Isogeometric collocation for acoustic problems with higher-order boundary conditions

Abstract

Boundary value problems (BVPs) for the Helmholtz equation in an infinite domain include the requirement on the asymptotic behavior of the solution at infinity, the so-called Sommerfeld radiation condition. When the BVP is solved by any domain type numerical method, the infinite domain is truncated by an artificial boundary, where the Sommerfeld radiation condition is approximated by the absorbing boundary condition (ABC). Therefore, in addition to the pollution and discretization errors, numerical solution is affected by the domain truncation error.In this work, we develop a numerical framework based on isogeometric collocation to treat BVPs with ABCs of two types: Bayliss–Gunzburger–Turkel (BGT) ABC, formulated in terms of derivatives of arbitrary high order (in this work we considered up to degree 4 in 2D and up to degree 2 in 3D), and Karp’s (2D) and Wilcox (3D) farfield expansions (KFE, WFE) ABCs, formulated in terms of one or two families of unknown boundary functions. The approach inherits all main features of isogeometric collocation, such as lower computational cost in comparison with Galerkin methods and reduced pollution error due to higher order and higher continuity of NURBS. The latter also facilitates evaluation of higher order derivatives appearing in the ABCs. The tensor-product structure of NURBS patches allows to use the same basis functions for discretizing both the solution inside the domain and the unknown boundary functions.We analyze the accuracy of the ABCs on two benchmark problems, for which the corresponding analytical solutions in the infinite and truncated domains are available. Comparison with the analytical solutions allows to estimate both the discretization and domain truncation errors. We conduct a detailed parametric study to demonstrate the performance of the ABCs depending on the wavenumber, radius of the truncation surface, the number of terms in the farfield expansion or the order of derivatives in the BGT ABC. Finally, the ability of the approach to handle complex geometries is also demonstrated.