Factorization method for inverse elastic cavity scattering

Shuxin Li, Junliang Lv, Yi Wang
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Abstract

This paper is concerned with the inverse elastic scattering problem to determine the shape and location of an elastic cavity. By establishing a one-to-one correspondence between the Herglotz wave function and its kernel, we introduce the far-field operator which is crucial in the factorization method. We present a theoretical factorization of the far-field operator and rigorously prove the properties of its associated operators involved in the factorization. Unlike the Dirichlet problem where the boundary integral operator of the single-layer potential involved in the factorization of the far-field operator is weakly singular, the boundary integral operator of the conormal derivative of the double-layer potential involved in the factorization of the far-field operator with Neumann boundary conditions is hypersingular, which forces us to prove that this operator is isomorphic using Fredholm's theorem. Meanwhile, we present theoretical analyses of the factorization method for various illumination and measurement cases, including compression-wave illumination and compression-wave measurement, shear-wave illumination and shear-wave measurement, and full-wave illumination and full-wave measurement. In addition, we also consider the limited aperture problem and provide a rigorous theoretical analysis of the factorization method in this case. Numerous numerical experiments are carried out to demonstrate the effectiveness of the proposed method, and to analyze the influence of various factors, such as polarization direction, frequency, wavenumber, and multi-scale scatterers on the reconstructed results.
反弹性空腔散射的因式分解法
本文关注反弹性散射问题,以确定弹性空腔的形状和位置。我们提出了远场算子的理论因式分解,并严格证明了因式分解所涉及的相关算子的性质。与Dirichlet问题中远场算子因式分解所涉及的单层势的边界积分算子是弱单数不同,带Neumann边界条件的远场算子因式分解所涉及的双层势的常导数的边界积分算子是超单数,这迫使我们用Fredholm定理证明这个算子是同构的。同时,我们提出了针对各种照明和测量情况的因式分解方法理论分析,包括压缩波照明和压缩波测量、剪切波照明和剪切波测量以及全波照明和全波测量。此外,我们还考虑了有限孔径问题,并对这种情况下的因式分解方法进行了严谨的理论分析。为了证明所提方法的有效性,我们进行了大量的数值实验,分析了极化方向、频率、文波数和多尺度散射体等各种因素对重建结果的影响。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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