Gauss–Galerkin quadrature rules for quadratic and cubic spline spaces and their application to isogeometric analysis
© 2016 Elsevier LtdWe 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 (Barton and Calo, 2016) act on spac...
| Main Authors: | , |
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| Format: | Journal Article |
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Elsevier
2017
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| Online Access: | http://hdl.handle.net/20.500.11937/51172 |
| _version_ | 1848758633502867456 |
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| author | Barton, M. Calo, Victor |
| author_facet | Barton, M. Calo, Victor |
| author_sort | Barton, M. |
| building | Curtin Institutional Repository |
| collection | Online Access |
| description | © 2016 Elsevier LtdWe 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 (Barton 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 Barton 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. |
| first_indexed | 2025-11-14T09:47:05Z |
| format | Journal Article |
| id | curtin-20.500.11937-51172 |
| institution | Curtin University Malaysia |
| institution_category | Local University |
| last_indexed | 2025-11-14T09:47:05Z |
| publishDate | 2017 |
| publisher | Elsevier |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | curtin-20.500.11937-511722018-03-29T09:09:27Z Gauss–Galerkin quadrature rules for quadratic and cubic spline spaces and their application to isogeometric analysis Barton, M. Calo, Victor © 2016 Elsevier LtdWe 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 (Barton 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 Barton 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. 2017 Journal Article http://hdl.handle.net/20.500.11937/51172 10.1016/j.cad.2016.07.003 Elsevier restricted |
| spellingShingle | Barton, M. Calo, Victor Gauss–Galerkin quadrature rules for quadratic and cubic spline spaces and their application to isogeometric analysis |
| title | Gauss–Galerkin quadrature rules for quadratic and cubic spline spaces and their application to isogeometric analysis |
| title_full | Gauss–Galerkin quadrature rules for quadratic and cubic spline spaces and their application to isogeometric analysis |
| title_fullStr | Gauss–Galerkin quadrature rules for quadratic and cubic spline spaces and their application to isogeometric analysis |
| title_full_unstemmed | Gauss–Galerkin quadrature rules for quadratic and cubic spline spaces and their application to isogeometric analysis |
| title_short | Gauss–Galerkin quadrature rules for quadratic and cubic spline spaces and their application to isogeometric analysis |
| title_sort | gauss–galerkin quadrature rules for quadratic and cubic spline spaces and their application to isogeometric analysis |
| url | http://hdl.handle.net/20.500.11937/51172 |