System Of Boundary Integral Equations Research Articles

On the solution of direct and inverse multiple scattering problems for mixed sound-soft, sound-hard and penetrable objects

In this work we consider a scattering problem governed by the two-dimensional Helmholtz equation, where some objects of different nature (sound-hard, sound-soft and penetrable) are present in the background medium. First we propose and analyze a system of boundary integral equations to solve the direct problem. After that, we propose a numerical method based on the computation of a multifrequency topological energy based imaging functional to find the shape of the objects (without knowing their nature) from measurements of the total field at a set of observation points. Numerical examples show that the proposed indicator function is able to detect objects of different nature and/or shape and size when processing noisy data for a rich enough range of frequencies.

Method of Boundary Integral Equations in the Problem of Diffraction of a Monochromatic Electromagnetic Wave by a System of Perfectly Conducting and Piecewise Homogeneous Dielectric Objects

We consider the three-dimensional problem of diffraction of a monochromatic electromagnetic wave by a system of objects of various physical nature, including dielectric bodies (domains), perfectly conducting bodies, and perfectly conducting screens. The perfectly conducting bodies may be placed in an exterior medium or immersed into the dielectric domains. In addition, the perfectly conducting screens may reside at the interface of the dielectric domains, being part of each of those. We give a boundary value problem for the Maxwell equations that describes the electromagnetic field under consideration. Integral representations are derived for the electromagnetic field in terms of surface integrals, and the boundary value problem is reduced to a system of boundary integral equations containing weakly and strongly singular surface integrals. The integral equations are written on the dielectric and perfectly conducting parts of the interface between the dielectric domains, on the surfaces of the perfectly conducting bodies, and on the perfectly conducting screens.

Inverse Problem of Identifying a Small Defect Based on an Asymptotic Method

We consider the inverse problem of reconstructing a cavity of small characteristic size in an orthotropic layer based on information about the field of layer surface displacements measured within the framework of frequency sensing. Resolving equations in the inverse problem are based on the system of boundary integral equations formulated only along the cavity boundary. In the case of antiplane oscillations, based on an asymptotic approach, we derive formulas for characterizing a small defect. Results of computational experiments are presented.

Boundary integral equations in the frequency domain for interface linear cracks under impact loading

The linear crack between two dissimilar elastic isotropic half-spaces under normal pulse loading is considered. The system of boundary integral equations for displacements and tractions in the frequency domain is derived from the dynamic Somigliana identity and adapted to solve the problem in the time domain. The numerical convergence of the method with respect to the number of the Fourier coefficients is proved. The effects of material properties of the bimaterial on the distribution of stress intensity factors (opening and transverse shear modes) are presented and analysed.

On the solution of Laplace’s equation in the vicinity of triple junctions

In this paper we characterize the behavior of solutions to systems of boundary integral equations associated with Laplace transmission problems in composite media consisting of regions with polygonal boundaries. In particular we consider triple junctions, i.e. points at which three distinct media meet. We show that, under suitable conditions, solutions to the boundary integral equations in the vicinity of a triple junction are well-approximated by linear combinations of functions of the form $t^\beta,$ where $t$ is the distance of the point from the junction and the powers $\beta$ depend only on the material properties of the media and the angles at which their boundaries meet. Moreover, we use this analysis to design efficient discretizations of boundary integral equations for Laplace transmission problems in regions with triple junctions and demonstrate the accuracy and efficiency of this algorithm with a number of examples.

Numerical solutions of the forward and inverse problems arising in diffuse optical tomography

Optical tomography is a typical non-invasive medical imaging technique, aiming to reconstruct the optical properties of tissues from boundary measurements by passing near infrared light through the tissues. We are concerned with the forward and inverse problems for a time-dependent diffusion equation model arising in diffuse optical tomography. To solve the forward problem, we develop a boundary integral equation method, and then derive a discretization scheme for the resulting system of boundary integral equations. As for the inverse problem, rather than the precise values of the optical parameters, we recover geometric information on unknown inclusions as anomalies where the optical properties are significantly different from the background medium. A noniterative reconstruction method with its rigorous justification is established. Finally, we present some numerical results that illustrate the efficiency and effectiveness of the proposed methods for both the forward and inverse problems.

Asymptotic Solution to Scattering Problem on a Set of Small Particles and Application to Creating the Materials with a Peculiar Refractive Index

The asymptotic solution to the scattering problem on a set of small particles, supplemented into homogeneous material, is used for modeling the materials with the desired refractive index. The consideration concerns the case of acoustic scalar scattering and the solution to initial scattering problem is built using an asymptotic approach. The closed form solution is reduced for the scattering problem. This is significant advantage of approach because there is no need to solve the respective system of boundary integral equations. High accuracy of solving the scattering problem is achieved by choosing the optimal parameters of the domain with small particles. The approach allows obtaining an explicit formula for the refractive index of the resulting inhomogeneous material. The numerical calculations show the possibility to get the specific values of refractive index including its negative values.

A double layer potential approach for planar Cauchy problems for the Laplace equation

We derive a double-layer potential approach which recasts the ill-posed Cauchy problem for the Laplace equation in a planar bounded doubly-connected domain as a system of boundary integral equations. It is shown that the system has a unique solution for a dense set of data. For the integral equations involved having a hypersingular kernel the Nystr\"om method based on trigonometrical quadratures is applied. The resulting fully discrete linear system is solved by Tikhonov regularization. Results included of numerical experiments show that a stable approximation can be obtained with the double-layer method to the solution of the Cauchy problem.

Boundary integral approach for the mixed Dirichlet–Robin boundary value problem for the nonlinear Darcy–Forchheimer–Brinkman system

The purpose of this paper is to provide the solvability and uniqueness result to the mixed Dirichlet–Robin boundary value problem for the nonlinear Darcy–Forchheimer–Brinkman system in a bounded, two-dimensional Lipschitz domain. First we obtain a well-posedness result for the linear Brinkman system with Dirichlet–Neumann boundary conditions, by reducing the problem to the system of boundary integral equations based on the fundamental solution of the Brinkman system and by analyzing this system employing a variational approach. The result is extended afterwards to the Poisson problem for the Brinkman system and to Dirichlet–Robin boundary conditions, using the Newtonian potential and the linearity of the solution operator. Further, we study the nonlinear Darcy–Forchheimer–Brinkman boundary value problem of Dirichlet and Robin type.

Чисельний аналіз деформації в’язкого тіла під діею сил поверхневого натягу

Among the boundary value problems of hydrodynamics of viscous fluid, an important from a practical point of view the class of tasks in the region with a free (previously unknown) boundary, which is determined in the process of direct solving, is distinguished. One of the possible approaches to solving such problems in such an area is a method of hydrodynamic potentials, which translates the basic complexity of research and numerical calculations into certain boundary integral equations, which relate only to the boundary of a given region and take into account the boundary conditions directly. This approach allows you to immediately identify unknown values at the boundary without calculating them in the entire area. This advantage distinguishes the method of boundary integral equations from other methods. The aim of the research was to develop a method for numerical solution of the problems of the viscous fluid movement with a previously unknown boundary. The problem of deformation of a viscous elliptical cylinder under the action of forces of surface tension is formulated and solved in the article. The boundary integral equations are used, which are considered together with the kinematic contour condition. An algorithm for the steps of the numerical solution of the system of boundary integral equations for the problem of viscous fluid with a previously unknown boundary was implemented and computational features were analyzed. On the basis of the conducted research the regularities of the deformation process of the viscous elliptical cylinder under the influence of the forces of surface tension, depending on the viscosity of the liquid and the coefficient of surface tension , were established. A convex contour with continuous curvature deforms into a circle, carrying out fading oscillations around the equilibrium position. Zones of stable synchronous oscillations of points of the contour around the equilibrium position are observed, if , and when , ; in both cases, the density of the liquid was taken equal to one. The liquid body reaches the equilibrium position without oscillation, if . Key words: nucleus, contour discretization interval, curvature, value of the Cauchy integral, surface tension forces, elliptic cylinder

Methods of Potential Theory in a Filtration Problem for a Viscous Fluid

Integral representations for the velocity and pressure fields are constructed in the problem on a three-dimensional filtration flow of a viscous fluid obeying the Darcy-Brinkman law in a piecewise homogeneous medium. This problem is reduced to a system of boundary integral equations.

Wave propagation from lateral Cauchy data using a boundary element method

The wave equation with lateral Cauchy data is considered in three-dimensional annular domains. It is a linear inverse ill-posed problem, where data in the form of the function values and normal derivative is given on the outer boundary surface. To reconstruct the missing data on the inner surface a recent boundary integral approach is applied and extended to three-dimensions. A transformation in time (semi-discretization) is employed via the Laguerre transform resulting in a Cauchy problem for a system consisting of a sequence of inhomogeneous elliptic equations. The solution to this system is represented as a sequence of single-layer potentials invoking a fundamental sequence of the elliptic equations, with the benefit of avoiding the use of any volume potentials or domain discretization. Matching the data against the single-layer representation of the solution gives a system of boundary integral equations to be solved for a sequence of densities. A Galerkin boundary element discretization is applied to this latter system, triangulating the boundary surfaces and using piecewise constant approximating functions. Tikhonov regularization is used when solving the sequence of discrete linear equations. Numerical examples show the feasibility of the proposed approach rendering accurate and stable approximations of the missing lateral data with little computational effort.

The scattering waves from an object in an elastic‐fluid layered structure

AbstractIn this work, we have investigated the scattered elastic waves from an object in a system consisting of an elastic layer above a fluid half‐space. A time‐harmonic load is applied upon the upper boundary surface, while the object is free of stresses. By constructing the corresponding Green's functions, a coupled system of boundary integral equations (BIE) over the contour of the object is formulated. The scattered waves are obtained by using the collocation technique which reduced the BIE to a system of linear algebraic equations. The results are compared with the case without the fluid to show the influence of the fluid pressure on the scattered waves. Numerical examples are carried to demonstrate the scattered waves from an object in different geometrical shapes.

Combination of the Laguerre Transform with the Boundary-element Method for the Solution of Integral Equations with Retarded Kernel

We apply the Laguerre transform with respect to time to a time-dependent boundary-value integral equation encountered in the solution of three-dimensional Dirichlet initial-boundary-value problems for the homogeneous wave equation with homogeneous initial conditions by using the retarded potential of single layer. The obtained system of boundary integral equations is reduced to a sequence of Fredholm integral equations of the first kind that differ solely by the recursively dependent right-hand sides. To find their numerical solution, we use the boundary-element method. We establish an asymptotic estimate of the error of numerical solution and present the results of numerical simulations aimed at finding the solutions of retarded-potential integral equations for model examples.

Shape optimization of conductive-media interfaces using an IGA-BEM solver

In this paper, we present a method that combines the Boundary Element Method (BEM) with IsoGeometric Analysis (IGA) for numerically solving the system of Boundary Integral Equations (BIE) arising in the context of a 2-D steady-state heat conduction problem across a periodic interface separating two conducting and conforming media. Our approach leads to a fast solver with high convergence rate when compared with low-order BEM. Additionally, an optimization framework comprising a parametric model for the interface’s shape, our IGA-BEM solver, and evolutionary and gradient-based optimization algorithms is developed and tested. The optimization examples demonstrate the efficiency of the framework in generating optimum interfaces for maximizing heat transfer under various geometric constraints.

A boundary integral equation method for numerical solution of parabolic and hyperbolic Cauchy problems

We present a unified boundary integral approach for the stable numerical solution of the ill-posed Cauchy problem for the heat and wave equation. The method is based on a transformation in time (semi-discretisation) using either the method of Rothe or the Laguerre transform, to generate a Cauchy problem for a sequence of inhomogeneous elliptic equations; the total entity of sequences is termed an elliptic system. For this stationary system, following a recent integral approach for the Cauchy problem for the Laplace equation, the solution is represented as a sequence of single-layer potentials invoking what is known as a fundamental sequence of the elliptic system thereby avoiding the use of volume potentials and domain discretisation. Matching the given data, a system of boundary integral equations is obtained for finding a sequence of layer densities. Full discretisation is obtained via a Nyström method together with the use of Tikhonov regularization for the obtained linear systems. Numerical results are included both for the heat and wave equation confirming the practical usefulness, in terms of accuracy and resourceful use of computational effort, of the proposed approach.

A boundary integral equation for the transmission eigenvalue problem for Maxwell equation

We propose a new integral equation formulation to characterize and compute transmission eigenvalues in electromagnetic scattering. As opposed to the approach that was recently developed by Cakoni, Haddar and Meng (2015) which relies on a two‐by‐two system of boundary integral equations, our analysis is based on only one integral equation in terms of the electric‐to‐magnetic boundary trace operator that results in a simplification of the theory and in a considerable reduction of computational costs. We establish Fredholm properties of the integral operators and their analytic dependence on the wave number. Further, we use the numerical algorithm for analytic nonlinear eigenvalue problems that was recently proposed by Beyn (2012) for the numerical computation of the transmission eigenvalues via this new integral equation.

Micro-structured materials: Inhomogeneities and imperfect interfaces in plane micropolar elasticity, a boundary element approach

In this paper we tackle the simulation of microstructured materials modelled as heterogeneous Cosserat media with both perfect and imperfect interfaces. We formulate a boundary value problem for an inclusion of one plane strain micropolar phase into another micropolar phase and reduce the problem to a system of boundary integral equations, which is subsequently solved by the boundary element method. The inclusion interface condition is assumed to be imperfect, which permits jumps in both displacements/microrotations and tractions/couple tractions, as well as a linear dependence of jumps in displacements/microrotations on continuous across the interface tractions/couple traction (model known in elasticity as homogeneously imperfect interface). These features can be directly incorporated into the boundary element formulation.The BEM-results for a circular inclusion in an infinite plate are shown to be in an excellent agreement with the analytical solutions. The BEM-results for inclusions in finite plates are compared with the FEM-results obtained with FEniCS.

Nonlinear integral equations for Bernoulli’s free boundary value problem in three dimensions

In this paper we present a numerical solution method for the Bernoulli free boundary value problem for the Laplace equation in three dimensions. We extend a nonlinear integral equation approach for the free boundary reconstruction (Kress, 2016) from the two-dimensional to the three-dimensional case. The idea of the method consists in reformulating Bernoulli’s problem as a system of boundary integral equations which are nonlinear with respect to the unknown shape of the free boundary and linear with respect to the boundary values. The system is linearized simultaneously with respect to both unknowns, i.e., it is solved by Newton iterations. In each iteration step the linearized system is solved numerically by a spectrally accurate method. After expressing the Fréchet derivatives as a linear combination of single- and double-layer potentials we obtain a local convergence result on the Newton iterations and illustrate the feasibility of the method by numerical examples.

Numerical solution of the Neumann boundary value problem for an infinite convolutional system of elliptic equation via the boundary elements method

We consider a Neumann boundary value problem for an infinite convolutional system of elliptic equations obtained as a result of the application of the Laguerre transform in the time domain to initial-boundary value problems for evolution equations in 3d Lipschitz domains. With help of q-convolution of sequences the integral representation of the generalized solution of the problem is built and an equivalent system of boundary integral equations is obtained. The existence and uniqueness of the numerical solution obtained by the boundary elements method are proven. A priory error estimates are investigated. Numerical results obtained for the modal problems are given.