Gauss-Galerkin quadrature rules for quadratic and cubic spline spaces and their application to isogeometric analysis

Michael Barton; Victor M. Calo

doi:10.1016/j.cad.2016.07.003

Gauss-Galerkin quadrature rules for quadratic and cubic spline spaces and their application to isogeometric analysis

Michael Barton, Victor M. Calo

Research output: Contribution to journal › Article › peer-review

62 Scopus citations

Abstract

We introduce Gaussian quadrature rules for spline spaces that are frequently used in Galerkin discretizations to build mass and stiffness matrices. By definition, these spaces are of even degrees. The optimal quadrature rules we recently derived (Bartoň and Calo, 2016) act on spaces of the smallest odd degrees and, therefore, are still slightly sub-optimal. In this work, we derive optimal rules directly for even-degree spaces and therefore further improve our recent result. We use optimal quadrature rules for spaces over two elements as elementary building blocks and use recursively the homotopy continuation concept described in Bartoň and Calo (2016) to derive optimal rules for arbitrary admissible numbers of elements.We demonstrate the proposed methodology on relevant examples, where we derive optimal rules for various even-degree spline spaces. We also discuss convergence of our rules to their asymptotic counterparts, these are the analogues of the midpoint rule of Hughes et al. (2010), that are exact and optimal for infinite domains.

Original language	English (US)
Pages (from-to)	57-67
Number of pages	11
Journal	Computer-Aided Design
Volume	82
DOIs	https://doi.org/10.1016/j.cad.2016.07.003
State	Published - Jul 22 2016

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: This publication was made possible in part by a National Priorities Research Program grant 7-1482-1-278 from the Qatar National Research Fund (a member of The Qatar Foundation), by the European Unions Horizon 2020 Research and Innovation Program under the Marie Sklodowska-Curie grant agreement No. 644602, and the Center for Numerical Porous Media at King Abdullah University of Science and Technology (KAUST). The first author has been partially supported by Bizkaia Talent under Grant AYD-000-270, by the Basque Government through the BERC 2014–2017 program, and by Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2013-0323. The J. Tinsley Oden Faculty Fellowship Research Program at the Institute for Computational Engineering and Sciences (ICES) of the University of Texas at Austin has partially supported the visits of VMC to ICES.

ASJC Scopus subject areas

Computer Graphics and Computer-Aided Design
Industrial and Manufacturing Engineering
Computer Science Applications

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10.1016/j.cad.2016.07.003

https://repository.kaust.edu.sa/bitstream/10754/617532/1/1-s2.0-S0010448516300665-main.pdf

Cite this

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title = "Gauss-Galerkin quadrature rules for quadratic and cubic spline spaces and their application to isogeometric analysis",

abstract = "We introduce Gaussian quadrature rules for spline spaces that are frequently used in Galerkin discretizations to build mass and stiffness matrices. By definition, these spaces are of even degrees. The optimal quadrature rules we recently derived (Barto{\v n} and Calo, 2016) act on spaces of the smallest odd degrees and, therefore, are still slightly sub-optimal. In this work, we derive optimal rules directly for even-degree spaces and therefore further improve our recent result. We use optimal quadrature rules for spaces over two elements as elementary building blocks and use recursively the homotopy continuation concept described in Barto{\v n} and Calo (2016) to derive optimal rules for arbitrary admissible numbers of elements.We demonstrate the proposed methodology on relevant examples, where we derive optimal rules for various even-degree spline spaces. We also discuss convergence of our rules to their asymptotic counterparts, these are the analogues of the midpoint rule of Hughes et al. (2010), that are exact and optimal for infinite domains.",

author = "Michael Barton and Calo, {Victor M.}",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: This publication was made possible in part by a National Priorities Research Program grant 7-1482-1-278 from the Qatar National Research Fund (a member of The Qatar Foundation), by the European Unions Horizon 2020 Research and Innovation Program under the Marie Sklodowska-Curie grant agreement No. 644602, and the Center for Numerical Porous Media at King Abdullah University of Science and Technology (KAUST). The first author has been partially supported by Bizkaia Talent under Grant AYD-000-270, by the Basque Government through the BERC 2014–2017 program, and by Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2013-0323. The J. Tinsley Oden Faculty Fellowship Research Program at the Institute for Computational Engineering and Sciences (ICES) of the University of Texas at Austin has partially supported the visits of VMC to ICES.",

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N2 - We introduce Gaussian quadrature rules for spline spaces that are frequently used in Galerkin discretizations to build mass and stiffness matrices. By definition, these spaces are of even degrees. The optimal quadrature rules we recently derived (Bartoň and Calo, 2016) act on spaces of the smallest odd degrees and, therefore, are still slightly sub-optimal. In this work, we derive optimal rules directly for even-degree spaces and therefore further improve our recent result. We use optimal quadrature rules for spaces over two elements as elementary building blocks and use recursively the homotopy continuation concept described in Bartoň and Calo (2016) to derive optimal rules for arbitrary admissible numbers of elements.We demonstrate the proposed methodology on relevant examples, where we derive optimal rules for various even-degree spline spaces. We also discuss convergence of our rules to their asymptotic counterparts, these are the analogues of the midpoint rule of Hughes et al. (2010), that are exact and optimal for infinite domains.

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Gauss-Galerkin quadrature rules for quadratic and cubic spline spaces and their application to isogeometric analysis

Abstract

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