Abstract
Consider the scattering of a time-harmonic elastic plane wave by a bi-periodic rigid surface. The displacement of elastic wave motion is modeled by the three-dimensional Navier equation in an unbounded domain above the surface. Based on the Dirichlet-to-Neumann (DtN) operator, which is given as an infinite series, an exact transparent boundary condition is introduced and the scattering problem is formulated equivalently into a boundary value problem in a bounded domain. An a posteriori error estimate based adaptive finite element DtN method is proposed to solve the discrete variational problem where the DtN operator is truncated into a finite number of terms. The a posteriori error estimate takes account of the finite element approximation error and the truncation error of the DtN operator which is shown to decay exponentially with respect to the truncation parameter. Numerical experiments are presented to illustrate the effectiveness of the proposed method.
Highlights
This paper concerns the scattering of a time-harmonic elastic plane wave by a bi-periodic surface in three dimensions
We intend to address both of these two issues by proposing an a posteriori error estimate based adaptive finite element method with the transparent boundary condition
Since there is no exact solution for this example, we plot in Figure 6 the curves of log }∇puukq}0,Ω versus log Nk for the a posteriori error estimates with different frequencies, where Nk is the total number of DoFs of the mesh
Summary
This paper concerns the scattering of a time-harmonic elastic plane wave by a bi-periodic surface in three dimensions. We intend to address both of these two issues by proposing an a posteriori error estimate based adaptive finite element method with the transparent boundary condition. The a posteriori error estimates based adaptive finite element PML methods take account of the finite element discretization errors and the PML truncation errors which decay exponentially with respect to the PML parameters Another effective approach to truncate the unbounded domain is to construct the Dirichlet-toNeumann (DtN) map and introduce the transparent boundary condition to enclose the domain of interest [21]. By the Helmholtz decomposition, a DtN operator is constructed in terms of Fourier series expansions for the compressional and shear wave components, an exact transparent boundary condition is introduced to reduce the unbounded domain problem into an equivalent boundary value problem in a bounded domain.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
