Exploring Matrix Generation Strategies in Isogeometric Analysis

Abstract

An important step in simulation via isogeometric analysis (IGA) is the assembly step, where the coefficients of the final linear system are generated. Typically, these coefficients are integrals of products of shape functions and their derivatives. Similarly to the finite element analysis (FEA), the standard choice for integral evaluation in IGA is Gaussian quadrature. Recent developments propose different quadrature rules, that reduce the number of quadrature points and weights used. We experiment with the existing methods for matrix generation. Furthermore we propose a new, quadrature-free approach, based on interpolation of the geometry factor and fast look-up operations for values of B-spline integrals. Our method builds upon the observation that exact integration is not required to achieve the optimal convergence rate of the solution. In particular, it suffices to generate the linear system within the order of accuracy matching the approximation order of the discretization space. We demonstrate that the best strategy is one that follows the above principle, resulting in expected accuracy and improved computational time.