Conventional Integral Equation Research Articles

A Modified HODLR Solver Based on Higher Order Basis Functions for Solving Electromagnetic Scattering Problems

In this letter, we present a novel fast direct solver based on a modified hierarchically off-diagonal low-rank (M-HODLR) matrix format with higher order basis functions for analyzing electromagnetic scattering by electrically large objects. The M-HODLR matrix compresses the far-group submatrices by an extended admissibility condition, which enhances the efficiency and accuracy of the original HODLR. To further alleviate the burden of the fast direct solution of the conventional surface integral equation (SIE), the higher order hierarchical vector basis (HOHVB) functions defined over a curved surface are employed. To account for HOHVB, an efficient assembly scheme based on patch coupling is developed to accelerate the adaptive across approximation implementation in M-HODLR. The proposed method can significantly reduce the computational time and memory cost of a conventional SIE solver. Several numerical examples are presented to validate the accuracy and efficiency of the proposed method in solving electrically large objects. Owing to the characteristics of HOHVB, different patch sizes or basis functions with different orders can be chosen according to the local geometry of the object. Therefore, the proposed method is very appealing for large- and multiscale simulations.

CNN-Based Deep Learning Architecture for Electromagnetic Imaging of Rough Surface Profiles

A convolutional neural network (CNN)-based deep learning (DL) technique for electromagnetic (EM) imaging of rough surfaces separating two dielectric media is presented. The direct scattering problem is formulated through the conventional integral equations, and the synthetic scattered field data are produced by a fast numerical solution technique, which is based on method of moments (MoM). Two different special CNN architectures are designed and implemented for the solution of the inverse rough surface imaging problem, wherein both random and deterministic rough surface profiles can be imaged. It is shown by a comprehensive numerical analysis that the proposed DL inversion scheme is very effective and robust.

Work-Energy Principle Based Characteristic Mode Theory With Solution Domain Compression for Metal-Material Composite Scattering Systems

This article focuses on further improving the characteristic mode theory (CMT) for metal-material composite scattering objects. The improved CMT is alternatively established in a novel <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">work-energy principle</i> (WEP) framework. Under the WEP framework, a novel <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">driving power operator</i> (DPO) is introduced as the generating operator for characteristic modes (CMs). Then, a novel <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">diagonalizing DPO method</i> is proposed to construct the CMs. In addition, a novel <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">solution domain compression</i> (SDC) scheme is developed to suppress spurious modes. Compared with the conventional integral equation (IE) framework, the novel WEP framework more clearly reveals the core physical feature of CMs, and then easily derives the famous Parseval’s identity. Compared with the conventional impedance matrix operator (IMO), the novel DPO is more advantageous in the aspect of distinguishing independent variables from dependent variables, and has a smaller computational burden, and has a more acceptable numerical performance in the aspect of calculating CMs. Compared with the conventional dependent variable elimination (DVE) scheme, the novel SDC scheme is equally effective in the aspect of suppressing spurious mode, and reconfirms the same conclusion that the spurious modes result from ignoring the dependence relationships among the currents contained in DPO.

Improved Integral Equation Method for Rapid 3-D Forward Modeling of Magnetotelluric

Computational cost tremendously restricts the wide application of conventional integral equation (IE) method in large-scale magnetotelluric (MT) modeling. A couple of obstacles limit the developments of traditional MT modeling based on the IE method. They are: O (N2) space complexity of memory requirements for storing coefficients of dense matrix; singularity of Dyadic Green’s function; low efficiency of using digital filtering, such as Hankel transform, to calculate the Bessel function integral within the dyadic Green’s function, as well as inefficiency of accumulative calculation of 3-D discrete convolution. To solve these problems, we use an analytical formula instead of the Hankel transform to compute the integral of the Bessel function and replace a block cell by a spherical cell with the same volume to integrate through the singularity. Because the coefficient matrices are symmetric and antisymmetric three-level block-Toeplitz (BT) and Toeplitz + Hankel matrices, only non-redundant entities of the matrix are computed and stored. Afterwards, 3-D fast Fourier transform (FFT) is used to expedite matrix–vector multiplication at each successive iteration when using the contraction iterative method to solve the system of equations, which decreases memory and time consumption sharply compared with the traditional IE method.

A Spurious-Free Characteristic Mode Formulation Based on Surface Integral Equation for Patch Antenna Structures

Conventional surface integral equation (SIE)-based characteristic mode formulation for the patch antenna structure with a finite substrate is susceptible to the spurious (nonphysical) modes due to the dielectric part. To avoid the contamination of spurious modes, we propose a novel generalized eigenvalue formulation based on the electric field integral equation coupled Poggio-Miller-Chang-Harrington-Wu-Tsai (EFIE-PMCHWT) equation. In this formulation, the real and imaginary parts of the exterior integral operators are chosen to construct the finalized weighting matrices, to connect radiated power of the characteristic current. Compared with other SIE-based methods, this equation doesn't need additional post-processing since it can effectively avoid spurious modes. Numerical results compared with the volume-surface integral equation (VSIE) are investigated to validate the accuracy.

Image Recovery of Inaccessible Rough Surfaces Profiles Having Impedance Boundary Condition

This letter addresses a reconstruction algorithm of locally rough inaccessible surface profiles via the knowledge of the scattered field data under the consideration of the impedance boundary condition (IBC). To this aim, first, the synthetic scattered field data are obtained through the solution of the conventional surface integral equation (SIE) written on the rough surface. Then, the same SIE together with the data equation is solved iteratively via Newton’s method to obtain the image of the rough surface profile. In the numerical implementation, the nonlinear ill-posed inverse problem is linearized in an iterative fashion via the Newton method and regularized by Tikhonov in the least-squares sense. The feasibility of the algorithm is provided via numerical examples, which shows that the method is effective and promising.

Accurate and Efficient Solutions of Densely Discretized Closed Conductors Using a Combined Potential-Field Formulation

We present an accurate, efficient, and stable formulation for rigorous analyses of electromagnetic problems involving closed conductors. The formulation, namely the combined potential-field formulation (CPFF), is constructed from the conventional potential integral equations and the magnetic-field integral equation, together with an additional integral equation using the boundary condition for the normal component of the magnetic vector potential. Being both low-frequency-stable and resonance-free, CPFF is a broadband formulation, which enables accurate and efficient solutions of objects with diverse dimensions and discretization sizes.

Contraction Integral Equation for Modeling of EM Scattering by 3-D Bodies Buried in Multilayered Anisotropic Medium

Integral equation (IE) is regarded as an efficient method for 3-D modeling of electromagnetic (EM) responses. The main contribution of this article is to develop a contraction integral equation (CIE) algorithm with a high performance for modeling of EM scattering by 3-D bodies buried in multilayered anisotropic medium. The introducing of contraction operator can ensure that the discretized IE will be iteratively convergent in large conductivity contrast between abnormal bodies and background medium. Moreover, the contraction operator can also accelerate the iteration convergence of the IE. During the discretization process of CIE, direct waves are extracted from dyadic Green's functions (DGFs) of multilayered medium, and the singular integrals of direct waves are eliminated using equivalent sphere and ellipsoid methods. By splitting DGF into convolution terms and correlation terms, 3-D FFT can be used to accelerate the product evaluation of DGFs integral and scattering current encountered in each iteration. Validity of the proposed CIE is verified by comparing with the existing finite volume method. Modeling results show that the conventional IE will be iteratively divergent when the conductivity contrast between bodies and background is large, but the CIE is always iteratively convergent even under very large conductivity contrast.

Single-source formulation for the analysis of electromagnetic penetrable objects

Modified global surface impedance (MGSI) formulation, a fast and efficient single-source integral equation for analyzing penetrable objects including plasmonic circuits and dielectric periodic structures, is proposed. Using the equivalence principle, a single-source formulation for the exterior region of the desired problem is constructed. Then the assumed equivalent source (surface electric current) is written as the sum of two surface electric currents, one related to the other based on the global surface impedance (GSI) model. Thus, the conventional surface integral equation (SIE) for the exterior problem will be modified by this new single-source formulation. The interior problem is also formulated using the GSI model resulting in our proposed MGSI formulation. MGSI eliminates the need for equivalent magnetic current in the SIE formulation. It also helps one to formulate the interior and exterior problems independently. Moreover, in our approach material property does not affect the problem convergence behavior, while the convergence behavior of the conventional SIE deteriorates for a plasmonic structure and also for a dielectric object with high contrast to its surrounding medium. The accuracy and efficiency of the proposed formulation are demonstrated by comparing its results with the conventional SIE in solving several two-dimensional examples.

Solving EM Scattering From Complex Thin Dielectric/PEC Composite Targets by a VSIE-Based Method

A nonconformal and nonoverlapping domain decomposition method (DDM) based on hybrid volume and surface integral equation (VSIE) is presented for modeling complex thin dielectric/perfect electric conductor (PEC) composite targets. To reduce the unknown amount of the conventional volume integral equation (VIE) for thin dielectric bulks, the improved simplified prism vector (ISPV) basis function is developed. By using the basis function, the volume integrals can be simplified into surface integrals or line integrals, which significantly reduce the simulation time. Prism grids can also avoid the overmeshing problem for thin dielectrics. Moreover, they can be perfectly matched to the triangular meshes widely used in surface integral equation (SIE) hence assures a simple and robust VSIE implementation. Next, in order to increase the flexibility of geometrical modeling and accelerate the convergence of the presented VSIE system for electrically large and multiscale composite targets, a DDM is further developed. In this DDM scheme, discontinuous Galerkin approaches are, respectively, employed for VIE and SIE, accounting for conformal/nonconformal volume grids and surface grids between adjacent subdomains. In addition, an interior penalty term is introduced to stabilize the DDM solution. Finally, some numerical results are considered and analyzed to show the validity, efficiency, and applicability of the presented scheme.

Reduced density profile of small particles near a large particle: Results of an integral equation theory with an accurate bridge function and a Monte Carlo simulation.

Solute-solvent reduced density profiles of hard-sphere fluids were calculated by using several integral equation theories for liquids. The traditional closures, Percus-Yevick (PY) and the hypernetted-chain (HNC) closures, as well as the theories with bridge functions, Verlet, Duh-Henderson, and Kinoshita (named MHNC), were used for the calculation. In this paper, a one-solute hard-sphere was immersed in a one-component hard-sphere solvent and various size ratios were examined. The profiles between the solute and solvent particles were compared with those calculated by Monte Carlo simulations. The profiles given by the integral equations with the bridge functions were much more accurate than those calculated by conventional integral equation theories, such as the Ornstein-Zernike (OZ) equation with the PY closure. The accuracy of the MHNC-OZ theory was maintained even when the particle size ratio of solute to solvent was 50. For example, the contact values were 5.7 (Monte Carlo), 5.6 (MHNC), 7.8 (HNC), and 4.5 (PY), and the first minimum values were 0.48 (Monte Carlo), 0.46 (MHNC), 0.54 (HNC), and 0.40 (PY) when the packing fraction of the hard-sphere solvent was 0.38 and the size ratio was 50. The asymptotic decay and the oscillation period for MHNC-OZ were also very accurate, although those given by the HNC-OZ theory were somewhat faster than those obtained by Monte Carlo simulations.

Reference-Based Integral MR-EPT:Simulation and Experiment Studies on the 9.4T MRI.

Current integral-equation (IE) based MR electrical properties tomography (EPT) methods utilize simulated incident radio-frequency (RF) fields, which are inaccurate and lead to reconstruction errors. To improve the accuracy and practicability of IE-based MR-EPT methods, a new approach is presented that obtains the incident fields using reference subjects and RF field mapping techniques. The Incident field approximation (IFA) is first demonstrated in this paper. This approximation assumes that two imaged subjects with similar coil/subject interactions will have similar incident RF fields, thus one can feed the estimation of the incident fields within the imaged subject into the calculation of those within a homogeneous subject (reference subject). This is done by measuring the total RF fields ( field) of the reference using field mapping techniques, using the known EPs of the reference subject and by rearranging Ampere's Law and the integral equations. The calculated incident RF fields are then used to reconstruct the EPs' distribution with a three-dimensional (3D) integral-based MR-EPT method. Numerical simulation results indicated that the incident RF fields obtained from the reference subject provide accurate 3D reconstruction of EPs with less than 16% root mean square error (RMSE) in noise-free scenario while the conventional IE method had more than 28% RMSE. The phantom-based experiments at 9.4T MRI system have also been conducted to evaluate the performance of the proposed method and the results indicated that the proposed method achieved desirable robustness against the noise in practical scenario with less than 21% RMSE while the conventional differential equation-based method showed worse than 37% RMSE.

Efficient method for evaluation of second-harmonic generation by surface integral equation

A frequency-domain method based surface integral equation of the tangential Poggio–Miller–Chang–Harrington–Wu–Tsai formulation is presented as a full wave analysis to evaluate surface second-harmonic generation from noble metal nanoparticles of virtually arbitrary shape. According to the similar solution of fields and boundary conditions at fundamental and second-harmonic frequency, we get the derivation of surface integral equation formulations with only half unknowns compared with the conventional surface integral equation method. Simultaneously, the condition number of impedance matrix has been sharply declined. Numerical examples of gold nanospheres of different radius are presented to demonstrate the accuracy and efficiency of the proposed method. To further research the distribution of surface nonlinear polarization and properties of the second-harmonic radiation, we apply our method to a noncentrosymmetric L-shaped gold nanoparticle studied experimentally. This method provides an efficient and promising approach for evaluation of nonlinear optical radiation generated from metal nanoparticles array and optimization design of nonlinear nanoantennas.

Accurate simulation of thin dielectric sheet by higher order surface integral equation

ABSTRACTA higher order approach for the accurate simulation of a thin dielectric sheet is proposed in this paper. Due to the small thickness of thin dielectric sheet, an equivalent surface integral equation containing tangential and normal fields can be derived from the conventional volume integral equation. Higher order bases are then applied to approximate both components. Although the programming complexity increases slightly compared with the traditional low-order thin dielectric sheet surface integral equation method, the number of unknowns required in the same problem can be drastically reduced in the proposed higher order approach. At the same time, a higher accuracy can be achieved. Numerical results are presented to show the high efficiency and stability of the proposed method.

Accuracy of Sources and Near-Zone Fields When Using Potential Integral Equations at Low Frequencies

We consider method-of-moments solutions of the recently developed potential integral equations (PIEs) for low-frequency electromagnetic problems involving perfectly conducting objects. The electric current density, electric charge density, and near-zone fields calculated by using PIEs are investigated at low frequencies, in contrast to those obtained via the conventional electric-field integral equation (EFIE). We show that: 1) the charge density can accurately be found by using EFIE despite the very poor accuracy in the current density; 2) using PIEs instead of EFIE leads to accurate computations of the dominating solenoidal part of the current density, while the vanishingly small irrotational part and the charge density become inaccurate; 3) it is possible to accurately compute the charge density, in addition to the current density, when using PIEs, but at the cost of a solution of an additional integral equation. Numerical examples involving spherical objects are presented to demonstrate the accuracy of sources and near-zone fields when PIEs and EFIE are used.

Generalized Hybrid Surface Integral Equations for Finite Periodic Perfectly Conducting Objects

Hybrid formulations that are based on simultaneous applications of diversely weighted electric-field integral equation (EFIE) and magnetic-field integral equation (MFIE) on periodic but finite structures involving perfectly conducting surfaces are presented. Formulations are particularly designed for closed conductors by considering the unit cells of periodic structures as sample problems for optimizing EFIE and MFIE weights in selected regions. Three-region hybrid formulations, which are designed by genetic optimizations, are demonstrated. Comparisons to the conventional combined-field integral equation show the benefits of the generalized formulations in terms of accuracy and efficiency.

Boundary element analysis of 2D and 3D thermoelastic problems containing curved line heat sources

AbstractTemperature and stress analysis of a medium with concentrated heat sources has some important applications in engineering. In this paper, a boundary element method for analysis of two- and three-dimensional thermoelastic problems containing curved line heat sources in isotropic media is presented. In these problems, the heat generation within the problem domain is concentrated over a curved path. In the conventional integral equations of thermoelasticity, the domain integrals are expressed in terms of the temperature function. In this work, modified integral equations, in which the domain integrals are expressed in terms of the heat source function is used. The shape of the curved line heat source and the intensity function along the source can be arbitrary. Temperature, displacement and stress analyses are performed without considering internal points and without any need to find the temperature distribution in the domain. Three numerical examples are presented to show the effectiveness and accur...

Theoretical and numerical investigations of inverse patchy colloids in the fluid phase.

We investigate the structural and thermodynamic properties of a new class of patchy colloids, referred to as inverse patchy colloids (IPCs) in their fluid phase via both theoretical methods and simulations. IPCs are nano- or micro- meter sized particles with differently charged surface regions. We extend conventional integral equation schemes to this particular class of systems: our approach is based on the so-called multi-density Ornstein-Zernike equation, supplemented with the associative Percus-Yevick approximation (APY). To validate the accuracy of our framework, we compare the obtained results with data extracted from NpT and NVT Monte Carlo simulations. In addition, other theoretical approaches are used to calculate the properties of the system: the reference hypernetted-chain (RHNC) method and the Barker-Henderson thermodynamic perturbation theory. Both APY and RHNC frameworks provide accurate predictions for the pair distribution functions: APY results are in slightly better agreement with MC data, in particular at lower temperatures where the RHNC solution does not converge.

Structure of discrete-potential fluids interacting via two piece-wise constant potentials with a hard-core

The structure of fluids whose molecules interact via potentials with a hard-core plus two piece-wise constant sections of different widths and heights has been studied by the perturbation theory based on the power series of an inverse temperature. The perturbation theory based on the reference system which incorporates both the repulsive and attractive potentials predicts accurate radial distribution function and direct correlation function of the discrete-potential fluids for the whole density regions. In particular, it is the most successful for the square-shoulder plus square-shoulder fluid and shifted square-shoulder fluid with a purely repulsive potential rather than the square-well plus square-well fluids with an attractive potential. It is found that the present theory predicts a more accurate structure than the rational-function approximation and conventional integral equations such as Percus–Yevick and hypernetted-chain theories in most of the cases.

Discretization of Volume Integral Equation Formulations for Extremely Anisotropic Materials

A stable volume integral equation formulation and its discretization for extremely anisotropic materials is presented. The volume integral equations are written in terms of the volume equivalent currents. The equivalent currents are expanded with piecewise constant basis functions, and the Galerkin's scheme is applied for testing the equations. Numerical results show that the behavior of the formulation is more stable than the behaviors of the more conventional volume integral equation formulations based on fluxes or fields, when the scatterer is extremely anisotropic. Finally, the developed method is applied to analyze a highly anisotropic material interface which approximates the ideal DB boundary.