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.
Accepté le :
Publié le :
DOI : 10.1051/m2an/2021074
Keywords: Elastic wave equation, scattering by biperiodic structures, adaptive finite element method, transparent boundary condition, DtN map, $$ error estimate
@article{M2AN_2021__55_6_2921_0,
author = {Bao, Gang and Jiang, Xue and Li, Peijun and Yuan, Xiaokai},
title = {An adaptive finite element {DtN} method for the elastic wave scattering by biperiodic structures},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {2921--2947},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {6},
doi = {10.1051/m2an/2021074},
mrnumber = {4347303},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2021074/}
}
TY - JOUR AU - Bao, Gang AU - Jiang, Xue AU - Li, Peijun AU - Yuan, Xiaokai TI - An adaptive finite element DtN method for the elastic wave scattering by biperiodic structures JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 2921 EP - 2947 VL - 55 IS - 6 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2021074/ DO - 10.1051/m2an/2021074 LA - en ID - M2AN_2021__55_6_2921_0 ER -
%0 Journal Article %A Bao, Gang %A Jiang, Xue %A Li, Peijun %A Yuan, Xiaokai %T An adaptive finite element DtN method for the elastic wave scattering by biperiodic structures %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 2921-2947 %V 55 %N 6 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2021074/ %R 10.1051/m2an/2021074 %G en %F M2AN_2021__55_6_2921_0
Bao, Gang; Jiang, Xue; Li, Peijun; Yuan, Xiaokai. An adaptive finite element DtN method for the elastic wave scattering by biperiodic structures. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 6, pp. 2921-2947. doi: 10.1051/m2an/2021074
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