A Chebyshev-based rectangular-polar integral solver for scattering by geometries described by non-overlapping patches

Abstract

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Weighted integral solvers for elastic scattering by open arcs in two dimensions

“…As is the case for wave scattering problems in acoustics and electromagnetics, 6,7 the boundary integral equation methods in elasticity require discretization of domains of lower dimensionality than those required by volumetric discretization methods (such as finite difference or finite element methods 4,8 ). These equations can generally be treated effectively even for high frequencies 9‐12 . As is well known, however, the classical boundary integral equations for open surfaces (or, in two dimensions, open arcs) are not second‐kind Fredholm integral equations.…”

“…These equations can generally be treated effectively even for high frequencies. [9][10][11][12] As is well known, however, the classical boundary integral equations for open surfaces (or, in two dimensions, open arcs) are not second-kind Fredholm integral equations. This setting presents some difficulties.…”

A windowed Green function method for elastic scattering problems on a half-space

“…associated with the WGF method for the solution of the Dirichlet and Neumann problems. For the numerical implementation in 3D, in turn, we utilize the methods presented in [12,17].…”

“…The overall proposed procedure thus reduces the operator evaluation problem to evaluation of weakly singular operators and tangential differentiation of surface densities. The weakly-singular integration problem is tackled in this paper by means of the Chebyshev-based rectangular-polar discretization methodology introduced recently [12,17]-which can be readily applied in conjunction with geometry descriptions given by a set of non-overlapping logically-quadrilateral patches, and which, therefore, makes the algorithm particularly well suited for treatment of complex geometries. The needed tangential differentiations, in turn, can easily be produced by means of differentiation of corresponding truncated Chebyshev expansions, with evaluation either via FFT or, for sufficiently small expansions, via direct summation.…”

“…Section 3 then presents the proposed 3D WGF methodology, including a description of the windowed integral operators and a preliminary windowed integral formulation (Section 3.1), as well as a "corrected" windowed integral formulation which is uniformly accurate for all incident angles, up to grazing (Section 3.2). Section 4 introduces the proposed high order operator discretization methods we use in our 2D implementation; the 3D operator discretization methods we use are described in [12,17]. A variety of numerical examples in 2D and 3D, finally, are presented in Section 5-demonstrating the accuracy and efficiency of the overall proposed approach.…”