  This paper is concerned with the analysis of elastic wave scattering of a
time-harmonic plane wave by a biperiodic rigid surface, where the wave
propagation is governed by the three-dimensional Navier equation. An exact
transparent boundary condition is developed to reduce the scattering problem
equivalently into a boundary value problem in a bounded domain. The perfectly
matched layer (PML) technique is adopted to truncate the unbounded physical
domain into a bounded computational domain. The well-posedness and exponential
convergence of the solution are established for the truncated PML problem by
developing a PML equivalent transparent boundary condition. The proofs rely on
a careful study of the error between the two transparent boundary operators.
The work significantly extend the results from the one-dimensional periodic
structures to the two-dimensional biperiodic structures. Numerical experiments
are included to demonstrate the competitive behavior of the proposed method.
