This paper is concerned with the time-harmonic direct and inverse elastic scattering by an extended rigid elastic body surrounded by a finite number of point-like obstacles. We first justify the point-interaction model for the Lamé operator within the singular perturbation approach. For a general family of pointwise-supported singular perturbations, including anisotropic and non-local interactions, we derive an explicit representation of the scattered field.In the case of isotropic and local point-interactions, our result is consistent with the ones previously obtained by Foldy's formal method as well as by the renormalization technique. In the case of multiple scattering with pointwise and extended obstacles, we show that the scattered field consists of two parts: one is due to the diffusion by the extended scatterer and the other one is a linear combination of the interactions between the point-like obstacles and the interaction between the point-like obstacles with the extended one.As to the inverse problem, the factorization method by Kirsch is adapted to recover simultaneously the shape of an extended elastic body and the location of point-like scatterers in the case of isotropic and local interactions. The inverse problems using only one type of elastic waves (i.e. pressure or shear waves) are also investigated and numerical examples are presented to confirm the inversion schemes.

中文翻译：

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点状和扩展障碍物的直接和逆时谐弹性散射

本文涉及由有限数量的点状障碍物包围的扩展刚性弹性体引起的时谐正向和反向弹性散射。我们首先证明奇异摄动方法中Lamé算子的点相互作用模型是正确的。对于一般的点支撑奇异摄动族，包括各向异性和非局部相互作用，我们得到了散射场的显式表示;在各向同性和局部点相互作用的情况下，我们的结果与先前通过Foldy的形式方法以及重归一化技术。在具有点状和扩展障碍物的多重散射的情况下，我们表明散射场由两部分组成：一个是归因于扩展散射体的扩散，另一归因于点状障碍物之间的相互作用以及点状障碍物之间的相互作用与扩展的线性组合。在各向同性和局部相互作用的情况下，Kirsch的“ Kirsch弹性体”适用于同时恢复伸展的弹性体的形状和点状散射体的位置。还研究了仅使用一种类型的弹性波（即压力波或剪切波）的反问题，并给出了数值示例来确认反演方案。在各向同性和局部相互作用的情况下，Kirsch的因式分解方法适用于同时恢复伸展的弹性体的形状和点状散射体的位置。还研究了仅使用一种类型的弹性波（即压力波或剪切波）的反问题，并给出了数值示例来确认反演方案。在各向同性和局部相互作用的情况下，Kirsch的因式分解方法适用于同时恢复伸展的弹性体的形状和点状散射体的位置。还研究了仅使用一种类型的弹性波（即压力波或剪切波）的反问题，并给出了数值示例来确认反演方案。