Scattering of elastic waves by a 2-D crack using the Indirect Boundary Element Method (IBEM)

Abstract

S U M M A R YThe scattering of elastic waves by cracks is an old problem and various ways to solve it have been proposed in the last decades. One approach is using dual integral equations, another useful and common formulation is the Boundary Element Method (BEM). With the last one, the boundary conditions of the crack lead to hyper-singularities and particular care should be taken to regularize and solve the resulting integral equations.In this work, instead, the Indirect Boundary Element Method (IBEM) is applied to study problems of zero-thickness 2-D cracks. The IBEM yields the Crack Opening Displacement (COD) which is used to evaluate the solution away from the crack. We use a multiregional approach which consists of splitting a boundary S into two identical boundaries S + and S − chosen such that the cracks lie in the interface. The resulting integral equations are not hyper-singular and wave propagation within media that contain zero-thickness cracks can be rigorously solved.In order to validate the method, we deal with the scalar case, namely the scattering of antiplane SH waves by a 2-D crack. We compare results against a recently published analytic solution, obtaining an excellent agreement. This comparison gives us confidence to study cases where no analytic solutions exist. Some examples of incidence of P-or SV waves are depicted and the salient aspects of the method are also discussed.

An energy approach to space–time Galerkin BEM for wave propagation problems

“…The temporal profile of the incident wave, that strikes the crack and from which we have deduced the Neumann datum on , is shown in Figure 27 and it is similar to that one considered in [31,32].…”

Numerical integration schemes for space–time hypersingular integrals in energetic Galerkin BEM

“…The temporal profile of the incident wave, that strikes the crack and from which we have deduced the Neumann datum on , is shown in Fig. 8 and it's similar to that one considered in [7,12]. We have taken into account the curvilinear crack = {x ∈ R 2 : x = (α, 0.5 sin(π α 2 ))}, depending on the parameter α ∈ [−1, 1], struck perpendicularly by the plane wave.…”

“…At this stage, considering the derivative of (9) with respect to n x and operating with the same arguments as before, after a cumbersome but easy calculation one obtains (7). The energetic weak problem related to (6) will be of the form (see [3])…”

Scattering of elastic waves in cracked media using a finite difference method

“…Geophys. Geod., 51 (2007) 2004), the boundary element method (BEM) (Rodríguez-Castellanos et al, 2005;Pointer et al, 1998;Liu and Zhang, 2001;Iturrarán-Viveros et al, 2005), or the finite difference methods (FDM) (e.g. Frankel and Clayton, 1986;Roth and Korn, 1993;van Baren et al, 2001;Saenger and Shapiro, 2002;Krüger et al, 2005).…”