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

Handle URI:
http://hdl.handle.net/10754/617532
Title:
Gauss-Galerkin quadrature rules for quadratic and cubic spline spaces and their application to isogeometric analysis
Authors:
Barton, Michael ( 0000-0002-1843-251X ) ; Calo, Victor M. ( 0000-0002-1805-4045 )
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.
KAUST Department:
Center for Numerical Porous Media (NumPor)
Citation:
Gauss-Galerkin quadrature rules for quadratic and cubic spline spaces and their application to isogeometric analysis 2016 Computer-Aided Design
Publisher:
Elsevier BV
Journal:
Computer-Aided Design
Issue Date:
21-Jul-2016
DOI:
10.1016/j.cad.2016.07.003
Type:
Article
ISSN:
00104485
Sponsors:
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.
Additional Links:
http://linkinghub.elsevier.com/retrieve/pii/S0010448516300665
Appears in Collections:
Articles

Full metadata record

DC FieldValue Language
dc.contributor.authorBarton, Michaelen
dc.contributor.authorCalo, Victor M.en
dc.date.accessioned2016-07-26T09:27:00Z-
dc.date.available2016-07-26T09:27:00Z-
dc.date.issued2016-07-21-
dc.identifier.citationGauss-Galerkin quadrature rules for quadratic and cubic spline spaces and their application to isogeometric analysis 2016 Computer-Aided Designen
dc.identifier.issn00104485-
dc.identifier.doi10.1016/j.cad.2016.07.003-
dc.identifier.urihttp://hdl.handle.net/10754/617532-
dc.description.abstractWe 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.en
dc.description.sponsorshipThis 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.en
dc.language.isoenen
dc.publisherElsevier BVen
dc.relation.urlhttp://linkinghub.elsevier.com/retrieve/pii/S0010448516300665en
dc.rightsNOTICE: this is the author’s version of a work that was accepted for publication in Computer-Aided Design. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Computer-Aided Design, 21 July 2016. DOI: 10.1016/j.cad.2016.07.003en
dc.subjectOptimal quadrature rulesen
dc.subjectGalerkin methoden
dc.subjectGaussian quadratureen
dc.subjectB-splinesen
dc.subjectIsogeometric analysisen
dc.subjectHomotopy continuation for quadratureen
dc.titleGauss-Galerkin quadrature rules for quadratic and cubic spline spaces and their application to isogeometric analysisen
dc.typeArticleen
dc.contributor.departmentCenter for Numerical Porous Media (NumPor)en
dc.identifier.journalComputer-Aided Designen
dc.eprint.versionPost-printen
dc.contributor.institutionBCAM–Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Basque Country, Spainen
dc.contributor.institutionCSIRO Professorial Chair in Computational Geoscience, Western Australian School of Mines, Faculty of Science and Engineering, Curtin University, Kent Street, Bentley, Perth, Western Australia, 6102, Australiaen
dc.contributor.institutionMineral Resources, CSIRO, Kensington, Western Australia, 6102, Australiaen
dc.contributor.affiliationKing Abdullah University of Science and Technology (KAUST)en
kaust.authorBarton, Michaelen
kaust.authorCalo, Victor M.en
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