Reproducing kernel triangular B-spline-based FEM for solving PDEs

Research output: Contribution to journalArticleResearchpeer review

Authors

  • Yue Jia
  • Yongjie Zhang
  • Gang Xu
  • Xiaoying Zhuang
  • Timon Rabczuk

External Research Organisations

  • Tongji University
  • Carnegie Mellon University
  • Bauhaus-Universität Weimar
  • Hangzhou Dianzi University
  • Korea University
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Details

Original languageEnglish
Pages (from-to)342-358
Number of pages17
JournalComputer Methods in Applied Mechanics and Engineering
Volume267
Publication statusPublished - 11 Sept 2013
Externally publishedYes

Abstract

We propose a reproducing kernel triangular B-spline-based finite element method (FEM) as an improvement to the conventional triangular B-spline element for solving partial differential equations (PDEs). In the latter, unexpected errors can occur throughout the analysis domain mainly due to the excessive flexibilities in defining the B-spline. The performance therefore becomes unstable and cannot be controlled in a desirable way. To address this issue, the proposed improvement adopts the reproducing kernel approximation in the calculation of B-spline kernel function. Three types of PDE problems are tested to validate the present element and compare against the conventional triangular B-spline. It has been shown that the improved triangular B-spline satisfies the partition of unity condition even for extreme conditions including corners and holes.

Keywords

    Finite element method, Poisson's equations, Reproducing kernel approximation, Reproducing kernel triangular B-spline, Triangular B-spline

ASJC Scopus subject areas

Cite this

Reproducing kernel triangular B-spline-based FEM for solving PDEs. / Jia, Yue; Zhang, Yongjie; Xu, Gang et al.
In: Computer Methods in Applied Mechanics and Engineering, Vol. 267, 11.09.2013, p. 342-358.

Research output: Contribution to journalArticleResearchpeer review

Jia Y, Zhang Y, Xu G, Zhuang X, Rabczuk T. Reproducing kernel triangular B-spline-based FEM for solving PDEs. Computer Methods in Applied Mechanics and Engineering. 2013 Sept 11;267:342-358. doi: 10.1016/j.cma.2013.08.019
Jia, Yue ; Zhang, Yongjie ; Xu, Gang et al. / Reproducing kernel triangular B-spline-based FEM for solving PDEs. In: Computer Methods in Applied Mechanics and Engineering. 2013 ; Vol. 267. pp. 342-358.
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abstract = "We propose a reproducing kernel triangular B-spline-based finite element method (FEM) as an improvement to the conventional triangular B-spline element for solving partial differential equations (PDEs). In the latter, unexpected errors can occur throughout the analysis domain mainly due to the excessive flexibilities in defining the B-spline. The performance therefore becomes unstable and cannot be controlled in a desirable way. To address this issue, the proposed improvement adopts the reproducing kernel approximation in the calculation of B-spline kernel function. Three types of PDE problems are tested to validate the present element and compare against the conventional triangular B-spline. It has been shown that the improved triangular B-spline satisfies the partition of unity condition even for extreme conditions including corners and holes.",
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note = "Funding information: The authors thank the support by the European Union through the FP7-grant ITN (Marie Curie Initial Training Networks) INSIST (Integrating Numerical Simulation and Geometric Design Technology), the US Office of Navy Research through the ONR-YIP award N00014–10-1–0698, the NSFC (41130751), National Basic Research Program of China (973 Program: 2011CB013800) and Shanghai Pujiang Program (12PJ1409100), the Nature Science Foundation of China (Nos. 61272390 , 61004117 , 61211130103 ) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars from State Education Ministry.",
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AU - Xu, Gang

AU - Zhuang, Xiaoying

AU - Rabczuk, Timon

N1 - Funding information: The authors thank the support by the European Union through the FP7-grant ITN (Marie Curie Initial Training Networks) INSIST (Integrating Numerical Simulation and Geometric Design Technology), the US Office of Navy Research through the ONR-YIP award N00014–10-1–0698, the NSFC (41130751), National Basic Research Program of China (973 Program: 2011CB013800) and Shanghai Pujiang Program (12PJ1409100), the Nature Science Foundation of China (Nos. 61272390 , 61004117 , 61211130103 ) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars from State Education Ministry.

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N2 - We propose a reproducing kernel triangular B-spline-based finite element method (FEM) as an improvement to the conventional triangular B-spline element for solving partial differential equations (PDEs). In the latter, unexpected errors can occur throughout the analysis domain mainly due to the excessive flexibilities in defining the B-spline. The performance therefore becomes unstable and cannot be controlled in a desirable way. To address this issue, the proposed improvement adopts the reproducing kernel approximation in the calculation of B-spline kernel function. Three types of PDE problems are tested to validate the present element and compare against the conventional triangular B-spline. It has been shown that the improved triangular B-spline satisfies the partition of unity condition even for extreme conditions including corners and holes.

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