Optimal quadrature rules for odd-degree spline spaces and their application to tensor-product-based isogeometric analysis

Abstract

We introduce optimal quadrature rules for spline spaces that are frequently used in Galerkin discretizations to build mass and stiffness matrices. Using the homotopy continuation concept [6] that transforms optimal quadrature rules from source spaces to target spaces, we derive optimal rules for splines defined on finite domains. Starting with the classical Gaussian quadrature for polynomials, which is an optimal rule for a discontinuous odd-degree space, we derive rules for target spaces of higher continuity. We further show how the homotopy methodology handles cases where the source and target rules require different numbers of optimal quadrature points. We demonstrate it by deriving optimal rules for various odd-degree spline spaces, particularly with non-uniform knot sequences and non-uniform multiplicities. We also discuss convergence of our rules to their asymptotic counterparts, that is, the analogues of the midpoint rule of Hughes et al. [34], that are exact and optimal for infinite domains. For spaces of low continuities, we numerically show that the derived rules quickly converge to their asymptotic counterparts as the weights and nodes of a few boundary elements differ from the asymptotic values.

Gaussian quadrature rules for C1 quintic splines with uniform knot vectors

“…where r 4k−3 and r 4k−2 are the residues between the exact integrals, see (7) and (8), and the result of the rule when applied to D 4k−3 and D 4k−2 on the previous interval [x k−2 , x k−1 ], respectively. That is…”

“…Starting with a known Gaussian rule, e.g., a union of classical polynomial Gaussian rules, the underlying spline space is continuously transformed to the desired configuration and the root is numerically traced. Using this homotopy continuation concept, we derived numerically Gaussian rules for spline spaces of various degrees and continuities [6,8].…”

“…From the point of view of applications, certain types of quadratures (in terms of the relation between degrees and continuities of the underlying spline spaces) are very important in isogeometric analysis as these spaces appear in Galerkin discretizations when building the mass and stiffness matrices, see [6,8]. The quadrature rule for the spline spaces considered in this work, S 5,1 , can be used, e.g., to exactly integrate the products in the stiffness matrix when the original splines are C 2 cubics (S 3,2 ).…”

Optimal spectral approximation of2n-order differential operators by mixed isogeometric analysis

“…For simplicity, we denote by G l the l−point Gauss-Legendre quadrature rule, by L l the l−point Gauss-Lobatto quadrature rule, by R l the l−point Gauss-Radau quadrature rule, and by O p the optimal blending scheme for the p-th order isogeometric analysis with maximum continuity. In one dimension, G l , L l , and R l fully integrate polynomials of order 2l − 1, 2l − 3, and 2l − 2, respectively [9][10][11].…”

Matrix-free weighted quadrature for a computationally efficient isogeometric k-method

“…Computational limitations appear also in the solution phase of the linear system: the large support overlap associated with B-splines makes the associated Galerkin matrices less sparse than their finite element counterparts, which decreases the performance of direct solvers: it has been shown in [10] that a multifrontal direct method would require O(N 2 p 3 ) flops and O(N 4/3 p 2 ) memory to solve the system.Several alternatives to reduce this computational effort have been proposed in recent years. For what concerns the formation of isogeometric matrices, we refer to [2,8,25,20,22,3,30,16]. On the linear algebra side, the focus of research shifted to the development of preconditioned iterative approaches, see [7,15,13,17,29,31].The starting points of this paper are the results of [8], where a new strategy, named weighted quadrature, is proposed to form isogeometric Galerkin matrices.…”