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Register a new userPerfectly-matched-layer boundary integral equation method for wave scattering in a layered medium
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For scattering problems of time-harmonic waves, the boundary integral equation (BIE) methods are highly competitive, since they are formulated on lower-dimension boundaries or interfaces, and can automatically satisfy outgoing radiation conditions. For scattering problems in a layered medium, standard BIE methods based on the Greens function of the background medium must evaluate the expensive Sommefeld integrals. Alternative BIE methods based on the free-space Greens function give rise to integral equations on unbounded interfaces which are not easy to truncate, since the wave fields on these interfaces decay very slowly. We develop a BIE method based on the perfectly matched layer (PML) technique. The PMLs are widely used to suppress outgoing waves in numerical methods that directly discretize the physical space. Our PML-based BIE method uses the Greens function of the PML-transformed free space to define the boundary integral operators. The method is efficient, since the Greens function of the PML-transformed free space is easy to evaluate and the PMLs are very effective in truncating the unbounded interfaces. Numerical examples are presented to validate our method and demonstrate its accuracy.
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Numerical mode matching (NMM) methods are widely used for analyzing wave propagation and scattering in structures that are piece-wise uniform along one spatial direction. For open structures that are unbounded in transverse directions (perpendicular to the uniform direction), the NMM methods use the perfectly matched layer (PML) technique to truncate the transverse variables. When incident waves are specified in homogeneous media surrounding the main structure, the total field is not always outgoing, and the NMM methods rely on reference solutions for each uniform segment. Existing NMM methods have difficulty handing gracing incident waves and special incident waves related to the onset of total internal reflection, and are not very efficient at computing reference solutions for non-plane incident waves. In this paper, a new NMM method is developed to overcome these limitations. A Robin-type boundary condition is proposed to ensure that non-propagating and non-decaying wave field components are not reflected by truncated PMLs. Exponential convergence of the PML solutions based on the hybrid Dirichlet-Robin boundary condition is established theoretically. A fast method is developed for computing reference solutions for cylindrical incident waves. The new NMM is implemented for two-dimensional structures and polarized electromagnetic waves. Numerical experiments are carried out to validate the new NMM method and to demonstrate its performance.
In this article, several discontinuous Petrov-Galerkin (DPG) methods with perfectly matched layers (PMLs) are derived along with their quasi-optimal graph test norms. Ultimately, two different complex coordinate stretching strategies are considered in these derivations. Unlike with classical formulations used by Bubnov-Galerkin methods, with so-called ultraweak variational formulations, these two strategies in fact deliver different formulations in the PML region. One of the strategies, which is argued to be more physically natural, is employed for numerically solving two- and three-dimensional time-harmonic acoustic, elastic, and electromagnetic wave propagation problems, defined in unbounded domains. Through these numerical experiments, efficacy of the new DPG methods with PMLs is verified.
The optical resonance problem is similar to but different from time-steady Schr{o}dinger equation. One big challenge is that the eigenfunctions in resonance problem is exponentially growing. We give physical explanation to this boundary condition and introduce perfectly matched layer (PML) method to transform eigenfunctions from exponential-growth to exponential-decay. Based on the complex stretching technique, we construct a non-Hermitian Hamiltonian for the optical resonance problem. We successfully validate the effectiveness of the Hamiltonian by calculate its eigenvalues in the circular cavity and compare with the analytical results. We also use the proposed Hamiltonian to investigate the mode evolution around exceptional points in the quad-cosine cavity.
This paper studies the PML method for wave scattering in a half space of homogeneous medium bounded by a two-dimensional, perfectly conducting, and locally defected periodic surface, and develops a high-accuracy boundary-integral-equation (BIE) solver. Along the vertical direction, we place a PML to truncate the unbounded domain onto a strip and prove that the PML solution converges linearly to the true solution in the physical subregion of the strip with the PML thickness. Laterally, we divide the unbounded strip into three regions: a region containing the defect and two semi-waveguide regions, separated by two vertical line segments. In both semi-waveguides, we prove the well-posedness of an associated scattering problem so as to well define a Neumann-to-Dirichlet (NtD) operator on the associated vertical segment. The two NtD operators, serving as exact lateral boundary conditions, reformulate the unbounded strip problem as a boundary value problem onto the defected region. Due to the periodicity of the semi-waveguides, both NtD operators turn out to be closely related to a Neumann-marching operator, governed by a nonlinear Riccati equation. It is proved that the Neumann-marching operators are contracting, so that the PML solution decays exponentially fast along both lateral directions. The consequences culminate in two opposite aspects. Negatively, the PML solution cannot exponentially converge to the true solution in the whole physical region of the strip. Positively, from a numerical perspective, the Riccati equations can now be efficiently solved by a recursive doubling procedure and a high-accuracy PML-based BIE method so that the boundary value problem on the defected region can be solved efficiently and accurately. Numerical experiments demonstrate that the PML solution converges exponentially fast to the true solution in any compact subdomain of the strip.
This note is intended as a brief introduction to the theory and practice of perfectly matched layer (PML) absorbing boundaries for wave equations, originally developed for MIT courses 18.369 and 18.336. It focuses on the complex stretched-coordinate viewpoint, and also discusses the limitations of PML.
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