Higher order numerical methods for acoustic scattering problems with locally perturbed periodic structures

本课题致力于研究三维空间中不可穿透板上局部摄动周期性非均匀层的时谐声散射问题。散射问题是用无界域的亥姆霍兹方程来模拟的，其理论分析和数值求解都非常具有挑战性。在这个项目中涉及的主要工具是Floquet-Bloch变换，它已经被证明是非常强大的散射问题与周期结构在二维空间。第一个目标是分析布洛赫变换场的准周期性参数的连续性和规律性，其中Dirichlet-to-Neumann映射起着重要的作用。第二个目标是在拟周期Bloch变换问题的正则性结果的基础上，提出具有周期层的散射问题的高阶数值方法。与二维情况相反，Bloch变换场的奇异点不再局限于有限的点，而是覆盖了一个“奇异圆”的并集。因此，2D情况下的高阶数值方法的直接扩展可能不适用于3D情况，并且需要新的想法。第三个目标是发展一种有效的局部摄动周期层的数值方法。耦合有限元法或Lippmann-Schwinger方程的离散化将被应用。

This project is devoted to the investigation of time-harmonic acoustic scattering problems with locally perturbed periodic inhomogeneous layers above impenetrable plates in three dimensional spaces. The scattering problems are modelled by Helmholtz equations in unbounded domains, both the theoretical analysis and the numerical solution of which are very challenging. The main tool involved in this project is the Floquet-Bloch transform, which has been proven to be very powerful for scattering problems with periodic structures in two dimensional spaces. The first objective is to analyze continuity and regularity of the Bloch transformed field with respect to the quasi-periodicity parameter, where the Dirichlet-to-Neumann map plays an important role. The second goal is to propose a high order numerical method for scattering problems with periodic layers, based on the regularity results established for the quasi-periodic Bloch transformed problems. In contrast to the 2D case, the singularities of the Bloch transformed fields are no longer localized in a finite number of points, but cover a union of "singular circles". Thus a straightforward extension of the high order numerical method for the 2D case may not be appropriate for the 3D case, and new ideas will be required. The third goal is to develop an efficient numerical method for locally perturbed periodic layers. Either a coupled finite element method or a discretization of the Lippmann-Schwinger equation will be applied.