Communications in Mathematical Sciences

Volume 16 (2018)

Number 6

Direct and inverse elastic scattering from a locally perturbed rough surface

Pages: 1635 – 1658



Guanghui Hu (Beijing Computational Science Research Center, Beijing, China)

Xiaokai Yuan (Department of Mathematics, Purdue University, West Lafayette, Indiana, U.S.A.)

Yue Zhao (School of Mathematics and Statistics, Central China Normal University, Wuhan, China)


This paper is concerned with time-harmonic elastic scattering from a locally perturbed rough surface in two dimensions. We consider a rigid scattering interface given by the graph of a one-dimensional Lipschitz function which coincides with the real axis in the complement of some compact set. Given the incident field and the scattering interface, the direct problem is to determine the field distribution, whereas the inverse problem is to determine the shape of the interface from the measurement of the field on an artificial boundary in the upper half-plane. We propose a symmetric coupling method between finite element and boundary integral equations to show uniqueness and existence of weak solutions. The synthetic data is computed via the finite element method with the Perfectly Matched Layer (PML) technique. To investigate the inverse problem, we derive the domain derivatives of the field with respect to the scattering interface. An iterative continuation method with multi-frequency data is used for recovering the unknown scattering interface.


linear elasticity, Fredholm alternative, half-plane, rigid surface, inverse scattering, multi-frequency data, local perturbation

2010 Mathematics Subject Classification

35A15, 74B05, 74J20, 78A46

Received 7 January 2018

Accepted 8 June 2018

Published 7 February 2019