Local Dirichlet-type absorbing boundary conditions for transient elastic wave propagation problems

Abstract

In this paper a new collocation technique for constructing time-dependent absorbing boundary conditions (ABCs) applicable to elastic wave motion is devised. The approach makes use of plane waves which satisfy the governing equations of motion to construct the absorbing boundary conditions. The plane waves are adjusted so that they can cope with the satisfaction of radiation boundary conditions. The proposed technique offers some advantages and exhibits the following features: it is easy to implement; its approximation scheme is local in space and time and thus it does not deal with any routine schemes such as Fourier and Laplace transform, making the method computationally less demanding; as the employed basis functions used to construct the absorbing boundary condition are residual-free, it requires neither any differential operator (to approximate the wave dispersion relation), nor any auxiliary variables; it constructs Dirichlet-type ABCs and hence no derivatives of the field variables are required for the imposition of radiation conditions. In this study, we apply the proposed technique to the solution procedure of a collocation approach based on the finite point method which proceeds in time by an explicit velocity-Verlet algorithm. It contributes to developing a consistent meshless framework for the solution of unbounded elastodynamic problems in time domain. We also apply the proposed method to a standard finite element solver. The performance of the method in solution of some 2D examples is examined. We shall show that the method exhibits appropriate results, conserves the energy almost exactly, and it performs stably in time even in the case of long-term computations.