弹性板中小形状缺陷反演的Lamb模和Born近似
Lamb modes and Born approximation for small shape defects inversion in elastic plates
摘要：
这项工作的目的是提出理论工具来研究弹性波导中的波传播，并进行多频散射反演，以重建在二维和三维弹性板的小形状缺陷。给定表面多频波场测量，我们使用玻恩近似来重建板的几何形状中的局部缺陷。为了证明这种近似，我们引入了一个严格的框架来研究由任意源产生的弹性波场的传播。通过研究非均匀Lamb模序列的衰减率，证明了二维和三维平面波导中弹性波传播的偏微分方程的适定性。我们还描述了临界频率的兰姆分解是无效的。利用这些结果，我们已经开发的声波波导的形状重建方法推广到2D弹性波导，并提供了一个稳定的重建方法的基础上的模式由模式的空间傅立叶反演给出的散射场。
英文摘要：
The aim of this work is to present theoretical tools to study wave propagation in elastic waveguides and perform multi-frequency scattering inversion to reconstruct small shape defects in a 2D and 3D elastic plate. Given surface multi-frequency wavefield measurements, we use a Born approximation to reconstruct localized defect in the geometry of the plate. To justify this approximation, we introduce a rigorous framework to study the propagation of elastic wavefield generated by arbitrary sources. By studying the decreasing rate of the series of inhomogeneous Lamb mode, we prove the well-posedness of the PDE that model elastic wave propagation in 2D and 3D planar waveguides. We also characterize the critical frequencies for which the Lamb decomposition is not valid. Using these results, we generalize the shape reconstruction method already developed for acoustic waveguide to 2D elastic waveguides and provide a stable reconstruction method based on a mode-by-mode spacial Fourier inversion given by the scattered field.
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