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

Abstract

This work is concerned with a unique combination of high order local absorbing boundary conditions (ABC) with a general curvilinear Finite Element Method (FEM) and its implementation in Isogeometric Analysis (IGA) for time-harmonic acoustic waves. The ABC employed were recently devised by Villamizar, Acosta and Dastrup [J. Comput. Phys. 333 (2017) 331] . They are derived from exact Farfield Expansions representations of the outgoing waves in the exterior of the regions enclosed by the artificial boundary. As a consequence, the error due to the ABC on the artificial boundary can be reduced conveniently such that the dominant error comes from the volume discretization method used in the interior of the computational domain. Reciprocally, the error in the interior can be made as small as the error at the artificial boundary by appropriate implementation of p-and h-refinement. We apply this novel method to cylindrical, spherical and arbitrary shape scatterers including a prototype submarine. Our numerical results exhibits spectral-like approximation and high order convergence rate. Additionally, they show that the proposed method can reduce both the pollution and artificial boundary errors to negligible levels even in very lowand high-frequency regimes with rather coarse discretization densities in the IGA. As a result, we have developed a highly accurate computational platform to numerically solve time-harmonic acoustic wave scattering in two-and three-dimensions.

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

“…This fact confirms the arbitrary high order character of the KFE-ABC that was claimed in [33]. Actually, this high order property was already observed in [41] where the KFE-ABC was combined with a high order isogeometric finite element method employed as a Helmholtz solver for the interior.…”

“…They reached only fourth order convergence for circular obstacles. More recently, Tahsin and Villamizar [41], formulated an overall arbitrary high order method for acoustic scattering combining a finite element implementation in isogeometric analysis (IGA), which uses arbitrary high order non-uniform rational B-splines (NURBS) as a basis, with the high order local farfield expansions ABC employed in this study. Their experiments corroborated the overall high order convergence for high and very low frequencies (wavenumbers).…”

“…Their experiments corroborated the overall high order convergence for high and very low frequencies (wavenumbers). Our plan is to compare the accuracy and computational cost performances of our overall high order DC scheme combined with the KFE, constructed in this work, against the high order method introduced in [41] and report these results elsewhere. and are given by (u 4r ) i j ≈ D…”

Non Uniform Rational B-Splines and Lagrange approximations for time-harmonic acoustic scattering: accuracy and absorbing boundary conditions

“…Another approach is to use enrichment techniques [65,66], but from the numerical study it is shown that the truncation error had higher influence than the discretization error. Khajah et al [67] employed the recently developed higher order local ABC based on the Karp's far-field expansion (KFE) and Wilcox's far-field expansion (WFE) within IGA for low to high frequencies for 2D and 3D problems, respectively. Unlike other ABCs, the salient feature of these higher order boundary conditions is that they do not require higher order derivative evaluations and hence impose no additional constraint on the regularity of the basis functions.…”

“…Unlike other ABCs, the salient feature of these higher order boundary conditions is that they do not require higher order derivative evaluations and hence impose no additional constraint on the regularity of the basis functions. This implies that the same basis functions can be used to estimate the field and the auxiliary ABC functions [67]. This is accomplished by introducing two families of auxiliary variables on the fictitious boundary.…”