裂缝与腔体混合的声波逆散射问题的反演方法及其数值实现

The acoustic scattering problems have important applications in many fields, such as radar, medical imaging, non-destructive testing, geologic exploration, etc. This proposal studies the mixed scattering problems with the incident point sources emitted in a cavity and scattered by the cavity and a crack. To be specific, one of the mixed scattering problems is the crack located inside the cavity, and the other is the crack located outside the cavity. First, the boundary integral equation methods or the variational methods are used to solve the direct scattering problems, to prove the existence, uniqueness and continuity of the direct solutions. Second, to prove the factorization methods or the linear sampling methods can be used to recover the shape and position of the cavities and cracks from the knowledge of the measurement data measured on a closed curve inside the cavities. Finally, some numerical examples will be proposed to verify the feasibility of the numerical algorithms. The advantages of the factorization methods and the linear sampling methods lie in that we can reconstruct the cavities and cracks simultaneously using only the scattered field data without needing other priori knowledge. The innovation of this proposal is reconstructing two different scatterers simultaneously from the knowledge of the near field data by using the factorization methods or the linear sampling methods. We hope to apply these methods to some more complex scattering problems with incident point sources inside cavities.

声波散射问题在雷达、医学成像、无损探测、地质勘探等许多领域具有重要意义。 本项目将考虑从腔体内部发射的点源声波，被腔体和裂缝散射所产生的混合散射问题。具体分成两类：一类是裂缝位于不可穿透的腔体内部的混合散射问题；另一类是裂缝位于可穿透的腔体外部的混合散射问题。首先，将采用边界积分方程方法或者变分方法证明正散射问题解的存在性、唯一性、连续性等；然后，证明因式法或者线性抽样法可以利用腔体内部一条闭曲线上收集的散射场信息重构腔体和裂缝的形状与位置；最后，通过数值实验来检验重构方法的可行性。因式法与线性抽样法的优点是只需要利用散射场信息，而不需要其他先验条件，就能够快速准确地同时重构腔体和裂缝的位置与形状。直接利用散射波的近场信息，根据因式法或线性抽样法同时重构两种不同性质的散射体，是本项目的创新点。我们希望这些方法能够运用到更多的入射点源在内部的复杂散射问题中。

声波和电磁波的散射问题在雷达、医学成像、无损探测、地质勘探等许多领域具有重要意义。本项目对点源声波从腔体内部向外发射，被腔体和外部障碍物散射的问题进行了研究；同时，对裂缝与障碍物在电磁波场中的混合散射问题，以及裂缝在弹性波中的散射问题进行了研究。. 我们分别运用边界积分方程方法和变分方法证明了各正散射问题解的存在性和唯一性，更主要的工作是研究其逆散射问题，即根据散射波数据对散射体的位置和形状进行反演。所采用的逆散射方法是“因式法”这种定性的方法。定性的方法处理逆散射问题的优点是：不需要知道散射体的边界条件，只需要根据散射波的远场数据或者近场数据，就能快速准确地确定散射体的位置和形状。我们分别从理论上和数值模拟上，证明了定性的方法在解决这几种复杂逆散射问题的可行性。

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