12 research outputs found
Fast, higher-order direct/iterative hybrid solver for scattering by Inhomogeneous media -- with application to high-frequency and discontinuous refractivity problems
This paper presents a fast high-order method for the solution of
two-dimensional problems of scattering by penetrable inhomogeneous media, with
application to high-frequency configurations containing (possibly)
discontinuous refractivities. The method relies on a combination of a
differential volumetric formulation and a boundary integral formulation. Thus,
in the proposed approach the entire computational domain is partitioned into
large numbers of volumetric spectral approximation patches which are then
grouped into patch subsets for local direct solution; the interactions with the
exterior domain are handled by means of a boundary integral equation. The
resulting algorithm can be quite effective: after a modestly-demanding
precomputation stage (whose results for a given frequency can be repeatedly
used for arbitrarily chosen incidence angles), the proposed algorithm can
accurately evaluate scattering by configurations including large and complex
objects and/or high refractivity contrasts, including possibly refractive-index
discontinuities, in fast single-core runs
Improved convergence of fast integral equation solvers for acoustic scattering by inhomogeneous penetrable media with discontinuous material interface
In recent years, several fast solvers for the solution of the
Lippmann-Schwinger integral equation that mathematically models the scattering
of time-harmonic acoustic waves by penetrable inhomogeneous obstacles, have
been proposed. While many of these fast methodologies exhibit rapid convergence
for smoothly varying scattering configurations, the rate for most of them
reduce to either linear or quadratic when material properties are allowed to
jump across the interface. A notable exception to this is a recently introduced
Nystr\"{o}m scheme [J. Comput. Phys., 311 (2016), 258--274] that utilizes a
specialized quadrature in the boundary region for a high-order treatment of the
material interface. In this text, we present a solution framework that relies
on the specialized boundary integrator to enhance the convergence rate of other
fast, low order methodologies without adding to their computational complexity
of for an -point discretization. In particular, to demonstrate
the efficacy of the proposed framework, we explain its implementation to
enhance the order to convergence of two schemes, one introduced by Duan and
Rokhlin [J. Comput. Phys., 228(6) (2009), 2152--2174] that is based on a
pre-corrected trapezoidal rule while the other by Bruno and Hyde [J. Comput.
Phys., 200(2) (2004), 670--694] which relies on a suitable decomposition of the
Green's function via Addition theorem. In addition to a detailed description of
these methodologies, we also present a comparative performance study of the
improved versions of these two and the Nystr\"{o}m solver in [J. Comput. Phys.,
311 (2016), 258--274] through a wide range of numerical experiments
Spectral Volumetric Integral Equation Methods for Acoustic Medium Scattering in a Planar Homogeneous 3D Waveguide
Scattering of acoustic waves from an inhomogeneous medium can be described by the Lippmann-Schwinger integral equation. For scattering problems in free space, Vainikko proposed a fast spectral solution method that exploits the convolution structure of this equation's integral operator by using the fast Fourier transform. In a planar 3--dimensional waveguide, the integral operator of the Lippmann-Schwinger integral equation fails to be a convolution. In this paper, we show that the separable structure of the kernel nevertheless allows to construct fast spectral collocation methods similar to Vainikko's technique. The numerical analysis of this method requires smooth material parameters; if the material parameters are, say, discontinuous, no theoretical statement on convergence is available. We show how to construct a Galerkin variant of Vainikko's method for which a rigorous convergence analysis is available even for discontinuous materials. For several distant scattering objects inside the 3--dimensional waveguide this discretization technique leads to a computational domain consisting of one large box containing all scatterers, and hence many unnecessary unknowns. However, the integral equation can be reformulated as a coupled system with unknowns defined on the different parts of the scatterer. Discretizing this coupled system by a combined spectral/multipole approach yields an efficient method for waveguide scattering from multiple objects
High order methods for acoustic scattering: Coupling Farfield Expansions ABC with Deferred-Correction methods
Arbitrary high order numerical methods for time-harmonic acoustic scattering
problems originally defined on unbounded domains are constructed. This is done
by coupling recently developed high order local absorbing boundary conditions
(ABCs) with finite difference methods for the Helmholtz equation. These ABCs
are based on exact representations of the outgoing waves by means of farfield
expansions. The finite difference methods, which are constructed from a
deferred-correction (DC) technique, approximate the Helmholtz equation and the
ABCs, with the appropriate number of terms, to any desired order. As a result,
high order numerical methods with an overall order of convergence equal to the
order of the DC schemes are obtained. A detailed construction of these DC
finite difference schemes is presented. Additionally, a rigorous proof of the
consistency of the DC schemes with the Helmholtz equation and the ABCs in polar
coordinates is also given. The results of several numerical experiments
corroborate the high order convergence of the novel method.Comment: 36 pages, 20 figure
Improved convergence of fast integral equation solvers for acoustic scattering by inhomogeneous penetrable media with discontinuous material interface
In recent years, several fast solvers for the solution of the LippmannâSchwinger integral equation that mathematically models the scattering of time-harmonic acoustic waves by penetrable inhomogeneous obstacles, have been proposed. While many of these fast methodologies exhibit rapid convergence for smoothly varying scattering configurations, the rate for most of them reduce to either linear or quadratic when material properties are allowed to jump across the interface. A notable exception to this is a recently introduced Nyström scheme (Anand et al., 2016 [22]) that utilizes a specialized quadrature in the boundary region for a high-order treatment of the material interface. In this text, we present a solution framework that relies on the specialized boundary integrator to enhance the convergence rate of other fast, low order methodologies without adding to their computational complexity of O(N log N) for an N-point discretization. In particular, to demonstrate the efficacy of the proposed framework, we explain its implementation to enhance the order to convergence of two schemes, one introduced by Duan and Rokhlin (2009) [13] that is based on a pre-corrected trapezoidal rule while the other by Bruno and Hyde (2004) [12] which relies on a suitable decomposition of the Green's function via Addition theorem. In addition to a detailed description of these methodologies, we also present a comparative performance study of the improved versions of these two and the Nyström solver in Anand et al. (2016) [22] through a wide range of numerical experiments