Scattering of elastic waves by a 2-D crack using the Indirect Boundary Element Method (IBEM) (original) (raw)
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Boundary element simulation of scattering of elastic waves by 3-D cracks
Journal of Applied Geophysics, 2008
Numerical modelling techniques are now becoming common for understanding the complicated nature of seismic wave propagation in fractured rock. Here the Indirect Boundary Element Method (IBEM) is applied to study scattering of elastic waves by cracks. The problem addressed in this paper is the diffraction of P and S waves by open 3-D cracks of arbitrary shape embedded in a homogeneous isotropic medium. The IBEM yields the value of the jump of displacements between opposite surfaces of the crack, often called Crack Opening Displacement (COD). This is used to evaluate the solution away from the crack. We use a multi-regional approach which consists of splitting a surface S into two identical surfaces S + and S − chosen such that the crack lies at the interface. The resulting integral equations are not hypersingular and wave propagation within media that contain open cracks can be rigorously solved. In order to validate the method, we compare results of displacements of a penny-shaped crack for a vertical incident P-wave with the classic results by obtaining excellent agreement. This comparison gives us confidence to study cases where no analytic solutions exist. Some examples of incidence of P or S waves upon cracks with various shapes are depicted and the salient aspects of the method are also discussed. Both frequency and time-domain results are included.
Scattering Of 3D Acoustic Waves By Cracks UsingTwo Different Boundary Elements Formulations
2004
Two different Boundary Elements formulations are used to compute the threedimensional (3D) scattering of acoustic waves by two-dimensional (2D) cracks, which may be either empty or filled. The first is the Traction Boundary Element Method (TBEM), and the second is a mixed formulation involving both the TBEM and the Boundary Element Method (BEM). These models overcome the thin body difficulty for which the conventional direct BEM degenerates. Both formulations can be used to solve the case of an empty crack, while the mixed formulation finds the solution of a filled crack. The two models are formulated in the frequency domain. Because of the 2-1/2D geometry of the problem, the solution at each frequency is expressed in terms of waves with the varying wavenumber in the z direction, z k . All integrals with hypersingular kernels are evaluated analytically. Time responses in the space domain are computed by applying an inverse (Fast) Fourier Transform, using a Ricker pulse as the dynami...
Scattering By Cracks: Numerical Simulations Using A Boundary Finite Element Method
2002
feature of the far field pattern of the scattered wave. show that even with sparse meshes the method is able to give a global where it becomes clear the good convergence of the method. Moreover, we non planar cracks in the three dimensional case, presenting several examples information on the shape of the scattering obstacle. We include the case of potential. Simulations on the far field patterns allow to characterize some method applied to a variational formulation derived from the double layer in the resonance region. The results are obtained with a boundary element We present several simulations of the amplitude scattered by acoustic cracks 1 Introduction mainly penny-shaped flat cracks. non hypersingular kernel. First attempts considered only simple geometries, express the exterior domain problem in terms of an integral equation with a tried to solve this problem. In terms of integral equations, the difficultyis to materials. Early works by Bouwkamp[5]and Jones[S],in the acoustic case, plications in industrial problems, for instance, the detection of cracks in Acoustic and elastic scattering by cracks is an old issue, with many a p
Wave propagation in the presence of empty cracks in an elastic medium
Computational Mechanics, 2005
This paper proposes the use of a traction boundary element method (TBEM) to evaluate 3D wave propagation in unbounded elastic media containing cracks whose geometry does not change along one direction. The proposed formulation is developed in the frequency domain and handles the thin-body difficulty presented by the classical boundary element method (BEM). The empty crack may have any geometry and orientation and may even exhibit null thickness. Implementing this model yields hypersingular integrals, which are evaluated here analytically, thereby surmounting one of the drawbacks of this formulation. The TBEM formulation enables the crack to be modelled as a single line, allowing the computation of displacement jumps in the opposing sides of the crack. Furthermore, if this formulation is combined with the classical BEM formulation the displacements in the opposing sides of the crack can be computed by modelling the crack as a closed empty thin body.
Wave propagation in the presence of empty cracks in elastic slabs - TBEM and MFS Formulations
2007
This paper evaluates the 3D wave propagation in an elastic slab containing cracks whose geometry does not change along the direction parallel to the formation surfaces. Two different formulations are used and compared: the Traction Boundary Element Method (TBEM) and the Method of Fundamental Solutions (MFS). Both approaches are developed in the frequency domain and surmount the thin-body difficulty posed by the classical Boundary Element Method (BEM).
Journal of Computational Mathematics, 2010
The wave scattering problem by a crack Γ in R 2 with impedance type boundary is considered. This problem models the diffraction of waves by thin two-sided cylindrical screens. A numerical method for solving the problem is developed. The solution of the problem is represented in the form of the combined angular potential and single-layer potential. The linear integral equations satisfied by the density functions are derived for general parameterized arcs. The weakly singular integrals and the Cauchy singular integral arising in these equations are computed using a highly accurate scheme with a truncation error analysis. The advantage of the scheme proposed in this paper is, in one hand, the fact that we do not need the analytic property of the crack and we allow different complex valued surface impedances in both sides of the crack. In the other hand, we avoid the hyper-singular integrals. Numerical implementations showing the validity of the scheme are presented.
On the boundary integral equations for the crack opening displacement of flat cracks
Integral Equations and Operator Theory, 1992
The boundary integral equations for the crack opening displacement in acoustic and elastic scattering problems are discussed in the case of fiat cracks by means of the Fourier analysis technique.The pseudo-differential nature of the hypersingular integral operators is shown and their symbols explicited. It is then proved that the variational problems assocaited with these BIE are well-posed in a Sobolev functional framework which is closely linked with the elastic energy. A decomposition of the vector integral equation in the elastic case into scalar integral equations is obtained as a by-product of the variational formulation.
Wave propagation in cracked elastic slabs and half-space domains—TBEM and MFS approaches
Engineering Analysis with Boundary Elements, 2007
In this paper, the traction boundary element method (TBEM) and the method of fundamental solutions (MFS), formulated in the frequency domain, are used to evaluate the 3D scattered wave field generated by 2D empty cracks embedded in an elastic slab and a halfspace. Both models overcome the thin-body difficulty posed when the classical BEM is applied. The crack exhibits arbitrary cross section geometry and null thickness. In neither model are the horizontal formation surfaces discretized, since appropriate fundamental solutions are used to take them into consideration. The TBEM models the crack as a single line. The singular and hypersingular integrals that arise during the TBEM model's implementation are computed analytically, which overcomes one of the drawbacks of this formulation. The results provided by the proposed TBEM model are verified against responses provided by the classical BEM models derived for the case of an empty cylindrical circular cavity. The MFS solution is approximated in terms of a linear combination of fundamental solutions, generated by a set of virtual sources simulating the scattered field produced by the crack, using a domain decomposition technique. To avoid singularities, these fictitious sources are not placed close to the crack, and the use of an enriched function to model the displacement jumps across the crack is unnecessary. The performances of the proposed models are compared and their limitations are shown by solving the case of a C-shaped crack embedded in an elastic slab and a half-space domain. The applicability of these formulations is illustrated by presenting snapshots from computer animations in the time domain for an elastic slab containing an S-shaped crack, after applying an inverse Fourier transformation to the frequency domain computations.
The Indirect Boundary Integral Method for Curved Cracks in Plane Elasticity
Journal of the Korean Mathematical Society, 2002
For curved crack problems in plane elasticity, subjected to the traction conditions on the crack faces, we present a system of boundary integral equations. The procedure is based on the indirect boundary integral method in terms of real variables. For efficient mathematical analysis, we decompose the singular kernel into the Cauchy singular part and the regular one. As a result, solvability of the presented system is proved and availability of the present approach is shown by the numerical example of a circular arc crack.