Integral Equation Method Research Articles

An Adaptive Rational Fitting Technique of Sommerfeld Integrals for the Efficient MoM Analysis of Planar Structures

The integral equation method is one of the most successful computational models for microwave devices or integrated circuits in planar layered media. However, the efficient and accurate evaluation of the associated Green’s function consisting of Sommerfeld integrals (SIs) is still a remaining challenge. To mitigate this difficulty, this work proposes a spatial domain rational function fitting technique (RFFT) for SIs so that the approximation accuracy is controllable. In conjunction with an adaptive sampling strategy, the proposed RFFT minimizes the orders of rational functions, and the resultant SI evaluation efficiency is optimized. In addition, we investigate the semi-analytical singularity treatment for the rational expression of SIs in method of moment (MoM) implementation. Extensive simulation of representative planar devices validates the correctness of the proposed method and demonstrates its superior performance over conventional SI approximation methods.

Protein 3D Hydration: A Case of Bovine Pancreatic Trypsin Inhibitor

Characterization of the hydrated state of a protein is crucial for understanding its structural stability and function. In the present study, we have investigated the 3D hydration structure of the protein BPTI (bovine pancreatic trypsin inhibitor) by molecular dynamics (MD) and the integral equation method in the three-dimensional reference interaction site model (3D-RISM) approach. Both methods have found a well-defined hydration layer around the protein and revealed the localization of BPTI buried water molecules corresponding to the X-ray crystallography data. Moreover, under 3D-RISM calculations, the obtained positions of waters bound firmly to the BPTI sites are in reasonable agreement with the experimental results mentioned above for the BPTI crystal form. The analysis of the 3D hydration structure (thickness of hydration shell and hydration numbers) was performed for the entire protein and its polar and non-polar parts using various cut-off distances taken from the literature as well as by a straightforward procedure proposed here for determining the thickness of the hydration layer. Using the thickness of the hydration shell from this procedure allows for calculating the total hydration number of biomolecules properly under both methods. Following this approach, we have obtained the thickness of the BPTI hydration layer of 3.6 Å with 369 water molecules in the case of MD simulation and 3.9 Å with 333 water molecules in the case of the 3D-RISM approach. The above procedure was also applied for a more detailed description of the BPTI hydration structure near the polar charged and uncharged radicals as well as non-polar radicals. The results presented for the BPTI as an example bring new knowledge to the understanding of protein hydration.

A spectral collocation method for a nonlinear multidimensional Volterra integral equation

AbstractIn this paper, a spectral method is implemented to solve a nonlinear multidimensional Volterra integral equation. Under some condition, the spectral convergence analysis of the solution is investigated in both norm and norm. The results of a numerical example confirm the theoretical prediction.

An Efficient Volume Integral Equation Method for Analysis of Boresight Error of a Radome with Minor Ablation

In this paper, an efficient method based on volume integral equation is developed to analyze the effects of ablation of a radome on the boresight error. To avoid recalculating the whole impedance matrix when the permittivity of the radome or the shape of the top portion is slightly changed due to ablation, the radome is divided into unaffected and affected parts and the volume equivalent current instead of the displacement current is used as the unknown. This permits us to reassemble rather than recalculate the impedance matrix when the ablation condition is altered. Moreover, a viable preconditioning technique is introduced and integrated with the multilevel fast multipole algorithm (MLFMA) to cope with the electrically large antenna-radome system (ARS). Simulation results are provided for the boresight error (BSE) and boresight error slope (BSES) of the ARS at some different ablation states. The present approach is considerably faster than using the conventional methods.

Combined Legendre Spectral-Finite Element Methods for Two-Dimensional Fredholm Integral Equations of the Second Kind

In this paper, we will discuss on the Combined Legendre spectral-Finite element methods (CLSFEM) for the two-dimensional Fredholm integral equations with smooth kernel on the Banach spaces and the corresponding eigenvalue problem. In these methods, the approximated finite dimensional space is the cartesian product of spline space and Legendre polynomial space. The problem is approximated by the CLSFEM using orthogonal projection, which projects from the Banach space into the finite dimensional space. The convergence analysis for both Fredholm integral equations and the corresponding eigenvalue problem will be discussed in both L 2 and norms. The numerical results will be shown to validate the theoretical estimate.

A Review on Fast Direct Methods of Surface Integral Equations for Analysis of Electromagnetic Scattering from 3-D PEC Objects

This paper reviews a series of fast direct solution methods for electromagnetic scattering analysis, aiming to significantly alleviate the problems of slow or even non-convergence of iterative solvers and to provide a fast and robust numerical solution for integral equations. Then the advantages and applications of fast direct solution methods and the research trends are introduced in detail. Three different main methods are discussed, namely hierarchically off-diagonal low-rank matrices (HODLR) and skeletonization direct methods based on weak and strong admissibility condition. Numerical examples of computational complexity and electromagnetic scattering analysis of jet models are presented to demonstrate the efficiency and accuracy of each approach. Finally, a brief discussion is given on the main challenges and possible strategies of fast direct solution methods which still exist.

Application of the Deformation Fracture Criterion to Cracking of Disc Specimens with a Central Narrow Slot

Abstract Using the method of singular integral equations, the elastic-plastic problem for cracked Brazilian disk was solved. Based on the Dugdale model and deformation fracture criterion, the relationships between critical load, notch tip opening displacement and length of the plastic strips were established. Also, the comparison between the present solution for the finite domain and the known solution obtained for the semi-infinite notch in the elastic plane was performed.

New solutions for interfacial capillary waves of permanent form

Progressive capillary waves on the interface between two homogeneous fluids confined in a channel with rigid walls parallel to the undisturbed interface are investigated. This problem is formulated as a system of integrodifferential equations that can be solved numerically via a boundary integral equation method coupled with series expansions of the unknown functions. With this highly accurate scheme and numerical continuation, we explore the global bifurcation of periodic travelling waves. It is found that there are two types of limiting profile, self-intersecting and boundary-touching, which appear either along a primary branch bifurcating from infinitesimal periodic waves or on an isolated branch existing above a certain finite amplitude. For particular sets of parameters, these two types of bifurcation curves can intersect, which can be viewed as a secondary bifurcation phenomenon occurring on the primary branch. Based on asymptotic and numerical analyses of the almost limiting waves, it is found that the boundary-touching solutions feature a circular geometry, i.e. the interface is pieced together by circular arcs of the same radius. A theoretical investigation yields the necessary conditions for the existence of these extreme waves, whereby we can predict the limiting configurations for most parameter sets. The comparisons between theoretical predictions and numerical results show good agreement.

Windowed Green function method for wave scattering by periodic arrays of 2D obstacles

AbstractThis paper introduces a novel boundary integral equation (BIE) method for the numerical solution of problems of planewave scattering by periodic line arrays of two‐dimensional penetrable obstacles. Our approach is built upon a direct BIE formulation that leverages the simplicity of the free‐space Green function but in turn entails evaluation of integrals over the unit‐cell boundaries. Such integrals are here treated via the window Green function method. The windowing approximation together with a finite‐rank operator correction—used to properly impose the Rayleigh radiation condition—yield a robust second‐kind BIE that produces superalgebraically convergent solutions throughout the spectrum, including at the challenging Rayleigh–Wood anomalies. The corrected windowed BIE can be discretized by means of off‐the‐shelf Nyström and boundary element methods, and it leads to linear systems suitable for iterative linear algebra solvers as well as standard fast matrix–vector product algorithms. A variety of numerical examples demonstrate the accuracy and robustness of the proposed methodology.

General Approaches to Solving Problems of Analysis and Synthesis of Directional Properties of Antenna Arrays

The work carried out the calculation and synthesis of antenna arrays used in radio-electronic complexes on unmanned aerial vehicles. The analytical model is designed to find asymptotic estimates of the polarization components of the electric field of the grating in the far zone of the carrier surface; the results obtained with its use are the initial data for constructing a technique for the numerical solution of a boundary value problem for a grating on a carrier surface in the CST MWS electrodynamic simulation environment. The synthesis of gratings with the maximum achievable coefficient of directional action is carried out with the control of radiation patterns at a given set of angles. A two-mirror antenna system has been calculated. It is shown that with an increase in the number of re-reflections taken into account, the convergence of the result for the calculated characteristics of the antenna is observed. To test the proposed method, the same antenna was calculated using the integral equation method. The comparison showed a high degree of agreement between the results obtained by two different methods. The results of the simulation based on a software algorithm designed to quantify the input matching at the input of a multichannel frequency-scanning antenna array power divider are presented. It was found that when performing wide-angle scanning in a relative frequency band of more than a few percent, the disadvantage of the known method for eliminating the normal effect is a sharp deterioration in matching in the lower and upper frequencies of the operating range. A new method is proposed based on an automated iterative process of optimizing the divider geometry, which makes it possible to obtain an acceptable match over the entire operating frequency band. The feasibility of switching from a serial power divider construction scheme to a series-parallel scheme is analyzed for wide-angle scanning in a relative frequency band of about 5%.

A novel high‐order collocation indirect boundary element method based on the Leis formulation for three‐dimensional high frequency exterior acoustic problems

AbstractHigh‐order finite element method (FEM) has recently achieved significant success in solving acoustic problems and has proved to be accurate, robust, and capable of reducing the pollution effect. Inspired by the high‐order FEM, this study presents a novel high‐order collocation indirect BEM for solving three‐dimensional high‐frequency exterior acoustic problems. Due to the difficulty of evaluating hypersingular integrals on curved surface elements with complex density functions, the high‐order BEM scheme for acoustic problems has long been a challenging task, especially for relatively large‐scale problems. In the developed scheme, the Leis formulation is used to overcome the well‐known nonuniqueness problem in the indirect BEM, and an auxiliary integral equation method is presented and employed to regularize the hypersingular integrals in this formulation. The derived regularized boundary integral formulation is mathematically simple and amenable to the elements of arbitrary shape and arbitrary variable interpolation order, and leads to a simple, efficient, and robust algorithm for the calculation of all integrals. Several benchmark examples are investigated to verify the performance of the proposed method. Results demonstrate that the present method is very effective in stimulating the far‐field solution with large wavenumber.

Iterative solver with folded preconditioner for finite element simulation of magnetotelluric fields

We propose an iterative solver with a folded preconditioner for large sparse systems of linear algebraic equations (SLAEs), which are obtained as a result of a vector finite element approximation of electromagnetic fields in magnetotelluric problems. The solver is based on the computational scheme of the Conjugate A-Orthogonal Conjugate Residual method (COCR), diagonal preconditioner, and residual smoothing. To build a finite element approximation, the primary-secondary field approach is used. To construct a preconditioner, we use a matrix, the elements of which are the weights of the representation of gradients of nodal basis functions as a linear combination of edge basis functions. For Dublin Test Model 1 in a wide frequency range from 0.0001 Hz to 10 Hz, we compare the accuracy and computational efficiency of our code with the results of other authors, as well as compare the computational efficiency of the proposed diagonal folded preconditioned (DFP) COCR solver with other solvers. It is shown that the apparent resistivities obtained on the coarsest mesh with our code differ by no more than a few percent from the solution on a finer mesh and from some solutions obtained by other authors using the finite element, finite difference, and integral equation methods. The computational time required for our code is 8 times less than the minimum time presented by other authors (with a corresponding recalculation of the performance of the computers used). It is also shown that the DFP COCR solver practically does not increase the number of iterations at low frequencies and its computational time is about 20 times less than the time required for the COCR solver with a standard diagonal preconditioner. A comparison of the DFP COCR solver with the Parallel Direct Sparse solver (PARDISO) shows that even at low and medium frequencies, the computational time of the DFP COCR solver is about 20 times less than the computational time of the PARDISO, and at high frequencies this difference reaches already 50 times.

An Adaptive Irregular Grid Generation Method for Scattering Simulation of Electrically Large Terrains

This communication provides a novel adaptive irregular grid model (AIGM) to improve the scattering simulation efficiency of electrically large terrains. This model distinguishes the steep and flat regions of the measured topographic point cloud models by a threshold. To determine the optimal threshold, we present the height gradient as the criterion to describe the topographic undulation. The optimal threshold is located at the inflection point of the curve describing topographic similarity in the range of height gradient. This generated AIGM can reduce the grid quantity and maintain consistency with the original point cloud model. It is also valuable in the popular coherent and incoherent (C&I) scattering model of global navigation satellite system reflectometry (GNSS-R). In this communication, the coherent component can be expressed by a simple product of the electric field of the conductor grid, the characteristic function of roughness, and the Fresnel reflection coefficient. The incoherent component comes from the diffuse scattering simulated by the integral equation method (IEM). Compared with the traditional solution to C&I scattering by using the Kirchhoff approximation (KA), the modified C&I model has a more extensive application. The precision of the AIGM-based-C&I model is verified by traditional physical optics (PO). The excellent simulation efficiency of the AIGM-based-C&I model is also demonstrated.

Initial stages of a three dimensional dam break flow

Short time behavior of a three dimensional, gravity-driven free surface flow is studied analytically and numerically. Initially the fluid is at rest, held by a vertical wall. A rectangular section of the wall suddenly disappears and the gravity driven three-dimensional flow starts. In order to describe the flow in the early stage, the potential theory is employed. Viscous effects are ignored for small times. The leading order problem is solved by using the Fourier series method and an integral equation method. Local analysis of the flow field close to the side edges of the rectangular section reveals a square root singularity. The flow velocity is also log-singular at the bottom edge of the rectangular section. In the limiting case, as the width of the rectangular section approaches infinity, the results of the classical two-dimensional dam break flow are recovered. Three dimensional effects become important closer to the side edges of the rectangular section.

Acoustic scattering from dynamic rough ocean surfaces using finite-difference time-domain modeling techniques

Established models for underwater acoustic propagation and scattering typically assume the sea-surface to be either perfectly smooth or rough but static in time. When the sea-surface is rough and moving, received signals show anomalies such as additional transmission losses (due to scattering) and Doppler effects (due to surface motion). Understanding the mechanisms behind these anomalies leads to better sonar system designs without having to perform expensive at-sea experiments. The Finite-Difference Time-Domain (FDTD) method is ideal to predict these physical phenomena. This work presents a FDTD method implemented to model the impact of both sea-surface roughness and motion on underwater acoustic propagation. The FDTD method allows an arbitrary function to define the rough sea-surface and its time evolution. The surface characteristics are modeled using a Pierson-Moskowitz (PM) frequency spectrum. The PM surface model is simple to implement and fully defined by wind speed and direction. Time-domain results from FDTD simulations of static rough sea-surfaces are compared to an established Helmholtz integral equation (HIE) method to establish the validity of the approach. Broadband signals are used as the source waveform. Results demonstrate the anomalous transmission loss and Doppler in received signals. [Work supported by the Office of Naval Research.]

Numerical Methods for Integral Equations of the Second Kind with NonSmooth Solutions of Bounded Variation

Numerical Methods for Integral Equations of the Second Kind with NonSmooth Solutions of Bounded Variation

Solving magnetic induction heating problem with multidimensional Fredholm integral equation methods: Alternative approach for optimization and evaluation of the process performance

Induction heating is a frequently used technology in both fundamental and applied research. It is heavily exploited in the industry for processing materials by heat treatments. In addition, it is viewed as a promising tool in medicine, particularly as a part of therapeutic strategies for treating cancer diseases. Thus, in order to optimize (i.e., enhance and tune) the performance of the induction heating process, several aspects must be considered, including the design of the magnetic coils, features of the magnetic fields applied, coupling of magnetic and thermal fields, and the material’s characteristics. To tackle this complex problem, numerical mathematical models are often used. The results of which can help in understanding the role of the various parameters on the performance of the induction heating. Here, we present an alternative mathematical approach to solve the induction heating problem using Fredholm integral equations of the second kind with a singular kernel. To reduce the computation time, the Nyström method has been adopted. As the kernel function shows a singularity, a singularity subtraction has been involved in the developed mathematical procedure. Furthermore, the error features of the Nyström method with the singularity subtraction have been described, and convergence conditions of the proposed computational algorithm have been thoroughly identified. Although special conditions for the kernel function and the integration rule are needed, the method shows lower computing times, competing well with those of traditional finite-element based routines. The applicability of the developed methodology is demonstrated for the simulation of induction heating the body of a metal object.

Sound field reconstruction in urban environments: Application to enhance mapping of urban microspaces

In this work, reconstruction techniques for the spatial interpolation and extrapolation of sound fields in urban environments are presented. Gaussian processes are generally used for sound field reconstruction from limitedly observations of isotropic acoustic fields. However, this model is often not applicable for the anisotropic urban environments including urban street canyons and enclosed spaces, when the complexity of the sound field is high in the mid-frequency regime, unless diffusely reflecting boundaries are assumed. Two different techniques are compared for reconstructing the sound field: the least-squared method and the Kirchhoff-Helmholtz integral equation method. Of particular interest is the reconstruction of the sound field with a minimal number of irregularly and arbitrarily distributed microphone measurements. Therefore, the techniques will not require knowledge of the microphone positions. A successive series approximation approach is presented to enhance the microscale prediction of the Kirchhoff-Helmholtz integral equation method. The sound field reconstruction results from limited urban environment observations for both methods are presented and discussed.

Ray Optical Scattering From Uniform Reflective Cylindrical Metasurfaces Using Surface Susceptibility Tensors

A ray optical methodology based on the uniform theory of diffraction (UTD) is proposed to model electromagnetic (EM) field scattering from curved metasurfaces (MSs). The problem addressed is the illumination of a purely reflective uniform cylindrical MS by a line source, models the surface with susceptibilities and employs a methodology previously used for cylinders coated in thin dielectric layers [Kim and Wang (1989)]. The approach is fundamentally based on a representation of the MS using the generalized sheet transition conditions (GSTCs) which characterizes the surface in terms of susceptibility dyadics. An eigenfunction (EF) description of the MS problem is derived considering both tangential and normal surface susceptibilities, and used to develop a ray optics (RO) description of the scattered fields including the specular geometrical optical field, surface diffraction described by creeping waves and a transition region over the shadow boundary. The specification of the fields in the transition region is dependent on the evaluation of the Pekeris caret function integral and the method follows [Kim and Wang (1989)]. The proposed RO-GSTC model is then successfully demonstrated for a variety of cases and is independently verified using a rigorous EF solution (EF-GSTC) and full-wave Integral Equation method (IE-GSTC), over the entire domain from the deep lit to deep shadow.

Контактная задача об осесимметричном кручении упругого слоя посредством цилиндрического штампа

The paper studies the axis-symmetric contact problem between an elastic layer and a rigid cylindrical circular stamp under torque. The stamp adheres to the upper boundary of the layer whereas the lower boundary of the layer is rigidly fastened. With the application of Hankel integral transform solving the problem reduces to solving the first kind Fredholm integral equation (IE) with symmetrical kernel, represented as a sum of its principal part, Weber-Sonin integral, and the regular kernel. It is estimated that once its height attains a certain level, the layer actually deforms as a semi-space. In the process, through Abel IE method, the solution of the well-known Reissner-Sagoci problem is obtained once again and the original first kind Fredholm IE is reduced to the second kind Fredholm IE. Concurrently, using the collocation method combined with Gauss type quadrature formulas for integral estimation, the original IE reduces to a finite system of linear algebraic equations. To obtain this quadrature formula, properties of Gegenbauer and Chebyshev orthogonal polynomials are used. In the enough wide range of change of characteristic elastic and geometrical parameters of the problem numerical analysis is performed and patterns of changes of tangential contact stresses under the stamp as well as the angle of twist of the stamp are identified.