Fast Solution of Volume Integral Equations Based on Meshless Discretization

The skeletonization accelerated multilevel fast multipole algorithm (MLFMA) in terms of surface integral equation (SIE) is extended to the method of moments (MoM) solution of volume integral equation (VIE). Boxes at the finest MLFMA level are recursively divided into smaller ones and the associated rank-deficient submatrices are approximated by skeletonization. Numerical experiments are conducted to study the performance of the proposed algorithm.

Reconstruction of unknown objects requires efficient inversion for measured electromagnetic scattering data. In frequency-domain integral equation method for reconstructing dielectric objects, the volume integral equations (VIEs) are involved because the imaging domain with both true unknown objects and part of background is inhomogeneous. When solving the forward scattering integral equation (FSIE), the Nyström method is used since the traditional method of moments (MoM) with the Schaubert-Wilton-Glisson (SWG) basis function may not be convenient due to the inhomogeneity of the imaging domain. The benefits of the Nyström method include the simple implementation without using any basis and testing functions and lower requirement on geometrical discretization, so it is very suitable for inhomogeneous problems. When solving the inverse scattering integral equation (ISIE), the Gauss-Newton minimization approach (GNMA) with a multiplicative regularization method (MRM) is employed. Numerical examples for reconstructing three-dimensional dielectric objects are presented to illustrate the inversion approach.

A novel equivalence principle algorithm (EPA) domain decomposition method previously for the surface integral equation is explored for volume integral equation in this work. The whole object is decomposed into several subdomains, the equivalence spheres are automatically produced to enclose each subdomain. The Rao–Wilton–Glisson (RWG) and body of revolution (BoR) basis functions are defined on the equivalence sphere surface. The equivalence currents defined on the RWG basis functions are evaluated with multilevel fast multipole algorithm, and then the currents defined on the RWGs can be projected onto the BoR basis functions. Compared with surface integral equation EPA-BoR algorithm, the ratio of the number of average original unknowns in each subdomain with respect to the equivalence unknowns is relatively larger, which leads to significant reduction of computations. Numerical simulations validate the accuracy and efficiency of the proposed method.

Electromagnetic (EM) problems with conducting and penetrable media are formulated by volume-surface integral equations (VSIEs) which allow inhomogeneous and anisotropic materials. Traditionally, the VSIEs are solved by the method of moments (MoM) with Rao-Wilton-Glisson (RWG) and Schaubert-Wilton-Glisson (SWG) basis functions. We propose a point-matching scheme to discretize the VSIEs. The point-matching scheme does not rely on any basis and testing functions and allows the use of nonconforming meshes. Also, the integral kernels are free of material parameters by selecting current densities as unknowns. To accelerate the solution for electrically large problems, we incorporate the scheme with the multilevel fast multipole algorithm (MLFMA) and a typical numerical example is presented to validate its effectiveness.

The Numerical Solution of Integral Equations on the Half-Line

Skeletonization is employed to accelerate the multilevel fast multipole algorithm (MLFMA) solution of the method of moments matrix system arising from volume integral equations (VIEs). The acceleration is conducted in two aspects: 1) speedup of the calculation of near-field interactions of the MLFMA and 2) improvement of the iterative convergence. Numerical experiments indicate that the proposed algorithm can significantly reduce the runtime and memory usage without degrading the accuracy, in comparison with the VIE solver purely accelerated by the MLFMA. Additionally, for piecewise homogeneous complex plasmonic structures, the proposed algorithm can be more efficient and accurate than the commonly used surface integral equation solver.

We consider the method of evaluating matrix elements of electromagnetic volume integral equations with the help of suitably constructed Laplacian-type representations of singular kernels (Green functions) appearing in electromagnetic volume and surface integral equations. The method consists of representing the singular kernels (Green functions) in terms of a generalized Laplacian operator acting on some auxiliary functions. Such a representations allow us, by using Gauss divergence theorem, to convert volumetric and surface integrals representing matrix elements to integrals always involving only non-singular integrands. The task of of finding particular Laplacian representation of different kernels amounts to solving appropriate ordinary or partial inhomogeneous differential equations. We apply the Laplacian method to analytic evaluation of matrix elements of the volume integral equation by employing two different Laplacian representations of the integral equation kernels. The first representation allows us to reduce 6-dimensional volume integrals over tetrahedra to 2-dimensional surface integrals over tetrahedra faces. The second representation is subsequently used reducing surface integrals over tetrahedra faces to pairs of of 1-dimensional line integrals over tetrahedra edges. We also derive analytic expressions for the resulting line integrals given in terms of elementary functions.

A new volume integral equation (VIE) approach is introduced to study transcranial magnetic stimulation (TMS) and high-contrast media at low frequencies. This new integral equation offers a simple solution to the high-contrast breakdown observed in low-frequency electric field (E-field) dosimetry of conductive media. Specifically, we employ appropriate approximations that are valid for low frequencies and stabilize the VIE by introducing a basis expansion set that removes solutions associated with high eigenvalues in the equation. The new equation is devoid of high-contrast breakdown and does not require the use of auxiliary surface variables or projectors, providing a straightforward practical solution for the VIE analysis of TMS. Our results indicate that the novel VIE formulation matches boundary element, finite element, and analytical solutions. This new VIE represents a first step towards including anisotropy in integral equation E-field dosimetry for brain stimulation.

The improved hybrid discretisation of volume integral equation (VIE) combined with the characteristic basis function method is proposed to analyse the scattering from complex dielectric objects in this study. To mitigate the excessive number of discrete elements which arise from the conformal discretisation on the complex dielectric objects, the hybrid discretisation is developed. For the conventional hybrid discretisation, the conformal Schaubert–Wilton–Glisson (SWG) basis functions are used to discretise the homogeneous regions, meanwhile, the non‐conformal half‐SWG basis functions are employed to discretise the boundary regions which separate different materials or multiscale structures. The hybrid discretisation takes advantage of the merits of both conformal and non‐conformal basis functions and improves the computational efficiency for solving the VIE. In our study, a kind of edge basis function defined within boundary tetrahedral elements substitutes for the SWG basis functions to model the boundary regions, leading to the further reduction on the number of basis functions. Moreover, the characteristic basis functions extracted from the low‐level basis functions are employed to construct the reduced matrix of which the size is considerably smaller than that of the system matrix derived from the VIE. Several numerical results are presented to demonstrate the accuracy and efficiency of the proposed method.

Reconstruction of objects by microwave illumination can be described by integral equation approach and volume integral equations (VIEs) are indispensable for reconstructing penetrable objects due to the inhomogeneity of materials in the imaging domain. Traditionally, the objects are assumed to be nonmagnetic, so only the permittivity distribution is solved from the iterative process of reconstruction. This work considers the reconstruction of penetrable objects with both dielectric and magnetic materials and the full VIEs are needed to describe the problem. The forward scattering VIEs (FSVIEs) are solved by Nyström method while the inverse scattering VIEs (ISVIEs) are solved with a Gauss-Newton method. A numerical example is used to verify the approach.

Electromagnetic wave scattering by three dimensional inhomogeneous dielectric sphere is analyzed using Volume Integral Equation (VIE) and Method of Moments (MoM). The VIE/MoM technique deals with the formulation of an integral equation and then finding its solution by converting it into a system of linear equations with the help of set of pulse basis and delta testing functions. In this paper a simple object like dielectric sphere is considered and its scattering characteristics are determined through solving the linear equations by iteration based numerical procedure called Generalized Minimum Residual (GMRES) method. The bistatic Radar Cross Section (RCS) results of this implementation are verified with the theory based Mie series and commercially available HFSS results, and they are found to be in good agreement.

An efficient method of moments (MoM) solution of the volume-surface integral equation (VSIE) is presented for the analysis of electromagnetic (EM) scattering and radiation of composite objects comprising arbitrarily contacted dielectrics and conductors. In contrast to the conventional Schaubert-Wilton-Glisson (SWG) and Rao-Wilton-Glisson (RWG) basis functions, which were widely used in the MoM solutions of the VSIE, a new hybrid pulse-rooftop basis function is proposed to explicitly enforce the boundary condition of vanishing tangential electric field at the dielectric-conductor interface. As a consequence, the displacement currents in the dielectrics can be efficiently modeled by a reduced number of volume bases on the same set of meshes, particularly when thin dielectric slabs are involved in the composite structures. The numerical results of EM scattering and radiation of several composite objects are shown to illustrate the accuracy and efficiency of the proposed scheme. Since the new hybrid basis function can be easily implemented into the existing fast algorithms, the VSIE solution presented in this paper is promising for EM problems of composite targets with large size.

The fast direct solver based on skeletonization is extended to the method of moments (MoM) solution of volume integral equations (VIE). The impedance matrix is approximated by the hierarchical data sparse representation through skeletonization. The inverse matrix is then implicitly computed. Numerical experiments are conducted to study the performance of the direct solver.

This paper presents a fast solution of electromagnetic scattering problems using the volume integral equation (VIE) combined with the integral equation fast Fourier transform (IE-FFT). The tetrahedron cells are used to model the geometry of the scatterer and the Schaubert-Wilton-Glisson (SWG) basis function is used to expand the electric flux density. A floating stencil topology is introduced to reduce the memory requirement and computational complexity for the near-interaction. Numerical result shows that the method can reduce both memory requirement and computational complexity while maintains the same accuracy.

A multilevel fast multipole algorithm (MLFMA) for analyzing electromagnetic radiation and coupling characteristics of large-scale antenna arrays mounted on the platforms is presented in this paper. Compared with the method of moments (MoM), the MLFMA can be used to calculate larger scale problems with limited resources. First, waveport model of the MLFMA based on the equivalence principle and mode matching theory is established to efficiently and accurately simulate the antenna array. Then, a preconditioning approach for solving the radiation problems with the waveports is designed to improve convergence of the MLFMA. An initial guess construction method is proposed to accelerate the MLFMA computation for the multi-excitation problems, which can reduce the iteration time by at least 50%. Numerical results demonstrate accuracy and efficiency of the proposed method.