Assessment of Computational Efficiency of Numerical Quadrature Schemes in the Isogeometric Analysis

Isogeometric analysis (IGA) has been recently introduced as a viable alternative to the standard, polynomial-based finite element analysis. One of the fundamental performance issues of the isogeometric analysis is the quadrature of individual components of the discretized governing differential equation. The capability of the isogeometric analysis to easily adopt basis functions of high degree together with the (generally) rational form of those basis functions implies that high order numerical quadrature schemes must be employed. This may become computationally prohibitive because the evaluation of the high degree basis functions and/or their derivatives at individual integration points is quite demanding. The situation tends to be critical in three-dimensional space where the total number of integration points can increase dramatically. The aim of this paper is to compare computational efficiency of several numerical quadrature concepts which are nowadays available in the isogeometric analysis. Their performance is assessed on the assembly of stiffness matrix of B-spline based problems with special geometrical arrangement allowing to determine minimum number of integration points leading to exact results.