2,987 research outputs found
Recommended from our members
A Galerkin boundary element method for high frequency scattering by convex polygons
In this paper we consider the problem of time-harmonic acoustic scattering in two dimensions by convex polygons. Standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave. Here we present a novel Galerkin boundary element method, which uses an approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh, with smaller elements closer to the corners of the polygon. We prove that the best approximation from the approximation space requires a number of degrees of freedom to achieve a prescribed level of accuracy that grows only logarithmically as a function of the frequency. Numerical results demonstrate the same logarithmic dependence on the frequency for the Galerkin method solution. Our boundary element method is a discretization of a well-known second kind combined-layer-potential integral equation. We provide a proof that this equation and its adjoint are well-posed and equivalent to the boundary value problem in a Sobolev space setting for general Lipschitz domains
Recommended from our members
Boundary integral methods in high frequency scattering
In this article we review recent progress on the design, analysis and implementation of numerical-asymptotic boundary integral methods for the computation of frequency-domain acoustic scattering in a homogeneous unbounded medium by a bounded obstacle. The main aim of the methods is to allow computation of scattering at arbitrarily high frequency with finite computational resources
An efficient high-order Nystr\"om scheme for acoustic scattering by inhomogeneous penetrable media with discontinuous material interface
This text proposes a fast, rapidly convergent Nystr\"{o}m method for the
solution of the Lippmann-Schwinger integral equation that mathematically models
the scattering of time-harmonic acoustic waves by inhomogeneous obstacles,
while allowing the material properties to jump across the interface. The method
works with overlapping coordinate charts as a description of the given
scatterer. In particular, it employs "partitions of unity" to simplify the
implementation of high-order quadratures along with suitable changes of
parametric variables to analytically resolve the singularities present in the
integral operator to achieve desired accuracies in approximations. To deal with
the discontinuous material interface in a high-order manner, a specialized
quadrature is used in the boundary region. The approach further utilizes an FFT
based strategy that uses equivalent source approximations to accelerate the
evaluation of large number of interactions that arise in the approximation of
the volumetric integral operator and thus achieves a reduced computational
complexity of for an -point discretization. A detailed
discussion on the solution methodology along with a variety of numerical
experiments to exemplify its performance in terms of both speed and accuracy
are presented in this paper
Fast and Accurate Computation of Time-Domain Acoustic Scattering Problems with Exact Nonreflecting Boundary Conditions
This paper is concerned with fast and accurate computation of exterior wave
equations truncated via exact circular or spherical nonreflecting boundary
conditions (NRBCs, which are known to be nonlocal in both time and space). We
first derive analytic expressions for the underlying convolution kernels, which
allow for a rapid and accurate evaluation of the convolution with
operations over successive time steps. To handle the onlocality in space,
we introduce the notion of boundary perturbation, which enables us to handle
general bounded scatters by solving a sequence of wave equations in a regular
domain. We propose an efficient spectral-Galerkin solver with Newmark's time
integration for the truncated wave equation in the regular domain. We also
provide ample numerical results to show high-order accuracy of NRBCs and
efficiency of the proposed scheme.Comment: 22 pages with 9 figure
- …