Efficient mass and stiffness matrix assembly via weighted Gaussian quadrature rules for B-splines
Calabro et al. (2017) changed the paradigm of the mass and stiffness computation from the traditional element-wise assembly to a row-wise concept, showing that the latter one offers integration that may be orders of magnitude faster. Considering a B-spline basis function as a non-negative measure, e...
| Main Authors: | , , , |
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| Format: | Journal Article |
| Published: |
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2017
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| Online Access: | https://arxiv.org/abs/1710.01048 http://hdl.handle.net/20.500.11937/58067 |
| _version_ | 1848760169456992256 |
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| author | Bartoň, M. Puzyrev, Vladimir Deng, Quanling Calo, Victor |
| author_facet | Bartoň, M. Puzyrev, Vladimir Deng, Quanling Calo, Victor |
| author_sort | Bartoň, M. |
| building | Curtin Institutional Repository |
| collection | Online Access |
| description | Calabro et al. (2017) changed the paradigm of the mass and stiffness computation from the traditional element-wise assembly to a row-wise concept, showing that the latter one offers integration that may be orders of magnitude faster. Considering a B-spline basis function as a non-negative measure, each mass matrix row is integrated by its own quadrature rule with respect to that measure. Each rule is easy to compute as it leads to a linear system of equations, however, the quadrature rules are of the Newton-Cotes type, that is, they require a number of quadrature points that is equal to the dimension of the spline space. In this work, we propose weighted quadrature rules of Gaussian type which require the minimum number of quadrature points while guaranteeing exactness of integration with respect to the weight function. The weighted Gaussian rules arise as solutions of non-linear systems of equations. We derive rules for the mass and stiffness matrices for uniform C1 quadratic and C2 cubic isogeometric discretizations. Our rules further reduce the number of quadrature points by a factor of (p+12p+1)d when compared to Calabro et al. (2017), p being the polynomial degree and d the dimension of the problem, and consequently reduce the computational cost of the mass and stiffness matrix assembly by a similar factor. |
| first_indexed | 2025-11-14T10:11:30Z |
| format | Journal Article |
| id | curtin-20.500.11937-58067 |
| institution | Curtin University Malaysia |
| institution_category | Local University |
| last_indexed | 2025-11-14T10:11:30Z |
| publishDate | 2017 |
| publisher | - |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | curtin-20.500.11937-580672019-01-21T01:27:42Z Efficient mass and stiffness matrix assembly via weighted Gaussian quadrature rules for B-splines Bartoň, M. Puzyrev, Vladimir Deng, Quanling Calo, Victor Calabro et al. (2017) changed the paradigm of the mass and stiffness computation from the traditional element-wise assembly to a row-wise concept, showing that the latter one offers integration that may be orders of magnitude faster. Considering a B-spline basis function as a non-negative measure, each mass matrix row is integrated by its own quadrature rule with respect to that measure. Each rule is easy to compute as it leads to a linear system of equations, however, the quadrature rules are of the Newton-Cotes type, that is, they require a number of quadrature points that is equal to the dimension of the spline space. In this work, we propose weighted quadrature rules of Gaussian type which require the minimum number of quadrature points while guaranteeing exactness of integration with respect to the weight function. The weighted Gaussian rules arise as solutions of non-linear systems of equations. We derive rules for the mass and stiffness matrices for uniform C1 quadratic and C2 cubic isogeometric discretizations. Our rules further reduce the number of quadrature points by a factor of (p+12p+1)d when compared to Calabro et al. (2017), p being the polynomial degree and d the dimension of the problem, and consequently reduce the computational cost of the mass and stiffness matrix assembly by a similar factor. 2017 Journal Article http://hdl.handle.net/20.500.11937/58067 https://arxiv.org/abs/1710.01048 - restricted |
| spellingShingle | Bartoň, M. Puzyrev, Vladimir Deng, Quanling Calo, Victor Efficient mass and stiffness matrix assembly via weighted Gaussian quadrature rules for B-splines |
| title | Efficient mass and stiffness matrix assembly via weighted Gaussian quadrature rules for B-splines |
| title_full | Efficient mass and stiffness matrix assembly via weighted Gaussian quadrature rules for B-splines |
| title_fullStr | Efficient mass and stiffness matrix assembly via weighted Gaussian quadrature rules for B-splines |
| title_full_unstemmed | Efficient mass and stiffness matrix assembly via weighted Gaussian quadrature rules for B-splines |
| title_short | Efficient mass and stiffness matrix assembly via weighted Gaussian quadrature rules for B-splines |
| title_sort | efficient mass and stiffness matrix assembly via weighted gaussian quadrature rules for b-splines |
| url | https://arxiv.org/abs/1710.01048 http://hdl.handle.net/20.500.11937/58067 |