Gaussian quadrature for splines via homotopy continuation: Rules for C2 cubic splines
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We introduce a new concept for generating optimal quadrature rules for splines. To generate an optimal quadrature rule in a given (target) spline space, we build an associated source space with known optimal quadrature and transfer the rule from the source space to the target one, while preserving the number of quadrature points and therefore optimality. The quadrature nodes and weights are, considered as a higher-dimensional point, a zero of a particular system of polynomial equations. As the space is continuously deformed by changing the source knot vector, the quadrature rule gets updated using polynomial homotopy continuation. For example, starting with C1 cubic splines with uniform knot sequences, we demonstrate the methodology by deriving the optimal rules for uniform C2 cubic spline spaces where the rule was only conjectured to date. We validate our algorithm by showing that the resulting quadrature rule is independent of the path chosen between the target and the source knot vectors as well as the source rule chosen.
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Optimal quadrature rules for odd-degree spline spaces and their application to tensor-product-based isogeometric analysisBarton, M.; Calo, Victor (2016)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 ...
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Barton, M.; Ait-Haddou, R.; Calo, Victor (2017)We provide explicit quadrature rules for spaces of C1 quintic splines with uniform knot sequences over finite domains. The quadrature nodes and weights are derived via an explicit recursion that avoids numerical solvers. ...