Enriched isogeometric collocation for two-dimensional time-harmonic acoustics

Abstract

Nowadays, CAD and FEM softwares have a sustantial impact on engineering. Both of them are powerful tools for design and analysis. The main drawback is that FEM models and CAD geometries do not match, wich means that the generated CAD file needs to be translated into a suitable geometry , meshed and input into a large scale FEM code. This task takes most of the overall analysis time, and engineering designs are becoming increasingly more complex. Isogeometric Analysis (IGA) was proposed to build a direct link between CAD design and FEM analysis. The main idea of IGA is to substitute the shape functions used in FEM by the shape functions used in the CAD. Compared to the standard IGA, a much simpler method for approximating solutions to differential equations is called isogeometric collocation (IGA-C). Its simplicity follows from the fact that the differential equation (strong form) is only enforce at a discrete set of points called collocation points, thus the evaluation and assembly is much faster since there is no need of numerical integration. In this work, the IGA-C is paired with two types of enrichmentto in order to solve 2Dproblems for the Helmholtz equation: Plane Wave (PW-) and Generalized Harmonic Polynomial (GHP-) functions. 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 CG 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.