Integral Equation Method Research Articles

An alternating iterative method for solving the Cauchy problem for the Helmholtz equation in interior multiply connected domain

This paper is concerned with the Cauchy problem for the Helmholtz equation in interior multiply connected domain. We propose an alternating iterative method to solve the Cauchy problem and prove the convergence of the iterative procedure by using fixed point theory. To solve the two well-posed auxiliary problems in each iteration, we employ the boundary integral equation method. The effectiveness of our iterative method is demonstrated by some numerical tests.

An analytical Green’s function for Laplace operator in an infinite plane with two circular holes using degenerate kernels

In this paper, Green’s function problem of an infinite plane containing two circular boundaries is analytically studied by using the boundary integral equation method (BIEM). The original problem is decomposed into two parts by introducing the superposition technique. The first part is a free field caused by the concentrated force. The second part is a Laplace problem subjected to the corresponding boundary conditions. The second part can be solved by using the null-field boundary integral equation in conjunction with the degenerate kernel. Since the geometry of problem under study is composed by two circular boundaries, the kernel function is naturally expanded to series form in terms of the bipolar coordinates. The Green’s function is analytically expressed in three regions instead of two regions for the eccentric domain. To the authors’ best knowledge, this is the first time that the Green’s function of the problem is presented in this form. To show the validity of the present method, the contour result of the present method is compared with those obtained by the image method, the null-field BIE in conjunction with the adaptive observer of polar degenerate kernel.

Inverse Problems for the Equation of Vibrations of a Canister Beam to Find the Source

For the beam vibration equation, inverse problems are studied to find the right side, i.e. vibration source. Solutions of the problems by methods of spectral analysis and Volterra integral equations are constructed explicitly as sums of series, and the corresponding uniqueness and existence theorems are proved. When substantiating the existence of a solution to the inverse problem by determining the factor of the right-hand side, which depends on the spatial coordinate, the problem of small denominators arises. In this regard, estimates of the denominators are established that guarantee their separation from zero, with an indication of the corresponding asymptotics. On the basis of these estimates, the convergence of the series in the class of regular solutions of the beam oscillation equation is substantiated.

A Boundary Integral Equation Method for Computing Numerical Conformal Mappings onto the Disk with Rectilinear Slit and Spiral Slits Regions

This article proposes a boundary integral equation method for computing numerical conformal mappings of bounded multiply connected region Ω onto the disk with rectilinear slit and spiral slits regions, Ω1 and Ω2 Initially, the process involves calculating the boundary value of the canonical region. Cauchy’s integral formula can then be used to compute the mapping of the interior values. The effectiveness of the proposed method is demonstrated using several numerical examples.

On optimal radius of sub-domains in meshless LBIE method

Local weak based meshless methods construct weak form of governing equations on local sub-domains. In two dimensional domains, for the simplicity of computations, these sub-domains are taken as circles. In these methods, the optimal radius of sub-domains has been an open problem yet. This paper aims at solving this problem for meshless local boundary integral equation (LBIE) method to enhance its performance. It is proved that a sub-domain for which the Lebesgue constant takes its minimum over its boundary is the optimal sub-domain. In other words, the optimal sub-domain is one for which the solution of PDE is approximated on its boundary as accurate as possible. A comprehensive numerical study confirms the theory.

Numerical calculation of regular and singular integrals in boundary integral equations using Clenshaw–Curtis quadrature rules

The efficient and accurate calculation of integrals, especially singular and nearly singular integrals, poses a great challenge in boundary integral equation (BIE) methods. In this paper, the Clenshaw–Curtis integration technique is proposed to calculate both regular and singular integrals in the BIE in a unified way. Taking the meshless discretization of two-dimensional Helmholtz BIE by linear integration cell as an example, four types of Clenshaw–Curtis quadrature rules are established for regular, weakly singular, hypersingular, and nearly hypersingular integrals. Explicit expressions of integration weights in all quadrature rules are derived, and the degree of precision of these quadrature rules is analyzed theoretically. Integration points in all quadrature rules are the same, which is very useful for meshless analysis of BIEs. The values of integration points and weights for regular, weakly singular and hypersingular integrals are tabulated in the interval [−1, 1], which can be directly used to facilitate the practical application of BIE methods. Numerical results are provided to verify the effectiveness of these Clenshaw–Curtis quadrature rules.

Microwave Imaging of 3-D Dielectric–Magnetic Penetrable Objects Based on Integral Equation Method

Some objects like mineral substances may be both dielectric and magnetic and their high-resolution inner imaging is very desirable when detecting and analyzing their ingredients. An integral equation method is developed for reconstructing or imaging such objects by using microwave illumination. Since the objects are both dielectric and magnetic, full volume integral equations (VIEs) are needed to describe the problem. The objects can be reconstructed by alternatively solving the forward scattering VIEs (FSVIE) and the inverse scattering VIEs (ISVIEs) under the Born iterative method (BIM) or distorted BIM (DBIM). The Nyström method is used to solve the FSVIEs while a Gauss-Newton minimization method (GNMM) with a multiplicative regularization scheme (MRS) is employed to solve the ISVIE. The Nyström method is more suitable for solving the FSVIEs of inverse problems because it is a point-matching method. To accelerate the solution, the multilevel fast multipole algorithm (MLFMA) is incorporated with the Nyström method which is new for solving inverse problems. Also, using the MRS-based GNMM to solve the ISVIEs can facilitate the selection of regularization parameter and reduce the requirements on measured data. Numerical examples are provided to demonstrate the approach and good results have been achieved.

Модифицированный алгоритм трехмерной модели векторного магнитного гистерезиса применительно к методу пространственных интегральных уравнений

The paper proposes mathematical models of the scalar and vector magnetic hysteresis to solve the problems of calculating the three-dimensional magnetic field in electrical devices by the method of spatial integral equations. A system of spatial integral equations is presented describing processes in the electrical devices. Basic and modified algorithms of the Jiles --- Atherton scalar model implementation are considered. The model parameters were tuned using a genetic algorithm. Algorithms for constructing the Jiles --- Atherton direct and inverse vector model in the three-dimensional setting were presented in accordance with the Mayergoyz approach. Generalized model of the vector 3D hysteresis makes it possible to describe the nonlinear properties of ferromagnetic medium for the isotropic and anisotropic materials. An example of the magnetic system calculation is provided taking into account the vector magnetic hysteresis. The simulation model parameters were determined from experimental data of the limited scope. Adequacy of the computational models was checked using a test problem. Results of the numerical studies demonstrated that the models were quite accurately reproducing real hysteresis dependences for various ferromagnetic materials. Proposed models and numerical algorithms implementing them were used as the component models that specified nonlinear characteristics of the magnetic materials in a software package that implemented the spatial integral equations method

Расчет двузеркальной антенны c учётом влияния опор строгим методом

Calculation of a double reflector Cassegrain antenna with taking into account of struts was performed by electric field integral equation method with unknown surface electric current and physical optics method with multiple reflections. For the same Cassegrain antenna with circular polarization we compared characteristics (gain, efficiency, side-lobes) at frequencies 1.7 GHz, 3.4 GHz and 5.1 GHz for the structure with struts (quadripod) and without struts. Calculation of antenna characteristics is performed for variable diameter of quadripod at main reflector at frequencies 1.7 GHz and 5.1 GHz. Calculation by rigorous method showed increase of side-lobes in the planes of struts and increase of the cross-polarization. With this in mind, the side-lobes in diagonal planes are approximately the same as in structure without struts. Comparison of gain reduction due to the struts is done, when calculated by rigorous method and aperture method.

Dynamic Green’s Functions for an Infinite Acoustic Field with Multiple Spheres Subjected to the Robin Boundary Conditions

A semi-analytical approach is presented to solve three-dimensional dynamic Green’s function for an infinitely extended acoustic field with multiple spheres subjected to the Robin boundary conditions. The multipole expansions of the acoustic field induced by a time-harmonic point source are expanded with spherical wave functions. As an alternative to the complex addition theorem, the multipole expansion is computed in a straightforward way. By taking the finite number of terms, an algebraic system is constructed and is used to obtain Green’s function. This result of one sphere agrees with the available analytical solution. For the case of more than one sphere, the proposed results are verified by the numerical method such as the boundary element method (BEM). It indicates that the present solution is more accurate than that of the BEM and shows a fast convergence. Finally, the parameter study is performed to explore the influences of the exciting frequency of the point source, the surface admittance, the number and the separation of spheres on the dynamic Green’s functions. The proposed results can be applied to solve the acoustic scattering problems and to increase the application of boundary integral equation method in the way of numerical Green’s functions.

An Integral Equation Method for the Scattering by Core-Shell Structures in a Layered Medium

An Integral Equation Method for the Scattering by Core-Shell Structures in a Layered Medium

3D tomographic inversion for frequency semi-airborne EM under multinary constraints

The semi-airborne electromagnetic method (SAEM) has recently gained increased interest in geophysical exploration applications. For SAEM, ground-based transmitters are deployed within the area of interest or in its vicinity, and the induced magnetic is measured airborne. Because of the large footprint of extended transmitters, three-dimensional (3D) effects can have a significant impact on the gathered data and 3D modelling and inversion approaches must be employed for data analysis. Any 3D EM simulation is a resource-consuming task and efficient tools can therefore be useful to add data interpretation. In this study, 3D tomographic inversion of frequency-domain SAEM data was accomplished based on the integral equation (IE) method. The inversion was linearised by approximating the sensitivity matrix. We further employ the quasi-linear approximation to facilitate out quick forward modelling in the inversion which is tractable on field computers. While this approach sacrifices inversion accuracy for efficiency, we guide the inversion by introducing multinary constraints. These constraints impose a priori known conductivity values into the inversion process. Our results from synthetic and field data demonstrate the efficiency of our approach in achieving a coherent geophysical model that is consistent with geological data and findings from other geophysical techniques. This method provides a practical and capable solution for unveiling conductive 3D structures in the field.

Analysis of near-interface cracks in three-dimensional anisotropic multi-materials by efficient BIEM

This paper presents the full analysis of near-interface cracks in three-dimensional linear elastic multimaterial bodies by a weakly singular boundary integral equation method (BIEM). The proposed technique was implemented in a general framework that allowed the treatment of finite cracked bodies with general configurations, material anisotropy, and mode-mixity. A system of integral equations governing unknown data on the boundary, crack surface, and material interface was established using a pair of weakly singular weak-form displacement and traction integral equations together with continuity along the material interface. A symmetric Galerkin boundary element method along with the finite element technique were implemented to solve the governing integral equations. In addition, the special near-front interpolation was employed to enhance the approximation of the relative crack-face displacement in the neighborhood of the crack front. The solved crack-face data were used directly to post-process the stress intensity factors and T-stresses along the crack front. An extensive numerical study was carried out not only to demonstrate the accuracy, convergence, and capability of the proposed technique, but also to explore the influence of material stiffness and distance to the material interface on fracture data along the crack front.

GPU acceleration of Nystrom type method for solving singular integral equations in H-polarized EM waves scattering by strip gratings

Abstract In this paper, we propose the use of the graphics processing units (GPUs) for acceleration of solution of the H-polarized electromagnetic waves scattering by strip gratings. The approach combines the boundary integral equation method with the Nystrom-type discretization of singular integral equations and is realized with the use of C++ and OpenCL. The elements of the matrix of the resulting system of equations can be calculated independently and, thus, in parallel. The calculation time of the proposed parallelized algorithm on GPU is compared to standard serial algorithm executed on CPU. We considered gratings of graphene strips in vacuum and inside a dielectric slab. The acceleration is up to 35–55 times for the gratings in vacuum and up to 80–140 times for the gratings of 5–20 strips inside the dielectric slab.

Study of stress wave propagation across a non-persistent joint based on a boundary integral equation method

Studies on stress wave propagation across persistent joints have been conducted extensively. Nevertheless, there exists a consensus that non-persistent joints are widely and densely distributed, which have a profound impact on wave propagation in jointed rock masses. A boundary integral equation method is suggested in this paper to investigate the characteristics of transmitted wave field for the case of stress wave propagation across a single non-persistent joint. The displacement continuity and discontinuity boundaries are combined in the method. The method presented in the current study is applicable to the analysis of wave propagation across non-persistent joints with arbitrary incident angles. Then, taking a single non-persistent joint arranged with only one joint segment as an example, the applicability of the method in dealing with the problems of wave propagation is verified by comparing the results with those from the discrete element method and analytical methods. Subsequently, parametric studies are carried out, including the effects of joint-segment length, rock-bridge length, wave frequency and incident angle on the transmitted wave. The result indicates that the existence of non-persistent joint makes the transmitted displacement field different from that of persistent joint, because the scattered wave is produced during the process of wave propagation. The displacement amplitude may be amplified evidently in some regions and the spatial distribution pattern of the transmission coefficient is closely related to the joint-segment length, rock-bridge length and incident wavelength.

Thermoelastic instability on a frictional surface and its implication for size effect in friction experiments

Recent laboratory friction experiments on large rock samples revealed that dynamic weakening, a remarkable reduction in the friction coefficient at elevated slip rates, occurs at lower slip rates in larger samples. There is a large difference between the sizes of natural faults and those in laboratory experiments. Therefore, it is crucial to understand the effect of size on rock friction. In the field of tribology, the interaction between frictional heating and thermoelastic effect has long been investigated. It was shown that higher slip rates than the critical value {V}_{mathrm{cr}} causes growth of temperature and normal stress heterogeneity (thermoelastic instability), and {V}_{mathrm{cr}} is proportional to the wavenumber of the heterogeneity. Severely heterogeneous normal stress may cause concentration of frictional power, thus locally activating dynamic weakening and leading to macroscopic weakening. Because a larger sample hosts a perturbation of a smaller wavenumber, it is expected to weaken at a lower slip rate than a smaller sample. In this study, a new numerical method was developed for analysis of thermoelastic instability based on the definition of memory variables and numerical approximation to the integration kernel, for the 2-dimensional problem of a planar fault embedded in an infinite medium. This method was advantageous over the standard integral equation method in terms of numerical costs. Numerical solutions with the new method on sinusoidal perturbations in the normal stress were compared with previously derived steady-state solution and its stability for validation. The typical thermoelastic properties of gabbro yield {V}_{mathrm{cr}} in a range of experimentally adopted slip rates, indicating that the thermoelastic effect may play an important role in high-velocity friction experiments. Because the temperature rise and the resulting normal stress change smear out after the friction experiments, measurement of the temperature distribution in a sample during a friction experiment is important for further understanding the dynamic weakening and scale effect of rock friction.Graphical

Boundary-Value Problem for a Loaded Mixed-Type Equation with a Characteristic Line of Type Change

We prove the unique solvability of the boundary-value problem for a loaded third-order integrodifferential equation with parabolic-hyperbolic operator. By the method of integral equations, we prove the existence and uniqueness of solutions of boundary-value problems.

Accurate and Robust Surface MLFMA Solution for Scattering From a Large Plasma Object With Large Negative Permittivity

We present an accurate and robust surface integral equation (SIE) method for electromagnetic analysis of large-scale plasma media with large negative permittivity. To accelerate the matrix-vector multiplication (MVM) in the iterative solution of the matrix equation, the multilevel fast multipole algorithm (MLFMA) is generally preferred. Traditional implementation of MLFMA will lead to inaccurate results or poor convergence due to the numerical instability arising from the large negative permittivity. To tackle this problem, we analyze the formula of MLFMA in detail and propose a simple and efficient truncation strategy to improve its robustness. A practical and elegant truncation bound is derived based on the ratios of matrix elements between an ordinary permittivity background medium and the negative permittivity plasma. Applying the proposed truncation strategy, the performance of several commonly used SIE formulations is investigated comprehensively. Numerical results demonstrate that the proposed method can avoid numerical instability while maintaining high efficiency for large-scale plasma media with large negative permittivity.

A High-Order-Accurate 3D Surface Integral Equation Solver for Uniaxial Anisotropic Media

This paper introduces a high-order accurate surface integral equation method for solving 3D electromagnetic scattering for dielectric objects with uniaxially anisotropic permittivity tensors. The N-Müller formulation is leveraged resulting in a second-kind integral formulation, and a finite-difference-based approach is used to deal with the strongly singular terms resulting from the dyadic Green’s functions for uniaxially anisotropic media while maintaining the high-order accuracy of the discretization strategy. The integral operators are discretized via a Nyström-collocation approach, which represents the unknown surface densities in terms of Chebyshev polynomials on curvilinear quadrilateral surface patches. The convergence is investigated for various geometries, including a sphere, cube, a complicated NURBS geometry imported from a 3D CAD modeler software, and a nanophotonic silicon waveguide, and results are compared against a commercial finite element solver. To the best of our knowledge, this is the first demonstration of high-order accuracy for objects with uniaxially anisotropic materials using surface integral equations.

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ ДИНАМИЧЕСКИХ ЗАДАЧ МЕТОДОМ ГРАНИЧНЫХ УРАВНЕНИЙ.

The article describes the process of mathematical modeling of the dynamic problem of the theory of elasticity by the method of boundary integral equations. As a result of applying successive approximation, a system of algebraic equations is obtained, which is solved by a step method. The developed algorithm makes it possible to study the stress-strain state of a two-dimensional problem of the theory of elasticity from various dynamic influences.