A variety of problems in acoustic and electromagnetic scattering require the evaluation of impedance or layered media Green's functions. Given a point source located in an unbounded half-space or an infinitely extended layer, Sommerfeld and others showed that Fourier analysis combined with contour integration provides a systematic and broadly effective approach, leading to what is generally referred to as the Sommerfeld integral representation. When either the source or target is at some distance from an infinite boundary, the number of degrees of freedom needed to resolve the scattering response is very modest. When both are near an interface, however, the Sommerfeld integral involves a very large range of integration and its direct application becomes unwieldy. Historically, three schemes have been employed to overcome this difficulty: the method of images, contour deformation, and asymptotic methods of various kinds. None of these methods make use of classical layer potentials in physical space, despite their advantages in terms of adaptive resolution and high-order accuracy. The reason for this is simple: layer potentials are impractical in layered media or half-space geometries since they require the discretization of an infinite boundary. In this paper, we propose a hybrid method which combines layer potentials (physical-space) on a finite portion of the interface together with a Sommerfeld-type (Fourier) correction. We prove that our method is efficient and rapidly convergent for arbitrarily located sources and targets, and show that the scheme is particularly effective when solving scattering problems for objects which are close to the half-space boundary or even embedded across a layered media interface.
Taylor expansion based fast Multipole Methods for 3-D Helmholtz equations in Layered Media
Hunan Normal University（湖南师范大学）；Southern Methodist University（南方卫理公会大学）；University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校）；Indiana University（印第安纳大学）；University of Massachusetts Lowell（马萨诸塞大学洛厄尔分校）
From multi-layered problems to multiple two-layered problems: a novel frequency-time hybrid multiple-scattering integral equation solver
University of Electronic Science and Technology of China（电子科技大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Nanyang Technological University（南洋理工大学）
A Heterogeneous FMM for 2-D Layered Media Helmholtz Equation I: Two & Three Layers Cases
University of Massachusetts Lowell（马萨诸塞大学洛威尔分校）；University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校）；Beijing Computational Science Research Center（北京计算科学研究中心）；University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校）
Domain decomposition for quasi-periodic scattering by layered media via robust boundary-integral equations at all frequencies
MIT（麻省理工学院）；Pontificia Universidad Católica de Chile（智利天主教 Pontificia 大学）；Louisiana State University（路易斯安那州立大学）；New Jersey Inst. of Technology（新泽西理工学院）；Duke University（杜克大学）
A Geometrical Acoustics based Focusing Algorithm for Layered Media in Medical Ultrasound
Johannes Kepler University Linz（林茨约翰·开普勒大学）；Institute of Industrial Mathematics（工业数学研究所）；Johann Radon Institute for Computational and Applied Mathematics（约翰·拉德纳计算与应用数学研究所）
A Hybrid Physics-Informed Neural Network Framework for Computing Dispersion Relations of SH Waves in Generalized Hetrogeneous Layered Media with Applications
Indian Institute of Technology (Indian School of Mines)（印度理工学院（印度矿业学院））
