Perfect absorbing layers for wave scattering problems and related topics

• Main Author: Gao, Yang
• Other Authors: Wang Li-Lian
• Format: Thesis-Doctor of Philosophy
• Language: English
• Published: Nanyang Technological University 2020
• Subjects: Science::Mathematics::Applied mathematics::Numerical analysis
• Online Access: https://hdl.handle.net/10356/137391
• Institution: Nanyang Technological University

The main objective of this thesis is to design a truly exact and optimal perfect absorbing layer (PAL) method for domain truncation of the two-dimensional Helmholtz equation in an unbounded domain with bounded scatterers. This PAL is based on a complex compression coordinate transformation and a judicious substitution of the unknown field in the artificial layer. Compared with existing perfectly matched layer (PML) methods, the distinctive features of this technique lie in that (i) it is truly exact in the sense that the PAL-solution is identical to the original solution in the bounded domain enclosed by the truncation layer; (ii) with the substitution, the PAL-equation is free of singularity and the substituted unknown field is essentially non-oscillatory in the layer; and (iii) using the polar coordinates, the construction is valid for general star-shaped domain truncation. By formulating the final variational formulation in Cartesian coordinates, the implementation of this technique using standard spectral-element or finite-element methods can be made easy as a usual coding exercise. We provide ample numerical examples to demonstrate that this method is highly accurate, parameter-free and robust for very high wavenumber and thin layer. It outperforms the classical PML and the recently advocated PML using unbounded absorbing functions. Moreover, it can fix some flaws of the PML approach. To have more insights into the new PAL technique, we conduct the wavenumber explicit analysis of the associated PAL equation, and estimate the error of the spectral schemes with the characterisation of the convergence order on the wavenumber. We also extend the frequency domain PALs to the time domain, and show its difference and advantages over the time-domain PML methods, where we concentrate on the one-way and two-way wave equations. Finally, we propose the spectral approximation of multiple scattering problems, where through the Dirichlet-to-Neumann (DtN) artificial boundary conditions and an appropriate iterative solver, the multiple scattering can be decoupled into single scattering problems.