Article
Mathematics, Applied
J. Galkowski, E. A. Spence
Summary: This paper investigates the number of degrees of freedom required to approximate an arbitrary oscillating function with frequency less than or similar to k in d dimensions. It proves that the h-version of the Galerkin method applied to certain boundary integral equations does not suffer from the pollution effect when the obstacle does not trap geometric-optic rays.
Article
Mathematics, Applied
Carlos Borges, Manas Rachh, Leslie Greengard
Summary: The acoustic inverse obstacle scattering problem aims to determine the shape of a domain based on measurements of the scattered far field caused by a set of incident fields. Two approaches, treating the boundary alone or the entire object as unknown, are discussed. Both methods result in strongly nonlinear and nonconvex optimization problems, but recursive linearization provides a useful framework. A systematic study comparing the performance of these methods on different examples reveals that the volumetric approach is more robust, despite having a larger number of degrees of freedom. The phenomenon is discussed, along with potential directions for further research.
Article
Engineering, Electrical & Electronic
Ye Pan, Xiao-Wei Huang, Xin-Qing Sheng
Summary: The proposed AE-CFIE strategy enhances accuracy by replacing the original CFIE with EFIE on sharp edges, demonstrating improved performance in computing scattering from PEC objects.
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION
(2021)
Article
Engineering, Marine
Hongli Ge, Bingchen Liang, Libang Zhang
Summary: A novel waveguide type breakwater has been developed based on wave space modulation and grating control technologies. The paper analyzes the waveguiding effect on wave field distribution and studies the blocking effects of multiple vertical cylinders designed using combination grating theories. Numerical results show that a specific topology has better blocking effect. This is the first study to use topology control and combination grating theories in multiple cylinders.
Article
Mathematics, Applied
Habib Zribi
Summary: The paper establishes relationships between the shape deformation of an impenetrable obstacle and boundary measurements of scattering fields based on layer potential techniques and the field expansion method. It extends these techniques to derive asymptotic expansions of the Dirichlet-to-Neumann and Neumann-to-Dirichlet operators, providing effective algorithms for determining lower order Fourier coefficients of the shape perturbation of the obstacle.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Mathematics, Applied
Lu Zhao, Heping Dong, Fuming Ma
Summary: This paper discusses an inverse elastic scattering problem of determining a rigid obstacle from time domain scattered field data. By using Helmholtz decomposition and the retarded single layer potential, a set of coupled boundary integral equations is established and the uniqueness of the solution is proven through the energy method. The coupled boundary integral equations are reformulated using the convolution quadrature method and a convolution quadrature based nonlinear integral equation method is proposed for solving the inverse problem.
Article
Computer Science, Interdisciplinary Applications
Olha Ivanyshyn Yaman, Gazi Ozdemir
Summary: This study proposes two numerical schemes for solving the boundary value problem for the modified Helmholtz equation and generalized impedance boundary condition. The first scheme handles the hyper-singular integral operator by splitting off the singularity technique, while the second scheme employs the idea of numerical differentiation. Numerical examples demonstrate exponential convergence for analytic data in the first method.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2021)
Article
Mathematics, Applied
Ying Jiang, Bo Wang, Dandan Yu
Summary: This paper introduces a fast Fourier-Galerkin method for solving Dirichlet problems for the Helmholtz equation. By splitting integral operators and using a truncation strategy to compress the coefficient matrix, computational efficiency and accuracy can be improved. Exponential convergence is achievable with suitable assumptions.
NUMERICAL ALGORITHMS
(2021)
Article
Mathematics, Applied
Dries Bosman, Martijn Huynen, Daniel De Zutter, Hendrik Rogier, Dries Vande Ginste
Summary: This paper presents a novel technique for accurately modeling scattering phenomena at two-dimensional circular and rectangular structures, including magnetic materials, with the ability for broadband modeling. Proper selection of basis functions is critical for achieving convergence in the presence of magnetic contrast.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Computer Science, Interdisciplinary Applications
Bowei Wu, Min Hyung Cho
Summary: This work presents a boundary integral equation method for the Helmholtz equation in 3-D multilayered media, which is robust and efficient.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
Yaning Xie, Shuwang Li, Wenjun Ying
Summary: In this paper, a fourth-order Cartesian grid-based boundary integral method (BIM) is proposed for solving a multiple acoustic scattering problem on closely packed obstacles. The method does not require complex computations for nearly singular, singular or hyper-singular boundary integrals, but instead reinterprets them as solutions to equivalent simple interface problems. Extensive numerical experiments show that the method is formally high-order accurate, fast convergent and insensitive to complexity of scatterers.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Carlos Fresneda-Portillo, Zenebe Woldemicheal
Summary: In this study, the interior Dirichlet boundary value problem for the diffusion equation in nonhomogeneous media is transformed into a system of boundary-domain integral equations (BDIEs) using a parametrix from Portillo (2019). The research also extends previous results for the mixed problem in a smooth domain to Lipschitz domains and PDE right-hand side in the Sobolev space H-1(Ω), proving the equivalence between the system of BDIEs and the original BVP, as well as their solvability and solution uniqueness in appropriate Sobolev spaces.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Carlos Fresneda-Portillo, Zenebe Woldemicheal
Summary: This study simplifies the interior Dirichlet boundary value problem for the diffusion equation in nonhomogeneous media to a system of boundary-domain integral equations using a parametrix obtained in Portillo (2019). Results from Portillo (2019) are further extended to Lipschitz domains with a PDE right-hand side in the Sobolev space H-1(Omega), where neither classical nor canonical conormal derivatives are well-defined. Equivalence between the system of BDIEs and the original BVP is proven, along with their solvability and solution uniqueness in appropriate Sobolev spaces.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Rainer Kress
Summary: The vector Helmholtz equation is a generalization of the time-harmonic Maxwell equations for propagating time-harmonic electromagnetic waves. Reciprocity results for scattering of plane waves and point sources are obtained by reviewing the electric and magnetic boundary conditions. Uniqueness results for the inverse obstacle scattering problem and results on the far field operator are derived from the reciprocity results. These findings extend the corresponding results for the Maxwell equations and relate to the Dirichlet and Neumann boundary conditions for the scalar Helmholtz equation. The extension of the DB boundary condition from the Maxwell equations to the vector Helmholtz equation is also briefly considered.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2023)
Article
Mathematics, Applied
Ralf Hiptmair, Andrea Moiola, Euan A. Spence
Summary: This article considers the Helmholtz transmission problem with piecewise-constant material coefficients and the standard associated direct boundary integral equations. For certain coefficients and geometries, the norms of the inverses of the boundary integral operators increase rapidly through an increasing sequence of frequencies, even though this is not the case for the solution operator of the transmission problem. This phenomenon is referred to as spurious quasi-resonances. The article provides a rigorous explanation of why and when spurious quasi-resonances occur and proposes modified boundary integral equations that are not affected by them.
SIAM JOURNAL ON APPLIED MATHEMATICS
(2022)