. This class of methods contains established iterative methods as the Conjugate Orthogonal Conjugate Gradient (COCG) method, Bi-Conjugate Gradient Conjugate Residual (BiCGCR) method and Conjugate A-Orthogonal Conjugate Residual (COCR) method. The mathematical equivalence of the BiCGCR method and COCR method is shown and preconditioned variants of the various solvers are derived. An efficient kind of two-step preconditioning technique is also proposed. Numerical experiments involving e.g. electro-quasistatic frequency domain simulation are employed to show the difference in the convergence behaviors of these iterative methods and effectiveness of the two-step preconditioning techniques.

Application of the Jacobi–Davidson method for spectral low-rank preconditioning in computational electromagnetics problems

We consider the numerical solution of linear systems arising from computational electromagnetics applications. For large scale problems the solution is usually obtained iteratively with a Krylov subspace method. It is well known that for ill conditioned problems the convergence of these methods can be very slow or even it may be impossible to obtain a satisfactory solution. To improve the convergence a preconditioner can be used, but in some cases additional strategies are needed. In this work we study the application of spectral low-rank updates (SLRU) to a previously computed sparse approximate inverse preconditioner. The updates are based on the computation of a small subset of the eigenpairs closest to the origin. Thus, the performance of the SLRU technique depends on the method available to compute the eigenpairs of interest. The SLRU method was first used using the IRA’s method implemented in ARPACK. In this work we investigate the use of a Jacobi–Davidson method, in particular its JDQR variant. The results of the numerical experiments show that the application of the JDQR method to obtain the spectral low-rank updates can be quite competitive compared with the IRA’s method.

Accurate solutions of extremely large integral-equation problems in computational electromagnetics

Accurate simulations of real-life electromagnetics problems with integral equations require the solution of dense matrix equations involving millions of unknowns. Solutions of these extremely large problems cannot be achieved easily, even when using the most powerful computers with state-of-the-art technology. However, with the multilevel fast multipole algorithm (MLFMA) and parallel MLFMA, we have been able to obtain full-wave solutions of scattering problems discretized with hundreds of millions of unknowns. Some of the complicated real-life problems (such as scattering from a realistic aircraft) involve geometries that are larger than 1000 wavelengths. Accurate solutions of such problems can be used as benchmarking data for many purposes and even as reference data for high-frequency techniques. Solutions of extremely large canonical benchmark problems involving sphere and National Aeronautics and Space Administration (NASA) Almond geometries are presented, in addition to the solution of complicated objects, such as the Flamme. The parallel implementation is also extended to solve very large dielectric problems, such as dielectric lenses and photonic crystals.

Introducing a New Column on Open Problems in CEM [Open Problems in CEM

Computational electromagnetics (CEM) has become a big research area, which has seen rapid advances for solving more and more difficult problems in electromagnetics. We are trying to solve an incredibly wide range of real-life problems in diverse application areas, including but not limited to antennas, radar, optics, medical imaging, wireless communications, nanotechnology, metamaterials, photonic crystals, remote sensing, and electronic packaging. All of these involve electromagnetic waves and their interactions with matter via scattering, radiation, transmission, diffraction, reflection, and refraction phenomena. We are attacking these problems using our knowledge and implementing various algorithms on different sizes of computers, from notebooks to huge servers with thousands of cores. All these efforts are directed toward obtaining more-accurate and efficient solutions. However, as in all research areas, we often find ourselves sitting in front of a computer, scratching our heads when looking at the results on the screen, and asking the ultimate question: What is this?

The reduced basis method for the electric field integral equation

We introduce the reduced basis method (RBM) as an efficient tool for parametrized scattering problems in computational electromagnetics for problems where field solutions are computed using a standard Boundary Element Method (BEM) for the parametrized electric field integral equation (EFIE). This combination enables an algorithmic cooperation which results in a two step procedure. The first step consists of a computationally intense assembling of the reduced basis, that needs to be effected only once. In the second step, we compute output functionals of the solution, such as the Radar Cross Section (RCS), independently of the dimension of the discretization space, for many different parameter values in a many-query context at very little cost. Parameters include the wavenumber, the angle of the incident plane wave and its polarization.

A NOVEL LANCZOS-TYPE PROCEDURE FOR COMPUTING EIGENELEMENTS OF MAXWELL AND HELMHOLTZ PROBLEMS

We introduce a novel variant of the Lanczos method for computing a few eigenvalues of sparse and/or dense non- Hermitian systems arising from the discretization of Maxwell- or Helmholtz-type operators in electromagnetics. We develop a Krylov subspace projection technique built upon short-term vector recurrences that does not require full reorthogonalization and can approximate simultaneously both left and right eigenvectors. We report on experiments for solving eigenproblems arising in the analysis of dielectric waveguides and scattering applications from PEC structures. The theoretical and numerical results reported in this study will contribute to highlight the potential and enrich the database of this technology for solving generalized eigenvalue problems in Computational Electromagnetics.

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.

ACCELERATION OF SLOWLY CONVERGENT SERIES VIA THE GENERALIZED WEIGHTED-AVERAGES METHOD

A generalized version of the weighted-averages method is presented for the acceleration of convergence of sequences and series over a wide range of test problems, including linearly and logarithmically convergent series as well as monotone and alternating series. This method was originally developed in a partition- extrapolation procedure for accelerating the convergence of semi- inflnite range integrals with Bessel function kernels (Sommerfeld-type integrals), which arise in computational electromagnetics problems involving scattering/radiation in planar stratifled media. In this paper, the generalized weighted-averages method is obtained by incorporating the optimal remainder estimates already available in the literature. Numerical results certify its comparable and in many cases superior performance against not only the traditional weighted-averages method but also against the most proven extrapolation methods often used to speed up the computation of slowly convergent series.

A high order hybrid finite element method applied to the solution of electromagnetic wave scattering problems in the time domain

The development of a hybrid high order time domain finite element solution procedure for the simulation of two dimensional problems in computational electromagnetics is considered. The chosen application area is that of electromagnetic scattering. The spatial approximation adopted incorporates both a continuous Galerkin spectral element method and a high order discontinuous Galerkin method. Temporal discretisation is achieved by means of a fourth order Runge–Kutta procedure. An exact analytical solution is employed initially to validate the procedure and the numerical performance is then demonstrated for a number of more challenging examples.

Vector sequence accelerator for solving problems in computational electromagnetics using the method of moments

AbstractThe solution of an integral equation using the method of moments leads to a system of linear equations. The resulting system of equations can be solved by direct and iterative methods. This paper introduces an iterative method utilizing Brezinski's θ algorithm. The algorithm has previously been used in accelerating the convergence of a scalar sequence. With the aid of two integral equations, it is shown that the algorithm is able to accelerate the convergence of a vector sequence resulting in the solution vector. © 2008 Wiley Periodicals, Inc. Microwave Opt Technol Lett 50: 589–592, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.23155

Hierarchical parallelisation strategy for multilevel fast multipole algorithm in computational electromagnetics

A hierarchical parallelisation of the multilevel fast multipole algorithm (MLFMA) for the efficient solution of large-scale problems in computational electromagnetics is presented. The tree structure of MLFMA is distributed among the processors by partitioning both the clusters and the samples of the fields appropriately for each level. The parallelisation efficiency is significantly improved compared to previous approaches, where only the clusters or only the fields are partitioned in a level.