Consider the elastic scattering of a time-harmonic wave by multiple well-separated rigid particles with smooth boundaries in two dimensions. Instead of using the complex Green's tensor of the elastic wave equation, we utilize the Helmholtz decomposition to convert the boundary value problem of the elastic wave equation into a coupled boundary value problem of the Helmholtz equation. Based on single, double, and combined layer potentials with the simpler Green's function of the Helmholtz equation, we present three different boundary integral equations for the coupled boundary value problem. The well-posedness of the new integral equations is established. Computationally, a scattering matrix based method is proposed to evaluate the elastic wave for arbitrarily shaped particles. The method uses the local expansion for the incident wave and the multipole expansion for the scattered wave. The linear system of algebraic equations is solved by GMRES with fast multipole method (FMM) acceleration. Numerical results show that the method is fast and highly accurate for solving elastic scattering problems with multiple particles.
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An inverse acoustic-elastic interaction problem with phased or phaseless far-field data
Consider the scattering of a time-harmonic acoustic plane wave by a bounded elastic obstacle which is immersed in a homogeneous acoustic medium. This paper is concerned with an inverse acoustic-elastic interaction problem, which is to determine the location and shape of the elastic obstacle by using either the phased or phaseless far-field data. By introducing the Helmholtz decomposition, the model problem is reduced to a coupled boundary value problem of the Helmholtz equations. The jump relations are studied for the second derivatives of the single-layer potential in order to deduce the corresponding boundary integral equations. The well-posedness is discussed for the solution of the coupled boundary integral equations. An efficient and high order Nyström-type discretization method is proposed for the integral system. A numerical method of nonlinear integral equations is developed for the inverse problem. For the case of phaseless data, we show that the modulus of the far-field pattern is invariant under a translation of the obstacle. To break the translation invariance, an elastic reference ball technique is introduced. We prove that the inverse problem with phaseless far-field pattern has a unique solution under certain conditions. In addition, a numerical method of the reference ball technique based nonlinear integral equations is proposed for the phaseless inverse problem. Numerical experiments are presented to demonstrate the effectiveness and robustness of the proposed methods.
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- Award ID(s):
- 1912704
- NSF-PAR ID:
- 10182369
- Date Published:
- Journal Name:
- Inverse problems
- Volume:
- 36
- ISSN:
- 0266-5611
- Page Range / eLocation ID:
- 035014
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1088/1361-6420/ab693e
