Efficient quadrature for NURBS-based isogeometric analysis

Overlapping Schwarz Methods for Isogeometric Analysis

“…For instance, spaces of global C k regularity are easily built, thus allowing for fewer degrees of freedom, better performance in case of vibrations, easier approximation of higher order problems, and other advantages. IGA methodologies have been summarized in the recent book [18] and studied in, e.g., [2,4,9,10,19,23,28,29,11,5,8]. IGA methods are having a growing impact on fields as diverse as fluid dynamics [6,7,40,15,26], structural mechanics [3,1,12,20,30,39], and electromagnetics [17,16].…”

Computation of the Isogeometric Analysis Stiffness Matrix on Gpu

“…However, the standard element-wise Gauss rule requires extensive function evaluations due to increased support of the shape functions. According to [3], for the case of an one-dimensional function of order p the optimal (minimum exact) number of Gauss points per element is equal to ( p +1) /2 or ( p +2)/2 , for odd and even p respectively. For the computation of the stiffness matrix in 1D elasticity case, the integrand's order is equal to q=2 p−2 and the optimal number of Gauss points per element is equal to (q +2)/2= p .…”

“…The above rules are optimal for the case of minimum continuity. For higher continuity, new macro-element rules have been proposed [2], [3], which are more efficient, but also more involved and difficult to implement.…”

“…It was shown [2], [3] that standard element-wise Gauss rules are inefficient, because they do not take precise account of the preserved smoothness at the element boundaries in the case of higher-order NURBS and polynomial B-SPLines, and that the higher the inter-element regularity is the fewer the required number of Gauss points per element. However, recently proposed integration rules, although they are optimal or nearly optimal in terms of the number of function evaluations, they are either cumbersome to implement [2] or need special consideration to be given to the boundary elements [3].…”

“…However, recently proposed integration rules, although they are optimal or nearly optimal in terms of the number of function evaluations, they are either cumbersome to implement [2] or need special consideration to be given to the boundary elements [3]. In an effort to deal with the overhead in the computation of IGA characteristic matrices, collocation methods have been introduced for obtaining higher order accurate methods with a minimum number of quadrature points [5].…”