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The stability analysis problem is considered for multiarea load frequency control (LFC) systems with electric vehicles (EVs) and time delays. The novel linear operator inequality approach is proposed and a less conservative delay-dependent stability condition is achieved. First, the model of the multiarea LFC system with EVs and delays is expressed by the partial integral equation (PIE) at the first time. Then, a complete quadratic Lyapunov-Krasovskii functional is built in the form of an inner product of the linear partial integral (PI) operator. The novel stability criteria with less conservatism are proposed in the form of linear operator inequality. Moreover, the relationships between the delay margins and the controller parameters are shown. Finally, the simulations are conducted on both one-area and two-area LFC systems to show the effectiveness of the proposed approach.

International audience; This article proposes a strategy to consider the dielectric effects in the thin-wire integro-differential formulation and applies it to analyzing microstrip metasurfaces (MTSs) that support magnetic resonances. These structures comprise a periodic arrangement of subwavelength unit cells; therefore, the thin-wire approximation can be applied to analyze them. However, the substrate is usually disregarded. This work achieves more accurate results with a homogeneous equivalent medium approximation considering the dielectric properties. Moreover, an experimental methodology is presented to validate this implementation for each device and for the complete system. The proposed formulation is numerically solved with the method of moments (MoM), and the results show close agreement with the experimental data. Besides, it leads to a significant reduction in the computation time compared with a full-wave analysis, which makes this approach also compelling for designing inductive wireless power transfer (WPT) systems and magnetic resonators.

This work continues the development of the raytracing method of [1] for computing the scattered fields from metasurfaces characterized by locally periodic reflection and transmission coefficients. In this work, instead of describing the metasurface in terms of scattering coefficients that depend on the incidence direction, its scattering behavior is characterized by the surface susceptibility tensors that appear in the generalized sheet transition conditions (GSTCs). As the latter quantities are constitutive parameters, they do not depend on the incident field and thus enable a more compact and physically motivated description of the surface. The locally periodic susceptibility profile is expanded into a Fourier series, and the GSTCs are rewritten in a form that enables them to be numerically solved for in terms of the reflected and transmitted surface fields. The scattered field at arbitrary detector locations is constructed by evaluating critical-point contributions of the first and second kinds using a Forward Ray Tracing (FRT) scheme. The accuracy of the resulting framework has been verified with an Integral Equation based Boundary Element Method (BEM)-GSTC full-wave solver for a variety of examples such as a periodically modulated metasurface, a metasurface diffuser and a beam collimator., Comment: 12 pages, 9 figures

Estimation of subsurface properties by using the scattering integral equation is a method that finds increasing use in near-surface and shallow oil/gas exploration, based on seismic, low-frequency electromagnetic, and surface-radar surveys. The method can accurately simulate physical realizations induced by small-scale perturbations, but its accuracy depends on a suitable low-frequency property model. We propose a 3-D geological-structure-guided model building to provide a reliable low-frequency model and combine it with the Born-Wentzel-Kramers-Brillouin-Jeffreys (WKBJ)-approximation-based inversion algorithm. Instead of the traditional approach based on artificially interpreted horizons, we use 3-D seismic-slope attributes as lateral constraints, which contain more geological information. Plane-wave destruction (PWD) in 3-D is exploited to extract the 2-D slopes along the inline and crossline directions, which are the key factors in computing 3-D slopes. Then, by introducing the shaping regularization, we build low-frequency models by solving the inverse problem. Numerical analysis indicates that an appropriate background model is essential for seismic modeling with the Born-WKBJ approximation. The methodology is applied to synthetic and 3-D field data, and the examples show that it provides reliable background models and improves the inversion performance.

This work addresses an iteration scheme to observe the image of an inaccessible rough surface profile from the intensity of scattered field data. The solution of the problem is based on the integral equations written on the rough surface profile. By virtue of the surface integral equations, the phaseless scattered field is represented by a nonlinear ill-posed integral operator, which is linearized by Newton's method and regularized via Tikhonov. The surface image is reconstructed iteratively in the least-squares sense by utilizing spline-type basis functions. A detailed numerical assessment is provided, showing that the algorithm is very effective and promising.

We investigate an algorithm for the 3-D inversion of frequency-domain airborne electromagnetic (AEM) data based on the forward modeling and sensitivity calculation by footprint-guided compact finite element method (CFEM). Unlike the conventional approach, the modeling volume in our algorithm for each transmitter-receiver pair is a regular hexahedral that encloses the footprint, rather than a large mesh for the entire survey area or the local mesh with a number of grids extending from the footprint. After the electric fields in the modeling volume are solved by vector finite element method (FEM) with an integral equation boundary condition, the response and sensitivity are explicitly calculated by employing the product of the prepared Green's functions and the vector of electric fields. The accuracy of this footprint-guided CFEM is validated by comparing it against conventional CFEM, and different synthetic models are tested by our inversion algorithm. The inversion tests of synthetic models show the feasibility of the combination of footprint-guided CFEM and Gauss-Newton optimization in recovering models within an acceptable error level, and the inversion results show a good agreement with the true models on both the model geometry and recovered conductivity.

In this paper, a nonconformal domain decomposition method (DDM) based on the hybrid finite element method (FEM) and boundary element method (BEM) is proposed to solve arbitrary three-dimensional (3D) anisotropic (including bi-anisotropic and single-anisotropic) multiscale composite objects. The object is divided into interior and exterior regions. The interior region is solved by the bi-anisotropic finite element domain decomposition method (FEM-DDM), and the exterior region is solved by the discontinuous Galerkin (DG) method based on combined field integral equation (CFIE). In order to decouple the electric and magnetic fields of the bi-anisotropic object, a new 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, while the latter uses Robin transmission conditions (RTC) to ensure continuity. The BEM dense matrix vector multiplication is accelerated by multilevel fast multipole algorithm (MLFMA). The proposed FEM-BEM-DDM allows for nonconformal discretization between any touching sub-domains and provides an effective preconditioner. In numerical examples, the accuracy and effectiveness of the method are verified.

A single-source surface integral equation (SS-SIE) for transverse electric (TE) scattering from cylindrical multilayered objects is proposed in this article. By incorporating the differential surface admittance operator (DSAO) and recursively applying the surface equivalence theorem from innermost to outermost boundaries, an equivalent model with only electric current density on the outermost boundary can be obtained. In addition, an integration approach is proposed, where the small argument expansion of the Hankel function is used to evaluate the singular and nearly singular integrals. Compared with other SIEs, such as the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) formulation, the computational expenditure is reduced for multilayered structures because only a single source is needed on the outermost boundary. As shown in the numerical results, the proposed method generates only 19% of unknowns, uses 26% of memory, and requires 29% of the CPU time of the PMCHWT formulation.

An efficient EM scattering solution of geometrical uncertainty is fabricated to analyze the scattering from electrically large metallic/dielectric bodies of revolution (BoR). The algorithm started by describing the geometrical shape of the body of revolution with the two dimensional non-uniform rational B-spline curves. Then the relationship can be constructed between the local random variables and the spatial BoR basis functions. Thus, the surface integral equations were represented in terms of independent random variables. Radar cross section can be quickly obtained for perturbed geometrical shapes and the proposed algorithm can save computational resources when compared with the Monte Carlo method. The electrical field integral equation is applied to calculate the metallic BoR targets while the Poggio-Miller-Chang-Harrington-Wu-Tsai integral equation is used to analyze the dielectric BoR targets. Numerical results are given to demonstrate the accuracy and efficiency of the proposed method.

An algorithm for the fast computation of multi-static scattering patterns, produced by low-contrast inhomogeneous volumetric objects, which satisfy the Born approximation assumption, is presented. The algorithm relies on hierarchical interpolation and aggregation of optimally sampled phase-compensated partial contributions to the scattered field integral. The Born approximation enables the efficient simultaneous computation for ranges of incident and observation angles, as well as excitation frequencies. This leads to a reduction in the asymptotic computational complexity by several polynomial orders of the problem’s acoustical size. The algorithm’s error controllability and favorable computation time are demonstrated via representative examples.

We study the oblique scattering of arbitrarily polarized plane waves from radially inhomogeneous and bianisotropic cylinders in the general case where all the elements of the four constitutive tensors are nonzero. The analysis relies on an exact reformulation of Maxwell’s equations as a second-kind linear Volterra matrix integral equation. The discretization scheme uses a high order, entire-domain, Nystrom method that employs the Gauss–Legendre–Radau quadrature. The entries of the Nystrom matrix have simple closed-form expressions. The proposed algorithms have exponential convergence and can handle both single-layer and multilayer cylinders with either continuous or sectionally continuous inhomogeneity profiles.

Mutual coupling between the space-borne antenna has an essential impact on the communication characteristics of a wireless link to submerged submarines. In this article, the mutual coupling between antennas of a circular array in an anisotropic plasma over a very low-frequency (VLF: 3–30 kHz) range is investigated. An expansion of the currents into phase modes enables us to develop an efficient method to analyze the mutual coupling between antennas. The $N$ -coupled integral equations for antennas of an array are first derived with full consideration of the mutual coupling effect. By making use of the phase modes combined with the method of moment (MoM), the $N$ -coupled integral equations are decomposed into independent ones and solved for each phase mode. Computations and discussions show that the self and mutual impedances of antennas are increased with the geomagnetic field tilt angle. Moreover, the mutual impedance is decreased with the increasing electrical separation, thereby indicating that the mutual coupling effect is weakened as the separation increases. The accuracy of the proposed method is verified by comparing the computed results with experiments and simulation using commercial electromagnetics software FEKO.

The method based on the partitioned inverse matrix formula and the Sherman–Morrison–Woodbury formula has been developed to solve the partial modification problems efficiently. However, it requires the meshes of the mother-structure, the add-structure, and the subtract-structure are separated or conformal. It is inconvenient and inflexible for users. In this letter, the integral equation discontinuous Galerkin (IEDG) with a mesh modification algorithm (MMA) is proposed to mitigate this problem. With the help of the IEDG and the MMA, the mother-structure and the add-structure can be meshed independently, and the mesh errors caused by removing the subtract-structure from the mother-structure are reduced. Numerical results are given to demonstrate the advantages of the improved method.

Our numerical experiments show that the hierarchical matrix (H-matrix) solver is more efficient than the butterfly solver for electrically small problems, although the converse is true for sufficiently large problems. In this communication, we propose a hybrid matrix decomposition algorithm (HMDA) that combines the H-matrix and butterfly algorithms. In the HMDA, the H-matrix algorithm is employed to compress the lower–upper decomposition matrix in the lower levels, whereas the butterfly algorithm is adopted at the higher levels. Numerical examples demonstrate that the performance of the HMDA is similar to that of the H-matrix for electrically small problems and higher than the butterfly algorithm for electrically large problems.

A multibranch Rao–Wilton–Glisson (MB-RWG) basis function is proposed in this article, which is composed of one positive triangle with a larger edge length and several negative triangles with smaller edge lengths. All negative triangles are aggregated to replace the negative triangle in the traditional RWG basis function, with exactly the same function expression defined on the triangles. The number of negative triangles can be changed in different mesh structures. Similar to traditional RWG basis functions, the proposed MB-RWG basis functions guarantee normal current continuity across the common edges. No line charges exist and the total charge on one MB-RWG is zero. MB-RWG basis functions can be used very conveniently to connect one surface with a coarse mesh scheme to another with a fine mesh scheme and can be flexibly applied in the domain decomposition method (DDM). Numerical results validate the accuracy and demonstrate the versatility of the proposed basis function in modeling multiscale perfect electrically conducting (PEC) objects.

Uncertainty quantification related to nonlinear inverse scattering problems often involves the posterior covariance matrix that cannot be effectively estimated but still need to be considered as an important part of the nonlinear inverse scattering problem. The objective of this work is to show how to take advantage of a Bayesian framework to estimate the uncertainty of velocity reconstruction in the nonlinear inverse scattering imaging. The key ingredients are twofold. On the one hand, we rely on a Kalman Filter method, equipped with an optimization scheme, to solve the inverse problem. We use the distorted Born iterative method to formulate the sensitivity kernel. It directly uses an explicit representation of the data sensitivity function in terms of Green functions, rather than the indirect optimization approach based on the adjoint state method, in which the Green’s functions are based on integral equations. On the other hand, we use the direct envelope methods to provide the initial guess (large-scale smooth model) to overcome the nonlinearity in the inverse scattering problems. We first investigate the uncertainty of velocity imaging in high contrast media. Then we show by means of Lippmann-Schwinger-type equations that the uncertainty of multiparameter inverse scattering problems can be addressed. Numerical results dealing with the uncertainty of nonlinear inverse scattering problems in the case of both isotropic and anisotropic media highlight the important role played by uncertainty quantification.

Convergence studies of the Method of Moments solution of thin wires have long been handicapped by the issue of conditional convergence: when the segment length approaches the wire radius, the approximations typically made result in exponentially diverging solutions. In this communication, it is shown that it is the approximation of the singularity which is the main cause of this problem. Using the mixed potential integral equation, paralleling the classic RWG formulation for surfaces, the formulation in this communication approximates the surface (as opposed to equivalent filamentary) current, and applies Galerkin testing on the wire surface. The use of contemporary singularity-canceling integral transforms for highly efficient and accurate quadrature is demonstrated. Results are presented for typical thin-wire dipoles for both radiation and scattering problems. It is shown that with a rigorous treatment of the singularity, a far more comprehensive convergence study may be undertaken than previously demonstrated. Furthermore, the results shown justify the many years of successful use in practice of simple source models with modest mesh requirements.

The problem of identifying the load acting on the elements of structures belongs to the class of inverse problems of the mechanics of a deformable solid, which are often incorrect. Solving such problems is associated with the instability of the calculation results, which requires the development of special methods for their research. This predetermines the relevance of this study. The object of the study is a single-pass cylindrical shell consisting of two rigidly fastened butt-fastened sections made of different materials. Each of the shells is assumed to be elastic isotropic, having a cross-section of medium thickness. The equations of axisymmetric deformation of shells are used within the framework of Timoshenko hypotheses. An approach to solving direct and inverse problems for such discretely heterogeneous objects is proposed, which implies the conditional separation of a discretely heterogeneous cylindrical shell along the length, followed by the addition of functions of fictitious loads. The main analytical relationships for building a system of integral Volterra equations are given, for which an analytic-numerical solution is derived. The final ratios have been obtained, which make it possible to calculate the kinematic and force parameters of the study object in the process of non-stationary deformation. The inverse problem of identifying arbitrary loads acting on a shell that is heterogeneous in length is solved in a general form. An algorithm for the restoration of pulse loads has been developed, which is robust to errors in the initial data (about 5%). The material related to solving direct and inverse problems for shells that are discretely heterogeneous in length can significantly advance the methodology for identifying pulse loads acting on structural elements

In this communication, we present flexible and efficient solutions of large-scale electromagnetic scattering problems using the parallel discontinuous Galerkin (DG) surface integral equation (SIE) method. A simplified formulation of the conventional DG is used by removing the interior penalty term, thereby avoiding computational burden of the contour integral in the conventional DG. It is demonstrated numerically that the simplification causes convergence deficiency and changes the accuracy convergence rate of the solution for objects with sharp corners. The former is overcome by constructing preconditioners using the near-field matrix of the whole region, and the latter is proved to have a neglectable influence on the accuracy of the final solutions as long as normal mesh density requirement is satisfied. The wellscaling ternary parallelization approach of the multilevel fast multipole algorithm (MLFMA) is incorporated into the DG domain decomposition framework to reduce the complexity of the solution and strength its capability for large-scale problems, with the distribution of the nearfield-related calculation adjusted to adapt the nonuniform meshes for a better workload balance. Numerical results are included to validate the accuracy and demonstrate the scalability and versatility of the proposed method. In addition, we demonstrated the effectiveness of our algorithm by solving a high-definition complicated aircraft model with inlets and exhausts involving over one billion unknowns.

A technique of designing antenna excitation realizing orthogonal states is presented. It is shown that a symmetric antenna geometry is required in order to achieve orthogonality with respect to all physical quantities. A maximal number of achievable orthogonal states and a minimal number of ports required to excite them are rigorously determined from the knowledge of an antenna's symmetries. The number of states and number of ports are summarized for commonly used point groups (a rectangle, a square, etc.). The theory is applied to an example of a rectangular rim where the positions of ports providing the best total active reflection coefficient, an important metric in multi-port systems, are determined. The described technique can easily be implemented in existing solvers based on integral equations., Comment: 12 pages, 12 figures. Submitted to TAP

The singularity cancellation transformations are well-known for calculating weakly singular and near-singular integrals in integral equation solutions. However, some singularity cancellation methods suffer from the shape-dependent problem of inaccuracy and inefficiency for deformed triangles. By the theoretical analysis and numerical verification in this article, the relation between the near-singularity and shape-dependence of the singularity cancellation schemes is discussed. Moreover, a novel framework for devising cancellation transformations of weakly near-singular integrals is presented, which results in fast convergence for both regular and irregular triangular domains. Some numerical results are given to illustrate the validity of the theoretical framework and the efficiency of the proposed transformations.

For spaceborne transmitting systems, strip antennas provide a new scheme in boosting the radiation efficiency due to their availability in generating larger current moments. In this article, we investigate the theory of a very-low-frequency (VLF: 3–30 kHz) strip antenna immersed in a homogeneous anisotropic plasma. The focus lies on the analytical method for calculating the current distribution and input impedance of the antenna, where the current distribution, in this case, consists of two parts of distribution in both longitudinal and transverse directions. Kernel function for a strip antenna at arbitrary orientations is derived with the help of coordinate transformation, and the current integral equation is solved via the method of moments (MoM). Computations show that the surface currents on a strip antenna will concentrate near the edges of the strip, and the input impedance of the antenna will increase with either the length-to-width ratio or the dip angle. Simulation results using commercial software FEKO exhibit good consistency with the computed results, thus verifying the accuracy of the proposed method. By comparison of the input impedances with linear antennas, it is found that strip antennas are more efficient than linear antennas when the length-to-width ratio is selected properly.

An efficient and accurate hybrid spectral element-spectral integral (SESI) method is proposed to simulate electromagnetic (EM) scattering by objects in multiregion cylindrically layered media. The field in each of the multiple regions with multiple scatterers can be solved by the spectral element method (SEM), while the spectral integral method (SIM) is used as an exact boundary condition for the adjacent regions with multilayered media. This boundary condition between the electric field and the surface current density in each cylindrical layer can be obtained by using a recursive formula in SIM. The inverse matrix of a Toeplitz matrix arising from the SIM can be solved by the fast Fourier transform (FFT) with $O(N\log N)$ complexity. The hybrid SESI method is used to simulate homogeneous and/or inhomogeneous multiregion multilayered structures. Numerical examples, including the frequency selective surface (FSS) applications, demonstrate that the hybrid SESI method is efficient and accurate by comparing its results with analytical solutions and results from the conventional SEM and the finite element method (FEM).

In this article, the previously introduced finite-integration technique/method-of-moments (FIT-MoM) hybrid method, which is suited for the analysis of electromagnetic scattering by inhomogeneous bodies of revolution (BoRs), is generalized to problems involving multilayered environment. Due to the fact that BoR-typical azimuthal mode decomposition is used, appropriate modal Green’s functions for multilayered media are given. They are responsible for source-field interactions within the region that is external with regard to the FIT part. Formulas for modal decomposition of incident plane wave fields as well as the approximation enabling computing fields produced by sources associated with individual modes in the far-zone are also provided.

A hybrid discretization scheme for solution of volume integral equation (VIE) by method of moments (MoM) for electromagnetic scattering from dielectric objects is proposed in this article. The Schaubert-Wilton-Glisson and edge (SWG-Edge) hybrid basis functions are used in this discretization scheme. According to the divergence-free condition of electric displacement vector, a kind of edge basis functions defined in elements including boundary faces which separate a dielectric object from the background is derived. As a result, we get a SWG-Edge hybrid basis set. Details for the calculation of the corresponding matrix elements for the edge basis and testing functions are presented. Numerical results show the validity and accuracy of the hybrid discretization scheme. Finally, the proposed method is used for efficient solution of VIE for inhomogeneous dielectric objects with multiboundary. It is shown that for multiboundary problems, the number of unknowns of the hybrid basis is only about 71% of the traditional SWG basis. This means that the memory for solution of VIE by the traditional SWG basis functions can be reduced by half. Therefore, the SWG-edge hybrid basis is much more efficient than the traditional SWG basis for solution of VIE with multiboundary problems.

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.

The multitrace domain decomposition surface integral equation (MT-DD-SIE) originally developed to analyze electromagnetic scattering from dielectric composite objects is extended to efficiently account for perfect electrically conducting (PEC) bodies. This is achieved by adopting Robin transmission conditions (RTCs) to PEC surfaces. These PEC-RTCs, which are the only governing equations for a PEC body, are used to “couple” it to the dielectric bodies. Upon discretization, PEC-RTCs produce a well-conditioned matrix block and, therefore, does not negatively affect the convergence of the iterative solution of the MT-DD-SIE matrix equation. The resulting method is significantly faster and has a smaller memory footprint than the traditional globally coupled contact-region-modeling method that uses the coupled electric field and Poggio–Miller–Chang–Harrington–Wu–Tsai SIEs in analyzing electromagnetic scattering from composite PEC/dielectric objects. This is demonstrated by numerical examples involving electrically large scatterers.

Methods for modelling the radar scattering characteristics of resonance‐size airborne objects containing metallic and dielectric design elements are discussed. The developed algorithms are based on solving Fredholm second‐kind boundary integral equations (IEs) that place them within the context of the method of analytical regularisation. The magnetic field IE is used for metal elements, and the Müller IE set is used for dielectric elements. The proposed algorithms provide important novelties in calculating the radar scattering characteristics of objects with various curvature radii, including electrically thin scatterers. The description is accompanied by an analysis of developed algorithm convergence. For the validation, a comparison of results for model object obtained by different methods is presented. The results of modelling the radar scattering characteristics of an unmanned aerial vehicle and its separate metallic and dielectric elements within the VHF and S frequency bands are demonstrated and discussed.

In this article, a new inversion model for 2-D microwave imaging is introduced by means of a convenient rewriting of the usual Lippmann–Schwinger integral scattering equation. Such a model is derived by decomposing Green’s function and the corresponding internal radiation operator in two different contributions, one of them easily computed from the collected scattered data. In the case of lossless backgrounds, the resulting model turns out to be more convenient than the traditional one, as it exhibits a lower degree of nonlinearity with respect to parameters embedding the unknown dielectric characteristics. This interesting property suggests its exploitation in the solution of the inverse scattering problem. The achievable performance is tested by comparing the proposed model with the one based on the usual Lippman–Schwinger equation in both cases of linearly approximated and full nonlinear frameworks. Both numerical and experimental data are considered.

A novel electromagnetic modeling method for multidomain planar layered medium (MD-PLM) by surface integral equation (SIE) is proposed. The MD-PLM refers to the whole space that is separated into multiple PLM volumes, e.g., the infinite-sized PLM background with inserted finite PLM objects or multilayered parallel-plate structures. In the conventional SIE method for this scenario, only the exterior domain is considered as PLM that is modeled by layered medium Green’s functions (LMGFs), while the interior domain is treated by the multidomain homogeneous SIE. However, in this work, the surface equivalence formulation for this special scenario is reestablished by the LMGFs. Then, both the exterior and interior domains are considered as PLM that is modeled by the corresponding LMGFs and SIE in each subdomain. As a result, only the boundary of each sub-PLM domain is required to be discretized rather than the interfaces of each interior medium, which is different from the way in conventional SIE. Consequently, compared with the conventional SIEs, the proposed method can reduce a large quantity of unknowns and significantly increase the efficiency by reducing the memory consumption. By some numerical examples, the performance of this method is evaluated and compared with the conventional methods. The validity and efficiency of the newly proposed method are also well demonstrated in the examples.

Since the existing integral representations of the Sommerfeld integral (SI) are of the (semi) infinite type, a numerical error is inevitable due to the truncation of the integral range. In this article, a new finite range integral representation of the SI is formulated based on the exact image theory. To derive the new representation, a finite range integral expression of the exact image current is first derived. The radiated field by the exact image current is expressed in terms of a double integral in the complex domain where the exact image current flows. One of the two integrals is a semiinfinite line integral along the current path. This integral is deformed into a more convenient form as the exact image representation for the impedance plane. To convert the line source in the complex domain into a line source in the real domain, the path-deformation technique is again applied. The final line source consists of three line segments along which the exact image current flows vertically from the image source point to $\pm \infty $ . In addition to the line source, a lateral wave component is also obtained. The impedance exact image representation can be evaluated as a closed-form expression in terms of an incomplete cylindrical function. Thus, the SI can be represented as the finite range integral of the incomplete cylindrical function. To efficiently evaluate the proposed integral, a complete uniform asymptotic expansion is formulated. All the proposed formulations are numerically verified, including, in some cases, near-Earth propagation, and their behaviors are investigated. Also, the electric field is computed for a vertical and horizontal infinitesimal dipole and compared by the exact Sommerfeld and known approximate formulations.

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.

A higher order current and charge integral equation (CCIE) approach is developed for accurate electromagnetic scattering analysis of dielectric objects for a wide frequency band, that is from extremely low frequency to microwave frequency. Compared to the conventional CCIE method, its accuracy has been considerably improved by expanding the currents and charges using high-order hierarchical vector basis functions and higher order orthogonal scalar polynomials, respectively. The proposed higher order approach retains whole-frequency-range stability as the traditional low-order counterpart, but characterized with enhanced accuracy and higher convergence rate. Moreover, the consistency of the basis functions’ expansion order in currents and charges is not mandatorily required, which further increase the flexibility in electromagnetic modeling. The scattered fields of penetrable objects are analyzed to validate the effectiveness of this new higher order CCIE, as well as investigate its accuracy, stability, and flexibility.

In this paper, we study the bounded traveling wave solutions for the Boussinesq equation with dissipative term by using the theory of planar dynamical systems, and reveal the relationships between the behaviors of bounded traveling wave solutions and the dissipation coefficient. The bell profile solitary wave solution and kink profile solitary wave solution can be obtained according to the undermined coefficients method. Meanwhile, the approximate oscillatory damped solutions can be obtained. Furthermore, the global error estimation of the approximate oscillatory damped solutions is given by establishing the integral equation which reflects the relationships between approximate and exact oscillatory damped solutions.

Incorrect one- and two-dimensional inverse problems of reconstructing images of objects with angular resolutionexceeding the Rayleigh criterion are considered. The technique is based on the solution of inverse problems of source reconstruction signals described Fredholm integral equations. Algebraic methods and algorithms for processing dataobtained by measuring systems in order to achieve angular superresolution are presented. Angular superresolution allows you to detail images of objects, solve problems of their recognition and identification on this basis. The efficiency of using algorithms based on developed algebraic methods and their modifications in parameterization the inverse problems under study and further reconstructing approximate images of objects of various types is shown. It is shown that the noise immunity of the obtained solutions exceeds many known approaches. The results of numerical experiments demonstrate the possibility of obtaining images with a resolution exceeding the Rayleigh criterion by 2-6 times at small values of the signal-to-noise ratio. The ways of further increasing the degree of superresolution based on the intelligent analysis of measurement data are described. On the basis of the preliminary information on a source of signals algorithms allow to increase consistently the effective angular resolution before achievement greatest possible for a solved problem. Algorithms of secondary processing of the information necessary for it are described. It is found that the proposed symmetrization algorithm improves the quality of solutions to the inverse problems under consideration and their stability. The examples demonstrate the successful application of modified algebraic methods and algorithms for obtaining images of the objects under study in the presence of a priori information about the solution. The results of numerical studies show that the presented methods of digital processing of received signals allow us to restore the angular coordinates of individual objects under study and their elements with super-resolution with good accuracy. The adequacy and stability of the solutions were verified by conducting numerical experiments on a mathematical model. It was shown that the stability of solutions, especially at a significant level of random components, is higher than that of many other methods. The limiting possibilities of increasing the effective angular resolution and the accuracy of image reconstruction of signal sources, depending on the level of random components in the data utilized, are found. The effective angular resolution achieved in this case is 2—10 times higher than the Rayleigh criterion. The minimum required signal-to-noise ratio for obtaining adequate solutions with super-resolution is 13—16 dB for the described methods, which is significantly less than for the known methods. The relative simplicity of the presented methods allows you to use inexpensive computing devices and work in real time.

The reflection and transmission of guided mode in the 2-D single-mode waveguide whose boundary is random shape is numerically simulated. Since the boundary element method based on the guided-mode extracted integral equation (GMEIE) is employed, the boundary condition on the random boundary is rigorously satisfied. The dependence of the transmission coefficient on the amplitude of the random boundary is investigated. It is found that strong resonances with high ${Q}$ -factor occur even for slightly corrugated boundaries and cause the extremely small transmission (high reflection). These resonances are created by the existence of the region where the second mode can propagate in the random waveguide. The frequency characteristics of these resonances are examined in detail.

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.

In this work, we develop efficient algorithms to generate a rank-minimized ${\mathcal{ H}}^{2}$ -matrix to represent electrically large surface integral operators for a prescribed accuracy. We first generate an ${\mathcal{ H}}^{2}$ -matrix using the fast multipole method (FMM), and hence, the complexity for ${\mathcal{ H}}^{2}$ -construction is as low as $O(NlogN)$ for solving electrically large surface integral equations. We then develop fast algorithms to convert the FMM-based ${\mathcal{ H}}^{2}$ -matrix whose rank is full asymptotically to a new ${\mathcal{ H}}^{2}$ -representation, whose rank is minimized based on accuracy. The proposed algorithms cost $O(k^{3})$ in time for each cluster in cluster basis generation and $O(k^{2})$ in memory, where $k$ is the minimal rank of the cluster basis required by accuracy. When the rank of the ${\mathcal{ H}}^{2}$ -matrix is a constant, the complexity of the proposed algorithms is $O(N)$ in both time and memory consumption. When the rank is a variable dependent on electrical size, the total complexity can be evaluated based on the rank’s behavior. The resultant rank-minimized ${\mathcal{ H}}^{2}$ -matrix has been employed to accelerate both iterative and direct solutions. Numerical experiments on large-scale surface integral equation-based scattering analysis have demonstrated its accuracy and efficiency.

We consider the problem of planning the optimal route of an object moving at a constant speed on a plane trying to evade detection by a single sensor with the aim to minimize the integral level of useful signal arriving at the observer. It is shown that various mechanisms for detecting a moving object can be reduced to a general mathematical formalization, namely, to the study of the detection problem for different types of signal and a small signal-to-noise ratio. As a result of the study, the solution of the two-point variational problem with an integral constraint is reduced to the solution of three types of systems of algebraic integral equations, which include the normal elliptic Legendre integrals of the first and second kind, with the type of the system determined by various initial conditions and the values of the integral constraint. A numerical procedure for its solution is implemented for each of the systems described in the paper.

Borehole and fracture in well-logging applications are usually simulated by the finite-element method (FEM) or volume integral equation (VIE). In this work, we propose the surface integral equation (SIE) to simulate them together. Specifically, the borehole is treated by the Poggio, Miller, Chang, Harrington, Wu, and Tsai (PMCHWT) formulation, whereas the fracture is handled by the thin dielectric sheet (TDS)-based SIE, i.e., TDS-SIE. The notorious low-frequency breakdown of PMCHWT is cured by using loop and tree basis functions for discretization. The hybridization of the loop-tree enhanced PMCHWT and TDS-SIE may be referred to as LT-TDS, which is reported for the first time. Moreover, we extend LT-TDS into the case where the background is a layered medium (LM), leading to the more universal and powerful solver entitled LM-LT-TDS. We validate the proposed (LM-)LT-TDS by simulating several low-frequency examples and comparing with the reference results obtained by FEM. Of particular concern is a practical well-logging case where both the borehole and the tilted fracture straddle a three-layer medium.

In this letter, a high-order volume–surface integral equation (HOVSIE)-based reduced basis method (HOVSIE-RBM) is presented for the fast calculation of the wideband electromagnetic scattering of anisotropic-metallic objects. In the HOVSIE-RBM, two main processes, i.e., the offline process and the online process, are included. During the offline process, the sampling current vectors are first calculated by the HOVSIE, where a high-order hierarchical vector base (HOHVB) is applied to reduce unknowns in the anisotropic medium. Meanwhile, the hybrid multilevel accelerated Cartesian expansion multilevel fast multipole algorithm is used to accelerate the iteration of HOVSIE. During the online process, integrating with RBM can avoid the replicative near-region impedance matrix-vector multiplications (IMVMs). Moreover, the wideband scattering can be obtained mainly by calculating far-region IMVMs, rather than the classical iteration of HOVSIE. Therefore, deteriorated iteration caused by the increased basis order of HOHVB can also be avoided. Meanwhile, compared with the low-order volume–surface integral equation RBM, HOVSIE-RBM has higher interpolation efficiency and requires less memory cost, when solving the same wideband scattering problem. Interpolation accuracy, memory cost, and efficiency will be demonstrated by two numerical examples.

The numerical solution of the electric field integral equation in electromagnetic scattering problems involving metallic bodies, using the method of moments in conjunction with the fast multipole method, gives rise a full matrix with highly spread complex eigenvalues and consequently with a low condition number. Thus, iterative methods may not converge or it will require an extremely large number of iterations, which make this strategy unfeasible due to the high solution time and the expensive computational resources required. To overcome this drawback, the system matrix must be preconditioned to ensure the convergence in a reasonable number of iterations and with appropriate accuracy. In this work, some preconditioning techniques are revised and applied to a linear equation system derived from such type of problems, and the performance of these preconditioners is estimated and analyzed comparatively using the generalized minimal residual iterative method.

The WS time delay matrix relates a lossless and reciprocal system's scattering matrix to its frequency derivative, and enables the synthesis of modes that experience well-defined group delays when interacting with the system. The elements of the WS time delay matrix for surface scatterers and antennas comprise renormalized energy-like volume integrals involving electric and magnetic fields that arise when exciting the system via its ports. Here, direct and indirect methods for computing the WS time delay matrix are presented. The direct method evaluates the energy-like volume integrals using surface integral operators that act on the incident electric fields and current densities for all excitations characterizing the scattering matrix. The indirect method accomplishes the same task by computing scattering parameters and their frequency derivatives. Both methods are computationally efficient and readily integrated into existing surface integral equation codes. The proposed techniques facilitate the evaluation of frequency derivatives of antenna impedances, antenna patterns, and scatterer radar cross sections in terms of renormalized field energies derived from a single frequency characterization of the system., Submitted to the IEEE Trans. on Antennas and Propagation

In this letter, an efficient integral equation model-order reduction method is proposed to analyze complex radiation problem. In the offline stage, the randomized singular value decomposition and discrete empirical interpolation method are used to construct a reduced-order model (ROM) for the platform. In the online stage, the equivalence principle algorithm is used to transfer unknowns on the antenna to the unknowns on the equivalence surface. The modified tap basis scheme is applied to decompose the antenna from the platform and solve the current continuity challenge when the platform is cut off by equivalence surface. When the antenna position is changed in the input region, only the tap region needs to be remeshed and resolved. The ROM of platform, antenna, and equivalence surface are reusable in the online computation, which can reduce the computational complexity significantly. Numerical results demonstrate the accuracy and efficiency of the proposed method.

In this paper, we discuss the numerical solution of Adomian decomposition method and Taylor’s expansion method in Volterra linear integral equation. And we apply modified Adomian decomposition method and Newton Raphson method in Volterra nonlinear integral equation with the help of example and estimated an error in MATLAB 13 versions.

Equivalence theorems in electromagnetic field theory stipulate that farfield radiation pattern/scattering profile of a source/scatterer can be evaluated from fictitious electric and magnetic surface currents on an equivalent imaginary surface enclosing the source/scatterer. These surface currents are in turn calculated from tangential (to the equivalent surface) magnetic and electric fields, respectively. However, due to the staggered-in-space placement of electric and magnetic fields in FDTD Yee cell, selection of a single equivalent surface harboring both tangential electric and magnetic fields is not feasible. The work-around is to select a closed surface with tangential electric (or magnetic) fields and interpolate the neighboring magnetic (or electric) fields to bring approximate magnetic (or electric) fields onto the same surface. Interpolation schemes available in the literature include averaging, geometric mean and the mixed-surface approach. In this work, we compare FDTD farfield scattering profiles of a dielectric cube calculated from surface currents that are obtained using various interpolation techniques. The results are benchmarked with those obtained from integral equation solvers available in the commercial packages FEKO and HFSS.

To derive a less conservative stability criterion via Lyapunov-Krasovskii functional (LKF) method, in previous literature, multiple integral terms are usually introduced into the construction of LKFs. This article generalizes the results of previous literature by proposing a polynomial-type LKF, which contains the LKFs with multiple integral terms as special cases. In addition, a Jacobi–Bessel inequality is presented to bound the derivative of such LKF. As a result, an improved stability criterion of time-delay systems is established. Finally, two numerical examples are given to illustrate the effectiveness, and advantages of our method.

A coupled-field volume integral equation (CFVIE) method is developed for the calculation of the normalized cutoff wavenumbers of circular metallic walled waveguides having concentric continuously varying highly inhomogeneous gyrotropic (i.e., gyroelectric and gyromagnetic) infill. The normalized cutoff wavenumbers are obtained as the roots of a determinantal equation formed by solving the CFVIEs using the cylindrical Dini series expansion (CDSE) method where the unknown fields inside the waveguide are expanded by entire domain orthogonal Dini-type vectorial basis functions. To account for the electric boundary condition (BC) on waveguide’s circular perfect electric conducting (PEC) surface, two modified 2-D tensorial Green’s functions (GFs), expanded in cylindrical vector wave functions (CVWFs), are employed in the kernels of the CFVIEs. These modified 2-D tensorial GFs are constructed by enforcing, on their dyadic form, the satisfaction of the electric BC. The CDSE, along with the modified 2-D tensorial GFs, allow for the analytical integration of the volumetric-type integrals and the reduction of the CFVIEs to a set of algebraic equations. We exhaustively demonstrate the validity of the CFVIE-CDSE by a series of comparisons on the normalized cutoff wavenumbers: we first construct the solutions for obtaining the normalized cutoff wavenumbers in homogeneous gyrotropic waveguides by the separation of variables method (SVM), and second, we employ HFSS commercial software for two-layered isotropic and three-layered gyroelectric loaded waveguides. We characterize the type of modes, i.e., TE/TM or hybrid HE/EH, for each configuration presented, and discuss the efficiency of the CFVIE-CDSE method.