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

Handle URI:
http://hdl.handle.net/10754/601360
Title:
Optimal quadrature rules for odd-degree spline spaces and their application to tensor-product-based isogeometric analysis
Authors:
Barton, Michael ( 0000-0002-1843-251X ) ; Calo, Victor M. ( 0000-0002-1805-4045 )
Abstract:
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.
KAUST Department:
Center for Numerical Porous Media (NumPor)
Citation:
Optimal quadrature rules for odd-degree spline spaces and their application to tensor-product-based isogeometric analysis 2016 Computer Methods in Applied Mechanics and Engineering
Publisher:
Elsevier BV
Journal:
Computer Methods in Applied Mechanics and Engineering
Issue Date:
14-Mar-2016
DOI:
10.1016/j.cma.2016.02.034
Type:
Article
ISSN:
00457825
Additional Links:
http://linkinghub.elsevier.com/retrieve/pii/S0045782516300640
Appears in Collections:
Articles

Full metadata record

DC FieldValue Language
dc.contributor.authorBarton, Michaelen
dc.contributor.authorCalo, Victor M.en
dc.date.accessioned2016-03-15T13:51:12Zen
dc.date.available2016-03-15T13:51:12Zen
dc.date.issued2016-03-14en
dc.identifier.citationOptimal quadrature rules for odd-degree spline spaces and their application to tensor-product-based isogeometric analysis 2016 Computer Methods in Applied Mechanics and Engineeringen
dc.identifier.issn00457825en
dc.identifier.doi10.1016/j.cma.2016.02.034en
dc.identifier.urihttp://hdl.handle.net/10754/601360en
dc.description.abstractWe 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.en
dc.language.isoenen
dc.publisherElsevier BVen
dc.relation.urlhttp://linkinghub.elsevier.com/retrieve/pii/S0045782516300640en
dc.rightsNOTICE: this is the author’s version of a work that was accepted for publication in Computer Methods in Applied Mechanics and Engineering. 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 Methods in Applied Mechanics and Engineering, 14 March 2016. DOI: 10.1016/j.cma.2016.02.034en
dc.subjectOptimal quadrature rulesen
dc.subjectGalerkin methoden
dc.subjectGaussian quadratureen
dc.subjectB-splinesen
dc.subjectIsogeometric analysisen
dc.subjectHomotopy continuation for quadratureen
dc.titleOptimal quadrature rules for odd-degree spline spaces and their application to tensor-product-based isogeometric analysisen
dc.typeArticleen
dc.contributor.departmentCenter for Numerical Porous Media (NumPor)en
dc.identifier.journalComputer Methods in Applied Mechanics and Engineeringen
dc.eprint.versionPost-printen
dc.contributor.institutionBasque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Basque Country, Spainen
dc.contributor.institutionChair 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.affiliationKing Abdullah University of Science and Technology (KAUST)en
kaust.authorBarton, Michaelen
kaust.authorCalo, Victor M.en
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