Combined Field Integral Equation Research Articles

Solving Combined Field Integral Equations With Physics-Informed Graph Residual Learning for EM Scattering of 3-D PEC Targets

Solving Combined Field Integral Equations With Physics-Informed Graph Residual Learning for EM Scattering of 3-D PEC Targets

Integral Equation Residuals for Error Estimation and Internal Resonance Detection

The performance of the four basic residual error estimators for perfectly conducting targets is evaluated in connection with the Rao-Wilton-Glisson method of moments numerical solutions of the electric field, magnetic field, and combined field integral equations. Results for 17 perfectly conducting test targets are used to evaluate the performance. Tangential-field residuals appear to be more reliable than normal-field residuals for assessing accuracy. Residual estimates are shown to be poor when used to assess an equation based on the same boundary condition as the residual. Residuals are sensitive to internal resonance modes associated with the complementary boundary condition and can be used to detect spurious solutions due to their presence.

Analytic Preconditioners for Decoupled Potential Integral Equations and Wideband Analysis of Scattering From PEC Objects

Many integral equations used to analyze scattering, such as the standard combined field integral equation (CFIE), are not well-conditioned for a wide range of frequencies and multi-scale geometries. There has been significant effort to alleviate this problem. A more recent one is using a set of decoupled potential integral equations (DPIE). These equations have been shown to be robust at low frequencies and immune to topology breakdown. But they mimic the ill-conditioning behavior of CFIE at high frequencies. This paper addresses this deficiency through new Calderón-type identities derived from the Vector Potential Integral Equation (VPIE). We construct novel analytic preconditioners for the vector potential integral equation (VPIE) and scalar potential integral equation (SPIE) constrained to perfect electric conductors (PEC). These new formulations are wide-band well-conditioned and converge rapidly for multi-scale geometries. This is demonstrated though a number of examples that use analytic and piecewise basis sets.

Code-verification techniques for the method-of-moments implementation of the combined-field integral equation

Code verification plays an important role in establishing the credibility of computational simulations by assessing the correctness of the implementation of the underlying numerical methods. In computational electromagnetics, the numerical solution to integral equations incurs multiple interacting sources of numerical error, as well as other challenges, which render traditional code-verification approaches ineffective. In this paper, we provide approaches to separately measure the numerical errors arising from these different error sources for the method-of-moments implementation of the combined-field integral equation. We demonstrate the effectiveness of these approaches for cases with and without coding errors.

Multi-Mode Analysis of Scattering by Bodies of Revolutions via the Combined-Field Integral Equation

Multi-Mode Analysis of Scattering by Bodies of Revolutions via the Combined-Field Integral Equation

Improving the Solution Accuracy of the Volume-Surface Integral Equation for the Open PEC Structures Embedded Within Dielectrics

In the solution of volume-surface integral equation (VSIE) for open perfect electric conductor (PEC) structures embedded within dielectric substrates, how to explicitly enforce the continuity condition (CC) of electric flux on the PEC–dielectric interfaces to improve the numerical accuracy is investigated. In the traditional VSIE formed by combining the surface electric field integral equation (EFIE) and volume integral equation (VIE), the fictitious surface charge, which does not exist physically, is accumulated on the PEC–dielectric interfaces. This will lead to the inaccuracy of numerical results, which can be eliminated by the CC. To correctly enforce the CC, the equivalent surface current distribution on both sides of open PEC should be distinguished. For this purpose, the combined field integral equation (CFIE) consisting of electric field and magnetic field integral equations is used to model the open PEC part. In the proposed approach, the number of surface unknowns needs to be doubled, which can be balanced by reducing the volumetric unknowns through the enforcement of CC. Furthermore, by fully considering the numerical symmetry between both sides of open PEC surface, the filling process of the matrix as well as the excitation vector in implementing the method of moments (MoMs) can be simplified. Numerical experiment demonstrates that the enforcement of CC can improve the VSIE accuracy obviously. For example, when calculating the frequency response of frequency selective surfaces (FSSs), the results obtained from the traditional VSIE often shift to higher frequencies, while the proposed approach gives reasonably good results.

Surface Integral Equation With Multibranch RWG Basis Functions for Electromagnetic Scattering From Dielectric Objects

A surface integral equation method with multibranch Rao–Wilton–Glisson (MB-RWG) basis functions for scattering from dielectric objects is proposed based on the electric and magnetic current combined field integral equations (JMCFIE). The MB-RWGs are defined at the boundaries between dense and coarse meshes and can keep the current continuity. Thus, the JMCFIE with MB-RWGs allows different surfaces of the dielectric models to be independently discretized by triangles with different sizes. Compared with the conventional JMCFIE, the JMCFIE with MB-RWGs increases the flexibility of mesh generation, which is attractive for multiscale dielectric models. Furthermore, the iterative convergence of JMCFIE with MB-RWGs is similar to that of conventional JMCFIE. Numerical examples are presented to demonstrate the advantages of the proposed method.

The Applications of the Symmetric Layered Medium Green’s Function in Magnetic Field Integral Equation

A symmetry relation of the integral operators in a layered medium proved by the reciprocal theory can be applied to improve the accuracy of the magnetic-field integral equation (MFIE). Based on the symmetric layered medium Green’s function, the normal vector can be transferred into the inner integral to keep the field integral as the inner integral of the elements in the impedance matrix. When discretized by the divergence-conforming function, this field-extracted scheme shows better performance than the classic expression in accuracy. Meanwhile, this form of expression can be applied in the combined-field integral equation (CFIE). Several numerical results are provided to validate the formulation of the symmetry relations from the layered medium Green’s function to the impedance matrices and to reveal its advantages in enhancing the accuracy of the field-extracted form of MFIE.

Boundary integral equation methods for the solution of scattering and transmission 2D elastodynamic problems

Abstract We introduce and analyse various regularized combined field integral equations (CFIER) formulations of time-harmonic Navier equations in media with piece-wise constant material properties. These formulations can be derived systematically starting from suitable coercive approximations of Dirichlet-to-Neumann operators (DtN), and we present a periodic pseudodifferential calculus framework within which the well posedness of CIER formulations can be established. We also use the DtN approximations to derive and analyse OS methods for the solution of elastodynamics transmission problems. The pseudodifferential calculus we develop in this paper relies on careful singularity splittings of the kernels of Navier boundary integral operators, which is also the basis of high-order Nyström quadratures for their discretizations. Based on these high-order discretizations we investigate the rate of convergence of iterative solvers applied to CFIER and OS formulations of scattering and transmission problems. We present a variety of numerical results that illustrate that the CFIER methodology leads to important computational savings over the classical CFIE one, whenever iterative solvers are used for the solution of the ensuing discretized boundary integral equations. Finally, we show that the OS methods are competitive in the high-frequency high-contrast regime.

H-Matrix-Based Direct Solver of JMCFIE for the Analysis of Scattering From Penetrable Objects

Method of moments (MoMs) solution of the electric and magnetic current combined-field integral equation (JMCFIE) for penetrable objects generally resorts to iterative solvers, which suffer from the problems of convergence rate and redundant computation for multiple right-hand-sides (RHSs). In this communication, an efficient direct solver based on the hierarchical ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {H}$ </tex-math></inline-formula> -) matrix technique is presented to solve these problems. <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {H}$ </tex-math></inline-formula> -matrix formatted algorithms provide an efficient way to factorize the impedance matrix with low computational costs, which make the direct solution feasible. Examples of electromagnetic scattering from a dielectric object and optical scattering from a plasmonic nanostructure are shown to demonstrate the efficiency and accuracy of the proposed 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.

A Discontinuous Galerkin Combined Field Integral Equation Formulation for Electromagnetic Modeling of Piecewise Homogeneous Objects of Arbitrary Shape

We present a novel discontinuous Galerkin (DG) surface integral equation approach, based on the electric and magnetic current combined field integral equations (JMCFIE), for the electromagnetic analysis of arbitrarily shaped piecewise homogeneous objects. In the proposed scheme, nonoverlapping boundary surfaces and interfaces between materials can be handled independently, without any continuity requirement through multimaterial junctions and tear lines between surfaces in contact. The use of nonconformal meshes provides improved flexibility for computer-aided-design (CAD) prototyping and tessellation. The proposed formulation can readily address nonconformal multimaterial junctions, where three or more material regions meet. The continuity of the electric surface current across the junction contours is enforced by the combination of the boundary conditions implicit in the JMCFIE formulation and the weakly imposed interior penalty (IP) between the contacting surfaces within each region. This completely avoids the cumbersome junction problem, which no longer requires any special treatment. Numerical experiments are included to validate the accuracy and demonstrate the great versatility of the proposed JMCFIE-DG formulation for the management and solution of complex composite objects with junctions.

Multiresolution Preconditioners for Solving Realistic Multi-Scale Complex Problems

In this work, the hierarchic multiresolution (MR) preconditioner is combined with the multilevel fast multipole algorithm-fast Fourier transform (MLFMA-FFT) and efficiently parallelized in multicore computers for computing electromagnetic scattering and radiation from complex problems exhibiting deep multi-scale features. The problem is formulated using the thin-dielectric-sheet (TDS) approximation for thin dielectric materials and the electric and combined field integral equations (EFIE/CFIE) for conducting objects. The parallel MLFMA-FFT is tailored to accommodate the MR hierarchical functions, which provide vast improvement of the matrix system conditioning by accurately handling multi-scale mesh features in different levels of detail. The higher (coarser) level hierarchical functions are treated by an algebraic incomplete LU decomposition preconditioner, which has been efficiently embedded into the parallel framework to further accelerate the solution. Numerical examples are presented to demonstrate the precision and efficiency of the proposed approach for the solution of realistic multi-scale scattering and radiation problems.

High-efficient analysis of metal target electromagnetics above the half-space based on mixed field integral equation

A new acceleration method is proposed for efficiently solving the problem of electromagnetic scattering from metal targets in half-space. The analysis of electromagnetic problems in any structure can be settled by the electric field integral equation. But the generated matrix condition number is large and the iterative solution has poor convergence. The number of the matrix condition generated by the magnetic field integral equation is small and iterative convergence is good. But only the closed structure problems can be worked out. The combined field integral equation is adopted because of the universality of the electric field integral equation and the convergence of the magnetic field integral equation. The gradient term of Green's function is involved in the integral equation of the mixed field. In order to further enhance the calculation efficiency, an efficient four-dimensional spatial interpolation method is introduced for half-space Green's function. Tabulation and lagrange interpolations are performed in the Sommerfeld integrals for the half-space Green's function. The improved efficiency can be 7.5 times higher than that of the traditional combined field integral equation(CFIE). Numerical results show that the computational time can be reduced significantly by the proposed method with encouraging accuracy.

CFIE-Based DDM With Full Exploitation of BoR Basis for Scattering From Compound BoR-and-Non-BoR PEC Object

A true domain decomposition method based on a combined field integral equation has been developed for scattering analysis of a combinational PEC structure comprising body of revolution (BoR) and non-BoR parts. Utilizing the method of moments (MoM), the surface currents of BoR and non-BoR parts are expanded by the BoR and RWG basis functions, respectively. BoR basis functions are exploited in calculating both self-coupling and mutual-coupling matrices. The proposed method has the accuracy of MoM, while the number of unknowns is considerably reduced due to converting the 3-D problem to a 2-D one in the BoR part. The correctness and robustness of this approach are verified by numerical results, and significant improvement in the efficiency in terms of both the computational time and memory consumption in comparison with the conventional RWG-RWG IE-domain decomposition method (DDM) is observed.

A Flexible FEM-BEM-DDM for EM Scattering by Multiscale Anisotropic Objects

In this article, a nonconformal domain decomposition method (DDM) based on the hybrid finite element method (FEM) and boundary element method (BEM) is proposed to solve arbitrary 3-D anisotropic (including bianisotropic and single anisotropic) multiscale composite objects. The object is divided into interior and exterior regions. The interior region is solved by the bianisotropic finite element domain decomposition method (FEM-DDM), and the exterior region is solved by the discontinuous Galerkin (DG) method based on the combined field integral equation (CFIE). In order to decouple the electric and magnetic fields of the bianisotropic object, a new finite element–boundary element–domain decomposition method (FEM-BEM-DDM) formulation is derived. To ensure the continuity of electric current and electric field between anisotropic FEM subdomains, a new auxiliary current is introduced into the interface of different subdomains of FEM and the interface of FEM subdomains with BEM subdomains. The former uses second-order transmission conditions (SOTCs) to ensure continuity, whereas the latter uses Robin transmission conditions (RTCs) to ensure continuity. The BEM dense matrix–vector multiplication is accelerated by a multilevel fast multipole algorithm (MLFMA). The proposed FEM-BEM-DDM allows for nonconformal discretization between any touching subdomains and provides an effective preconditioner. In numerical examples, the accuracy and effectiveness of the method are verified.

A Modified Hybrid Integral Equation to Electromagnetic Scattering from Composite PEC-Dielectric Objects Containing Closed-Open PEC Junctions

To efficiently analyze the electromagnetic scattering from composite perfect electric conductor (PEC)-dielectric objects with coexisting closed-open PEC junctions, a modified hybrid integral equation (HIE) is established as the surface integral equation (SIE) part of the volume surface integral equation (VSIE), which employs the combined field integral equation (CFIE) and the electric field integral equation (EFIE) on the closed and open PEC surfaces, respectively. Different from the traditional HIE modeled for the objects whose closed and open PEC surfaces are strictly separate, the modified HIE can be applied to the objects containing closed-open junctions. A matrix equation is obtained by using the Galerkin’s method of moments (MoM), which is augmented with the spherical harmonics expansion-based multilevel fast multipole algorithm (SE-MLFMA), improved by the mixed-potential representation and the triangle/tetrahedron-based grouping scheme. Because in the improved SE-MLFMA, the memory usage for storing the radiation patterns of basis functions is independent of the SIE type in the VSIE, it is highly appropriate for the fast solution of the VSIE that contains the HIE. Various numerical experiments demonstrate that during the calculation of composite objects containing closed-open PEC junctions, the application of the modified HIE in the VSIE can give reliable results with fast convergence speed.

A High-Order Solver of EM Scattering From Uncertain Geometrical Targets

A high-order solution of electromagnetic (EM) scattering from electrically large targets with uncertain shapes is proposed in this letter. At first, the uncertain geometry can be described with several independent variables by using the nonuniform rational B-spline surfaces. Then, the second-order Taylor series expansion is used to present the geometrical uncertainty in terms of the combined field integral equation. As a result, higher accuracy can be obtained by using this high-order solver. Finally, the mean value and variance of the radar cross section (RCS) are derived and can be compared with the traditional Monte Carlo (MC) method. Numerical results show the proposed method can provide higher accuracy than the traditional one-order solver and higher efficiency than the traditional MC method.

On the Use of Hybrid CFIE–EFIE for Objects Containing Closed–Open Surface Junctions

To effectively solve the electromagnetic scattering or radiation properties from the perfect electric conductor objects containing closed-open surface junctions, how to establish the hybrid combined field integral equation-electric field integral equation (CFIE-EFIE) is studied, which is different from the existing scheme for the objects, where the closed and open parts are separate. Furthermore, it is found that when the integral equation is solved using the method of moments, if the widely used RWG basis functions are employed to expand the induced surface current, the CFIE-EFIE may give inaccurate numerical results for the objects containing fine structures. The numerical accuracy can be improved by introducing the linear-linear (LL) basis functions. Moreover, to pursue a high computational efficiency, the LL and RWG basis functions are simultaneously used to expand the current on the fine structures and other relatively smooth surfaces, respectively, whose validity is verified by the numerical results.

Radar Cross Section Reduction and Shape Optimization using Adjoint Method and Automatic Differentiation

An efficient Radar Cross Section (RCS) gradient evaluation method based on the adjoint method is presented. The Method of Moments is employed to solve the Combined Field Integral Equation (CFIE) and the corresponding derivatives computing routines are generated by the program transformation Automatic Differentiation (AD) technique. The differential code is developed using three kinds of AD mode: tangent mode, multidirectional tangent mode, and adjoint mode. The differential code in adjoint mode is modified and optimized by changing the “two-sweeps” architecture into the “inner-loop two-sweeps” architecture. Their efficiency and memory consumption are tested and the differential code using modified adjoint mode demonstrates the great advantages in both efficiency and memory consumption. A gradient-based shape optimization design method is established using the adjoint method and the mechanism of RCS reduction is studied. The results show that the sharp leading can avoid the specular back-scattering and the undulations of the surface could change the phases which result in a further RCS reduction.