Inverse obstacle scattering for elastic waves with phased or phaseless far-field data. (English) Zbl 1524.78044
Summary: This paper concerns an inverse elastic scattering problem which is to determine the location and the shape of a rigid obstacle from the phased or phaseless far-field data for a single incident plane wave. By introducing the Helmholtz decomposition, the model problem is reduced to a coupled boundary value problem of the Helmholtz equations. The relation is established between the compressional or shear far-field pattern for the elastic wave equation and the corresponding far-field pattern for the coupled Helmholtz equations. An efficient and accurate Nyström-type discretization for the boundary integral equation is developed to solve the coupled system. The translation invariance of the phaseless compressional and shear far-field patterns are proved. A system of nonlinear integral equations is proposed and two iterative reconstruction methods are developed for the inverse problem. In particular, for the phaseless data, a reference ball technique is introduced to the scattering system in order to break the translation invariance. Numerical experiments are presented to demonstrate the effectiveness and robustness of the proposed method.
MSC:
| 78A46 | Inverse problems (including inverse scattering) in optics and electromagnetic theory |
| 65N21 | Numerical methods for inverse problems for boundary value problems involving PDEs |
| 78M15 | Boundary element methods applied to problems in optics and electromagnetic theory |
| 65N38 | Boundary element methods for boundary value problems involving PDEs |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 65R20 | Numerical methods for integral equations |
| 74J20 | Wave scattering in solid mechanics |
Keywords:
elastic wave equation; inverse obstacle scattering; phaseless data; boundary integral equations; Helmholtz decompositionReferences:
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