Electromagnetic Integral Equations Research Articles

Iso-Geometric Integral Equation Solvers and Their Compression via Manifold Harmonics

The state of art of electromagnetic integral equations has seen significant growth over the past few decades, overcoming some of the fundamental bottlenecks: computational complexity, low frequency and dense discretization breakdown, preconditioning, and so on. Likewise, the community has seen extensive investment in development of methods for higher order analysis, in both geometry and physics. Unfortunately, these standard geometric descriptors are continuous, but their normals are discontinuous at the boundary between triangular tessellations of control nodes, or patches, with a few exceptions; as a result, one needs to define additional mathematical infrastructure to define physical basis sets for vector problems. In stark contrast, the geometric representation used for design are second order differentiable almost everywhere on the surfaces. Using these description for analysis opens the door to several possibilities, and is the area we explore in this paper. Our focus is on Loop subdivision based isogeometric methods. In this paper, our goals are two fold: (i) development of computational infrastructure for isogeometric analysis of electrically large simply connected objects, and (ii) to introduce the notion of manifold harmonics transforms and its utility in computational electromagnetics. Several results highlighting the efficacy of these two methods are presented.

Three-dimensional numerical modeling of direct current resistivity methods in mixed space-wavenumber domain

At present, in order to obtain highly accurate solution for complex (geo-electric) conditions, it is necessary to divide the model precisely in the numerical modeling of DC resistivity method, which results in greatly increases the amount of computation and memory. To solve this problem, we propose a 3D numerical simulation method in a mixed space-wavenumber domain. The method transforms the partial differential equation of DC abnormal potential into many independent 1D differential equations with different wavenumbers by performing a 2D Fourier transform along the horizontal direction. It decomposes a large-scale 3D numerical modeling problem into many 1D numerical modeling problems, which greatly reduces the computation and memory requirements of modeling, and each 1D differential equation is independent, so it has high parallelism. The vertical direction is reserved as the spatial domain, which is convenient for the subdivision, taking into account the calculation accuracy and efficiency. A finite-element algorithm combined with a chasing method is used to solve the 1D differential equations with different wavenumbers, and a contraction operator based on electromagnetic integral equation method is used to the iteration of the algorithm for the first time. Taking full advantage of the rapidity of Fourier transform, the high efficiency of solving 1D differential equations and the high parallelism of 1D problems with different wavenumbers, it can greatly reduce the amount of calculation and storage, thus greatly improves the efficiency of the algorithm. The accuracy and convergence of the algorithm are verified by using the model of a sphere in half space. Based on the model with five cubes in half space, the efficiency of the algorithm is verified by comparing with the academic Gimli package based on the finite element method. In addition, a parallel algorithm based on OpenMP is used to verify the parallelism of the proposed algorithm. The examples results show that, the 3D numerical modeling method of DC resistivity in the mixed space-wavenumber domain has the characteristics of high efficiency, high precision and highly parallel. This method provides a new way for 3D numerical simulation of DC resistivity method.

High-order Nyström solution for electromagnetic scattering by multiscale objects based on combined field integral equation

Electromagnetic (EM) interactions with multiscale objects widely exist in practice and high-order accuracy of solutions for the problem is very desirable for accurate analysis. The combined field integral equation (CFIE) is used to describe the problem in the integral equation approach since it can remove the internal resonance problem. The Nyström method is a robust solver for EM integral equations, but its high-order scheme is less extensively studied. This work aims at the multiscale EM problems and develops a high-order Nyström method (HONM) to solve the governing CFIE. The accurate evaluation of hypersingular and near-hypersingular integrals is a key in the HONM and new formulations are developed for multiscale structures based on Green's theorem. Numerical examples for EM scattering by multiscale conducting objects are provided to illustrate the approach and good results are acquired.

Augmented Hybrid Integral Equations for Low-Frequency Analysis of Lossy Conducting Objects

Electromagnetic (EM) integral equations include either $\mathcal{L}$ operator or $\mathcal{K}$ operator or both. The $\mathcal{L}$ operator has a low-frequency-breakdown (LFB) problem at low frequencies, and as a remedy, the augmented electric field integral equation (AEFIE) for conducting objects was proposed. Later on, the AEFIE was extended to AEFIEs for dielectric objects which include both $\mathcal{L}$ operator and $\mathcal{K}$ operator and the dual basis function (DBF) is used to represent the magnetic current density. In this work, the hybrid field integral equations (HFIEs) are employed for lossy conducting objects, and the HFIEs consist of the EFIE in the exterior and magnetic field integral equation (MFIE) in the interior of the lossy conducting object, respectively. Both electric and magnetic current densities can be expanded by the Rao-Wilton-Glisson (RWG) basis function which is also a testing function and the resultant impedance matrix is well-conditioned due to the feature of HFIEs. At low frequencies, the HFIEs are extended to the augmented HFIEs (AHFIEs) by introducing the continuity equation of magnetic current and magnetic charge density as an extra unknown function. A numerical example is shown to demonstrate the approach and good performance has been verified.

Electromagnetic Integral Equations: Insights in Conditioning and Preconditioning

Integral equation formulations are a competitive strategy in computational electromagnetics but, lamentably, are often plagued by ill-conditioning and by related numerical instabilities that can jeopardize their effectiveness in several real case scenarios. Luckily, however, it is possible to leverage effective preconditioning and regularization strategies that can cure a large majority of these problems. Not surprisingly, integral equation preconditioning is currently a quite active field of research. To give the reader a propositive overview of the state of the art, this paper will review and discuss the main advancements in the field of integral equation preconditioning in electromagnetics summarizing strengths and weaknesses of each technique. The contribution will guide the reader through the choices of the right preconditioner for a given application scenario. This will be complemented by new analyses and discussions which will provide a further and more intuitive understanding of the ill-conditioning of the electric field (EFIE), magnetic field (MFIE), and combined field integral equation (CFIE) and of the associated remedies.

Comparison of Wavelet Packet and Wavelet in Solving Arbitrary Array of Parallel Wires Integral Equations in Electromagnetics

In this paper, wavelets transformation (WT) and wavelet packet transformation (WPT) are used in solving, by the method of moments, a semicircular array of parallel wires electric field integral equation. First, the integral equation is solved by applying the direct method of moments via point-matching procedure, results in a linear system with a dense matrix. Therefore, wavelet transformation and wavelet packet transformation are used to sparsify the impedance matrix, using two categories of wavelets functions, Biorthogonal (bior2.2) and Orthogonal (db4) wavelets. The far-field scattering patterns and the comparison between wavelets transformation and wavelet packet transformation in term number of zeros in impedance matrix and CPU Time reduction are presented. Numerical results are presented to identify which technique is best suited to solve such scattering electromagnetic problems and compared with published results.

Individualized SAR calculations using computer vision-based MR segmentation and a fast electromagnetic solver.

We propose a fast, patient-specific workflow for on-line specific absorption rate (SAR) supervision. An individualized electromagnetic model is created while the subject is on the table, followed by rapid SAR estimates for that individual. Our goal is an improved correspondence between the patient and model, reducing reliance on general anatomical body models. A 3D fat-water 3T acquisition (~2minutes) is automatically segmented using a computer vision algorithm (~1minute) into what we found to be the most important electromagnetic tissue classes: air, bone, fat, and soft tissues. We then compute the individual's EM field exposure and global and local SAR matrices using a fast electromagnetic integral equation solver. We assess the approach in 10 volunteers and compare to the SAR seen in a standard generic body model (Duke). The on-the-table workflow averaged 7'44″. Simulation of the simplified Duke models confirmed that only air, bone, fat, and soft tissue classes are needed to estimate global and local SAR with an error of 6.7% and 2.7%, respectively, compared to the full model. In contrast, our volunteers showed a 16.0% and 20.3% population variability in global and local SAR, respectively, which was mostly underestimated by the Duke model. Timely construction and deployment of a patient-specific model is computationally feasible. The benefit of resolving the population heterogeneity compared favorably to the modest modeling error incurred. This suggests that individualized SAR estimates can improve electromagnetic safety in MRI and possibly reduce conservative safety margins that account for patient-model mismatch, especially in non-standard patients.

Upper bounds on absorption and scattering

A general framework for determining fundamental bounds in nanophotonics is introduced in this paper. The theory is based on convex optimization of dual problems constructed from operators generated by electromagnetic integral equations. The optimized variable is a contrast current defined within a prescribed region of a given material constitutive relations. Two power conservation constraints analogous to the optical theorem are utilized to tighten the bounds and to prescribe either losses or material properties. Thanks to the utilization of matrix rank-1 updates, modal decompositions, and model order reduction techniques, the optimization procedure is computationally efficient even for complicated scenarios. No dual gaps are observed. The method is well-suited to accommodate material anisotropy and inhomogeneity. To demonstrate the validity of the method, bounds on scattering, absorption, and extinction cross sections are derived first and evaluated for several canonical regions. The tightness of the bounds is verified by comparison to optimized spherical nanoparticles and shells. The next metric investigated is bi-directional scattering studied closely on a particular example of an electrically thin slab. Finally, the bounds are established for Purcell’s factor and local field enhancement where a dimer is used as a practical example.

Efficient Solution of Electromagnetic Scattering Problems Using Multilevel Adaptive Cross Approximation and LU Factorization

The multilevel adaptive cross approximation (MLACA), previously described in the literature, extends the single-level ACA with a recursive multilevel algorithm that significantly improves compression of off-diagonal matrix blocks resulting from electromagnetic integral equations (IE) discretized via the method of moments (MoM). In this article, the MLACA approach is extended and applied to a direct solution of the MoM matrix system via LU factorization. It will be shown through numerical experiments that the off-diagonal LU blocks are also compressible using MLACA, yielding a compression rate superior to the single-level ACA and a memory complexity of O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4/3</sup> log N). In addition, the MLACA LU block updates are performed in rank-reduced form, yielding a very efficient software implementation via a Level 3 BLAS optimized for the CPU or GPU.

A spectrally preconditioned and initially deflated variant of the restarted block GMRES method for solving multiple right-hand sides linear systems

The solution of large linear systems with multiple right-hand sides given simultaneously is required in many large-scale scientific and engineering applications modelled by either partial differential or boundary integral equations. Block Krylov subspace methods are attractive to use for this problem class as they can overcome the memory bottleneck of direct methods and they perform block matrix-vector products, achieving high computational efficiency on modern cache-based computer architectures. In this paper we introduce variants of the block GMRES method that combine initial deflation and eigenvalue recycling strategies to remedy some typical convergence problems of block Krylov solvers. The new class of block iterative solvers has the ability to handle the approximate linear dependence of the block of right-hand sides and exploits approximate invariant subspaces recycled over the iterations to mitigate the bad effects that small eigenvalues can have on the convergence, by adapting an existing preconditioner. We illustrate the numerical behavior of the spectrally updated and initially deflated block GMRES method on a set of linear systems arising from the discretization of the Dirac equation and of boundary integral equations in electromagnetics scattering.

Differential operator acting on accelerated Cartesian expansion algorithm

The multilevel accelerated Cartesian expansion algorithm (MLACEA) and its incorporation with multilevel fast multipole algorithm (MLFMA) have been studied and used for solving practical problems. The algorithms are successful, but the authors found the truncation error of accelerated Cartesian expansion (ACE) is sensitive with the differential operator. Integral equations in electromagnetics always contain differential operators, so the precision will be influenced when trying to accelerate the method of moments with MLACE algorithm. The reason of why differential operator can influence the precision of the ACE is given, and also proved by numerical experimentation. To meet the requirements of piratical application, electric field integral equation is deeply investigated. The studies also extend to the hybrid algorithm MLFMA–MLACEA and performances are tested on different implementation.

Reduction of Singular Surface Integrals to Nonsingular Line Integrals in Integral Equations for Planar Geometries

A novel procedure is presented for the evaluation of matrix elements of the tensor Green function with Rao–Wilton–Glisson basis functions appearing in surface integral equations in electromagnetics. The procedure, at this point applicable to planar geometries, or geometries composed of parallel planar sheets, reduces 4-D surface integrals with singular integrands to line integrals over triangle edges with regular integrands. The main advantage of the derived expressions is that they offer simplicity and easily controllable accuracy achieved at a computational cost significantly lower than for previously considered techniques, in particular the conventional singularity subtraction method.

Efficient Solution to Electromagnetic Scattering Problems of Bodies of Revolution by Compressive Sensing**Supported by the National Natural Science Foundation of China under Grant Nos 51477039 and 51207041, the Program of Hefei Normal University under Grant Nos 2014136KJA04 and 2015TD01, and the Key Project of Provincial Natural Science Research of University of Anhui Province of China under Grant No KJ2015A174.

Under the theory structure of compressive sensing (CS), an underdetermined equation is deduced for describing the discrete solution of the electromagnetic integral equation of body of revolution (BOR), which will result in a small-scale impedance matrix. In the new linear equation system, the small-scale impedance matrix can be regarded as the measurement matrix in CS, while the excited vector is the measurement of unknown currents. Instead of solving dense full rank matrix equations by the iterative method, with suitable sparse representation, for unknown currents on the surface of BOR, the entire current can be accurately obtained by reconstructed algorithms in CS for small-scale undetermined equations. Numerical results show that the proposed method can greatly improve the computational efficiency and can decrease memory consumed.

Cell Curvature and Far-Field Superconvergence in Numerical Solutions of Electromagnetic Integral Equations

Two curved targets are used to explore far-field superconvergence effects arising in numerical solutions of the electric-field and magnetic-field integral equations. Three different orders of basis and testing functions are used to discretize these equations, and three different types of target models (flat facets, quadratic-curved facets, and cubic-curved facets) are employed. Ideal far-field convergence rates are only observed when the model curvature is one degree higher than the basis order.

Efficient construction and solution of MoM matrix equation with compressed sensing technique

A novel method which employs the recently developed theory of compressed sensing (CS) in method of moments (MoMs) is presented. With linear transform, the construction of the impedance matrix and the solving of the unknown vector in MoM are converted into the construction of a sparse matrix and the solving of a sparse vector, respectively. Under this framework of sparsity, the CS technique can be introduced to construct highly underdetermined equations, which can be solved with the orthogonal matching pursuit algorithm. Thus, the computational cost of both impedance matrix construction and MoM equation solving can be saved dramatically. The performance of the proposed method is illustrated by numerical studies of electromagnetic field integral equations.

An MPI-OpenMP Hybrid ParallelH-LU Direct Solver for Electromagnetic Integral Equations

In this paper we propose a high performance parallel strategy/technique to implement the fast direct solver based on hierarchical matrices method. Our goal is to directly solve electromagnetic integral equations involving electric-large and geometrical-complex targets, which are traditionally difficult to be solved by iterative methods. The parallel method of our direct solver features both OpenMP shared memory programming and MPl message passing for running on a computer cluster. With modifications to the core direct-solving algorithm of hierarchical LU factorization, the new fast solver is scalable for parallelized implementation despite of its sequential nature. The numerical experiments demonstrate the accuracy and efficiency of the proposed parallel direct solver for analyzing electromagnetic scattering problems of complex 3D objects with nearly 4 million unknowns.

Preconditioning for Large-Scale Boundary Integral Equations in Electromagnetics [Open Problems in CEM

The boundary-element method is a popular approach for solving electromagnetic applications. It can reduce a three-dimensional problem to solving a two-dimensional surface problem, simplifying considerably the mesh generation, and limiting grid-dispersion errors. The Method of Moments discretization of boundary integral equations leads to a very large and dense linear systems. It is mandatory to use iterative methods and robust preconditioning for the solution. Identifying the best class of method with respect to numerical stability, inherent parallelism, and good scalability with respect to the frequency of the problem is still an open research issue.

A Projection Height Independent Adaptive Radial-Angular-&lt;formula formulatype="inline"&gt;&lt;tex Notation="TeX"&gt;$R^{{2}}$&lt;/tex&gt;&lt;/formula&gt; Transformation for Singular Integrals

A new radial-angular- R 2 adaptive singularity cancellation transformation is proposed. This new transformation is flexible and applicable to singular integrals over triangular domains. When the height tends to zero and the observation point is in the plane of the source domain, the transformation remains stable and approaches the corresponding transformation within the source plane. Usually the Green's function and its gradient appear in the integral kernels leading to first order and second order singularities. The newly derived transformation formula provides an efficient solution for second order singular coupling integral kernels and the formula is in particular more efficient than alternative schemes for singular vector integral kernels. However, it is also effective for the lower order singular integral kernels. In electromagnetic boundary integral equation formulations, the proposed transformation is efficient for all types of singular integrals.

A high order spectral algorithm for elastic obstacle scattering in three dimensions

In this paper we describe a high order spectral algorithm for solving the time-harmonic Navier equations in the exterior of a bounded obstacle in three space dimensions, with Dirichlet or Neumann boundary conditions. Our approach is based on combined-field boundary integral equation (CFIE) reformulations of the Navier equations. We extend the spectral method developed by Ganesh and Hawkins [20] – for solving second kind boundary integral equations in electromagnetism – to linear elasticity for solving CFIEs that commonly involve integral operators with a strongly singular or hypersingular kernel. The numerical scheme applies to boundaries which are globally parametrised by spherical coordinates. The algorithm has the interesting feature that it leads to solve linear systems with substantially fewer unknowns than with other existing fast methods. The computational performances of the proposed spectral algorithm are demonstrated using numerical examples for a variety of three-dimensional convex and non-convex smooth obstacles.

Differential-Forms-Motivated Discretizations of Electromagnetic Differential and Integral Equations

In this letter, we present a differential-forms-motivated procedure to unify and guide discretizations of differential and integral equations in computational electromagnetics (CEM). In order to solve such equations accurately, it is crucial to find an appropriate matrix representation of the governing differential or integral operator. Differential forms theory inspires a general procedure of selecting both expansion and test functions wisely. Many well-functioning discretizations in finite element method (FEM) and boundary element method (BEM) can be reinterpreted with this theory. Moreoever, our approach offers guidance for discretizing complicated problems where straightforward discretizations may not be available.