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

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• 作者：Michael Bartoň^b ; ^a ; ^{Michael.Barton@centrum.cz" class="auth_mail" title="E-mail the corresponding author} ; Victor Manuel Calo^c ; ^a ; ^{Victor.Calo@Curtin.edu.au" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Optimal quadrature rules ; Galerkin method ; Gaussian quadrature ; B-splines ; Isogeometric analysis ; Homotopy continuation for quadrature
• 刊名：Computer Methods in Applied Mechanics and Engineering
• 出版年：2016
• 出版时间：15 June 2016
• 年：2016
• 卷：305
• 期：Complete
• 页码：217-240
• 全文大小：1979 K

文摘

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 (Bartoň 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.