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

机译:奇数样条空间的最佳正交规则及其在基于张量积的等几何分析中的应用

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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 (Barton and Calo, 2016) 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. (2010), 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. (C) 2016 Elsevier B.V. All rights reserved.
机译:我们为样条空间引入了最佳正交规则,这些规则通常在Galerkin离散化中用于建立质量和刚度矩阵。使用同伦延续概念(Barton和Calo,2016)将最优正交规则从源空间转换为目标空间,我们导出了有限域上定义的样条曲线的最优规则。从多项式的经典高斯正交开始,这是不连续奇数空间的最佳规则,我们得出了更高连续性的目标空间的规则。我们进一步展示了同伦方法论如何处理源规则和目标规则需要不同数量的最佳正交点的情况。我们通过推导各种奇数样条空间的最优规则来证明这一点,特别是对于非均匀结序列和非均匀多重性。我们还讨论了我们的规则到渐近对应项的收敛,即,休斯等人的中点规则的类似物。 (2010年),这是无限域的精确和最优选择。对于连续性较低的空间,我们用数值方法表明,随着一些边界元素的权重和节点与渐近值不同,导出的规则会迅速收敛到它们的渐近对应项。 (C)2016 Elsevier B.V.保留所有权利。

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