Keywords: isogeometric analysis, Gaussian quadrature, half-point rule, B-spline, NURBS.

The concept of the isogeometric analysis (IGA), initially motivated by the gap between the computer aided design (CAD) and the finite element analysis (FEA), builds upon the concept of isoparametric elements, in which the same shape functions are used to approximate the geometry and the solution on a single finite element. The IGA, as its name suggests, goes one step further because it employs the same functions for the description of the geometry and for the approximation of the solution space on that geometry. This implies that the isogeometric mesh (discretization for computational purposes) of the CAD geometry encapsulates the exact geometry no matter how coarse the mesh actually is. As a consequence, the need to have a separate representation for the original CAD model and another one for the actual computational geometry is completely eliminated.

It has been shown that the IGA out performs the classical FEA in various aspects (accuracy, robustness, system condition number, etc.), which is the consequence of several important advantages of the IGA compared to the FEA. On the other hand, the computational effort of the IGA, especially when using higher order basis functions, seems to exceed that for the FEA. One of the fundamental performance issues of the IGA is the integration of individual components of the discretized governing differential equation (stiffness matrix, load vector, etc.). The capability of the IGA to 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 becomes computationally very prohibitive because the evaluation of the high degree basis functions and, or their derivatives is quite demanding. The situation tends to be critical in three-dimensions where the total number of integration points can increase dramatically. Currently there are three quadrature concepts available. The first, most natural one, performs the integration over individual non-zero knot spans (intervals of the underlying parametric space) on which the basis functions are continuous up to infinite order. The second approach attempts to profit from the continuity of the basis functions between the adjacent knot spans. This implies that it uses the multi-span (or macro-element) driven approach rather than the knot span driven scheme. The last concept using the Bezier extraction methodology tries to benefit from the fact that for the given order of integration, the values of individual shape functions and their derivatives are the same at individual integration points on all Bezier segments and can be therefore precomputed. Since there is typically as many Bezier segments as non-zero knot spans, the last concept resembles the first one in the sense that the non-zero knot spans are used as the basic integration units. The aim of this paper is to compare the performance of the above integration concepts on one-, two- and three-dimensional B-spline based problems.

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