Long-time stable high-order absorbing boundary conditions for elastodynamics

Two new high-order local absorbing boundary conditions (ABCs) are devised for use on an artificial boundary for time-dependent elastic waves in unbounded domains, in two dimensions. The elastic medium in the exterior domain is assumed to be homogeneous and isotropic. The first ABC is written directly with respect to the displacement vector field, using an operator involving high derivatives. The second ABC is applied to the problem written in a first-order conservation form, using stresses and velocities as variables, and is formulated as a sequence of recursive relations using auxiliary variables. The two ABCs are not equivalent but are closely related. In each case, the order of the ABC determines its accuracy and can be chosen to be arbitrarily high. Both ABCs involve a product of first-order differential operators; all of them are of the Higdon type, except one which is of the Lysmer–Kuhlemeyer type. The stability of both ABCs is analyzed theoretically. The second (stress–velocity) ABC is implemented, employing a finite difference discretization scheme in space and time. Numerical experiments demonstrate the performance of the scheme. To the best of the authors’ knowledge, these are the first existing local high-order ABCs for elastodynamics which are long-time stable.