Enriched isogeometric collocation for two-dimensional time-harmonic acoustics

Abstract

In this work, the isogeometric collocation (IGA-C) is paired with two types of enrichment: Plane Wave (PW-) and Generalized Harmonic Polynomial (GHP-) functions, to solve 2D-problems for the Helmholtz equation. A parametric study is conducted, and a detailed assessment of the performance of the method in a number of benchmark problems is provided. Three different collocation methods are tested for the non-enriched formulation, namely Greville abscissae (GA), Approximated Cauchy-Galerkin (ACG) and Superconvergent (SC) points, showing that the ACG scheme is the best choice in terms of overall error, convergence rate and ease of solving the linear system. Then, the influence of the number of shape functions in the original and enriched basis, location and number of collocation points and wave-number over both the convergence rate as well as the condition number of the stiffness matrix are studied. The numerical results show that: (1) there is an improvement over the non-enriched formulation, (2) the improvement depends on the choice of the number and type of enrichment, and (3) The pollution error is not completely alleviated with the enriched formulations.