Isogeometric collocation method based on residual parameterization of planar physical domain

Abstract

Isogeometric analysis (IGA) becomes an effective tool for solving partial differential equations (PDEs). Compared with isogeometric Galerkin method, isogeometric collocation (IGC) method has higher computational efficiency. In this paper, we aim to optimize the domain parameterization in IGC for better numerical accuracy of solving PDEs. Firstly, we present a new parameterization method of planar physical domain, called residual parameterization, which is obtained by minimizing the objective functions consisting of geometry-related functionals and the analysis-related residual norms in an unconstrained optimization problem. Secondly, the reduced quadrature rules are applied to residual norms due to high computational cost in evaluating the integration of residual terms. Finally, based on the residual parameterization, we solve boundary value problems (BVPs) by IGC with Greville points and superconvergent points to verify the strength of our proposed residual parameterization of planar physical domain. Several numerical examples show that the numerical accuracy of the proposed method is improved nearly half order of magnitude compared with standard IGC method.