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

本计画主要研究三维空间中不可穿透平板上局部扰动周期性不均匀层的时谐声散射问题。电磁散射问题一般用无界区域上的亥姆霍兹方程来描述，无论是理论分析还是数值求解都具有很大的挑战性。 这个项目中涉及的主要工具是Floquet-Bloch变换，它已被证明是非常强大的散射问题的周期性结构在二维空间。第一个目标是分析Bloch变换场关于准周期参数的连续性和正则性，其中Dirichlet-to-Neumann映射起着重要的作用。第二个目标是提出一个高阶数值方法的散射问题的周期层，准周期Bloch变换问题的正则性结果建立的基础上。与二维情形相反，布洛赫变换场的奇点不再局限于有限个点，而是覆盖了一个“奇异圆”的并集。因此，一个简单的扩展的高阶数值方法的二维情况下可能不适合的三维情况下，新的想法将是必要的。第三个目标是发展一种有效的数值方法，局部扰动周期层。将应用耦合有限元法或Lippmann-Schwinger方程的离散化。

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.