In this paper, we formulate mathematical foundations of applications boundary integral equations with strongly singular integrals understood in the sense finite Hadamard value to numerical solution certain boundary-value problems. We describe schemes for solving based on quadrature formulas and collocation method. Also, make references known results justification methods described paper.

A boundary integral operator method for stochastic ray tracing in billiards was recently proposed in [1]. In particular, a phase-space boundary integral model for propagating uncertain ray or particle flows was described and shown to interpolate between deterministic and random models of the flow propagation. In this work we describe discretisation schemes for this class of boundary integral op...

The boundary element method (BEM) for linear elasticity in its curent usage is based on the boundary integral equation for displacements. The stress field in the interior of the body is computed by differentiating the displacement field at the source point in the BEM formulation, via the strain field. However, at the boundary, this method gives rise to a hypersingular integral relation which be...

For analyzing diffraction gratings, a new method is developed based on dividing one period of the grating into homogeneous subdomains and computing the Neumann-to-Dirichlet (NtD) maps for these subdomains by boundary integral equations. For a subdomain, the NtD operator maps the normal derivative of the wave field to the wave field on its boundary. The integral operators used in this method are...

Abstract This expository paper is concerned with the direct integral formulations for boundary value problems of the Helmholtz equation. We discuss unique solvability for the corresponding boundary integral equations and its relations to the interior eigenvalue value problems of the Laplacian. Based on the integral representations, we study the asymptotic behaviors of the solutions to the bound...

The inverse eddy-current problem of fast transients is solved by a new boundary element formulation employing time domain surface impedance boundary conditions. The integral equation is transformed to invariant form and is solved only once for a given geometry of the problem. Numerical results are in good agreement with measured data.

The problem of large deflections of thin flat plates is mderived here using a novel integral equation approach. These plate deformations arc governed by the von Karman plate theory. The numerical solution that is implemented combines both boundary and interior elements in the discretization of the continuum. The formulation also illustrates the adaptability of the boundary element technique to ...

We obtained oo-plane stress expression for a planar fault using boundary integral equation (BIE) in 3D innnite elastic medium, which are consistent with those by Tada (199779], 1998810]). Assuming a 2D arbitrary shaped curved fault as a series of small planar fault patches, we discretized the kernels analytically for constant slip velocity within a patch. Taking a limit to the on-plane expressi...

For hypersingular integral equations in 2D and 3D, we analyze easy-to-implement error estimators like (h − h/2)-based estimators, two-level estimators, and averaging on large patches and prove their equivalence. Moreover, we introduce some ZZ-type error estimators. All of these a posteriori error estimators are analyzed within the framework of localization techniques for the energy norm.

The numerical construction of a Green’s function for multiple interacting planar cracks in an anisotropic elastic space is considered. The numerical Green’s function can be used to obtain a special boundary integral method for an important class of two-dimensional elastostatic problems involving planar cracks in an anisotropic body.