A finite element scheme with a high order absorbing boundary condition for elastodynamics

A high-order absorbing boundary condition (ABC) is devised on an artificial boundary for time-dependent elastic waves in unbounded domains. The configuration considered is that of a two-dimensional elastic waveguide. In the exterior domain, the unbounded elastic medium is assumed to be isotropic and homogeneous. The proposed ABC is an extension of the Hagstrom-Warburton ABC which was originally designed for acoustic waves, and is applied directly to the displacement field. The order of the ABC determines its accuracy and can be chosen to be arbitrarily high. The initial boundary value problem including this ABC is written in second-order form, which is convenient for geophysical finite element (FE) analysis. A special variational formulation is constructed which incorporates the ABC. A standard FE discretization is used in space, and a Newmark-type scheme is used for time-stepping. A long-time instability is observed, but simple means are shown to dramatically postpone its onset so as to make it harmless during the simulation time of interest. Numerical experiments demonstrate the performance of the scheme.