High order local absorbing boundary conditions for acoustic waves in terms of farfield expansions

Abstract

We devise a new high order local absorbing boundary condition (ABC) for radiating problems and scattering of time-harmonic acoustic waves from obstacles of arbitrary shape. By introducing an artificial boundary S enclosing the scatterer, the original unbounded domain Ω is decomposed into a bounded computational domain Ω − and an exterior unbounded domain Ω + . Then, we define interface conditions at the artificial boundary S , from truncated versions of the well-known Wilcox and Karp farfield expansion representations of the exact solution in the exterior region Ω + . As a result, we obtain a new local absorbing boundary condition (ABC) for a bounded problem on Ω − , which effectively accounts for the outgoing behavior of the scattered field. Contrary to the low order absorbing conditions previously defined, the order of the error induced by this ABC can easily match the order of the numerical method in Ω − . We accomplish this by simply adding as many terms as needed to the truncated farfield expansions of Wilcox or Karp. The convergence of these expansions guarantees that the order of approximation of the new ABC can be increased arbitrarily without having to enlarge the radius of the artificial boundary. We include numerical results in two and three dimensions which demonstrate the improved accuracy and simplicity of this new formulation when compared to other absorbing boundary conditions.

Highly accurate acoustic scattering: Isogeometric Analysis coupled with local high order Farfield Expansion ABC

“…The three dimensional scattering problem using Wilcox farfield expansion absorbing boundary condition (WFE) with L terms was also introduced in [18]. The corresponding equations in spherical coordinates are:…”

“…In this work, we propose a numerical method consisting of an IGA technique employing arbitrary high order NURBS bases combined with recently developed high order local Farfield Expansions absorbing boundary conditions [18]. These conditions are defined from truncated versions of exact series representations of the outgoing waves outside a circular and spherical artificial boundary in 2D and 3D, respectively.…”

High order methods for acoustic scattering: Coupling farfield expansions ABC with deferred-correction methods

“…The exterior problem (1)-(3) needs to be reformulated as an equivalent BVP on a bounded domain before a numerical scheme, based on volume discretization methods, can be applied. In Villamizar et al [33], such problem transformation was carried out by introducing a circular (2D) or spherical (3D) artificial boundary S that enclose all the scatterers, regarding of their particular shapes, and then by defining high order local ABCs on these artificial boundaries. Their definition is based on the following series representations of the exact solution u outside the region bounded by the artificial boundary S .…”

“…As a consequence, overall low order numerical methods (at most second order) are obtained. For other contributions on high order ABC, the reader is referred to the review article [32] and also to the introduction in [33].…”

“…Among the pioneer applications of deferred-corrections to differential equations are the works by Pereyra [36,37]. Our construction proceed by coupling arbitrary high order DC finite difference schemes for the Helmholtz equation with high order DC finite difference schemes corresponding to arbitrary high order ABCs based on farfield expansions, which were developed by Villamizar et al in [33]. As a result of combining these high order techniques (domain's interior and boundary), we obtain an overall arbitrary high order method for acoustic scattering.…”