ISSN 0253-2778

CN 34-1054/N

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Using splines on triangular mesh to solve PDE with nonhomogeneous boundary

  • 1.

    School of Mathematical Sciences,University of Science and Technology of China,Hefei 230026,China

  • 2.

    Department of Public Foundation, Shanghai University of Finance and Economics Zhejiang College, Jinhua 321013, China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2020.07.005
  • Received Date: 20 April 2020
  • Accepted Date: 08 June 2020
  • Rev Recd Date: 08 June 2020
  • Publish Date: 31 July 2020
    Abstract

  • Abstract

    The spline spaces S,02(Δ(2)mn) defined on type Ⅱ regular triangulation are directly used to solve the PDE problems with homogeneous boundary.For the PDE problems with nonhomogeneous boundary, the solutions obtained usually do not satisfy the convergence properties if they are still solved in the spline spaces S,02(Δ(2)mn),because the basis functions of S,02(Δ(2)mn) vanish on the boundary of the parameter domain. Here, based on the spline spaces S,02(Δ(2)mn) and S2(Δ(2)mn) defined on type Ⅱ regular triangulation,a set of blended spline basis functions were formed by combining the basis functions of S,02(Δ(2)mn) with the basis functions of S2(Δ(2)mn) whose support centers are all outside the boundary of the parameter domain.The blended spline functions were used to solve the PDE problem with nonhomogeneous boundary.Experiment results show that the solutions obtained by this method are convergent.

    Abstract

    The spline spaces S,02(Δ(2)mn) defined on type Ⅱ regular triangulation are directly used to solve the PDE problems with homogeneous boundary.For the PDE problems with nonhomogeneous boundary, the solutions obtained usually do not satisfy the convergence properties if they are still solved in the spline spaces S,02(Δ(2)mn),because the basis functions of S,02(Δ(2)mn) vanish on the boundary of the parameter domain. Here, based on the spline spaces S,02(Δ(2)mn) and S2(Δ(2)mn) defined on type Ⅱ regular triangulation,a set of blended spline basis functions were formed by combining the basis functions of S,02(Δ(2)mn) with the basis functions of S2(Δ(2)mn) whose support centers are all outside the boundary of the parameter domain.The blended spline functions were used to solve the PDE problem with nonhomogeneous boundary.Experiment results show that the solutions obtained by this method are convergent.
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    • References(9)
  • [1]
    POWELL M J D,SABIN M A.Piece-wise quadratic approximations on triangles[J].ACM Trans Math Software,1977,3: 316-325.
    [2]
    PAUL D.On calculating normalized Powell-Sabin B-splines[J].Computer Aided Geometric Design,1997,15: 61-78.
    [3]
    SPELEERS H,DIERCKX P,VANDEWALLE S.Quasi-hierarchical Powell-Sabin B-splines[J].Computer Aided Geometric Design,2009,26:174-191.
    [4]
    SPELEERS H,MANNI C,FRANCESCA P,et al.Isogeometric analysis with Powell-Sabin splines for advection-diffusion-reaction problems[J].Comput Methods Appl Mech Engrg,2012,221-222: 132-148.
    [5]
    WANG R.Multivariate Splines and Its Application[M].Beijing：Science Publishers,1994.
    [6]
    KANG H M,CHEN F L,DENG J S.Hierarchical B-splines on regular triangular partition[J].Graphical Models,2014,76(5): 289-300.
    [7]
    冯玉瑜,曾芳玲,邓建松.样条函数与逼近论[M].合肥：中国科学技术大学出版社，2013.
    [8]
    QU K.Multivariate splines and some application[D].Dalian: Dalian University of Technology,2010.
    [9]
    KANG H M.Splines suitable for analysis and modeling[D].Hefei: University of Science and Technology of China,2014.)
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    [1]
    POWELL M J D,SABIN M A.Piece-wise quadratic approximations on triangles[J].ACM Trans Math Software,1977,3: 316-325.
    [2]
    PAUL D.On calculating normalized Powell-Sabin B-splines[J].Computer Aided Geometric Design,1997,15: 61-78.
    [3]
    SPELEERS H,DIERCKX P,VANDEWALLE S.Quasi-hierarchical Powell-Sabin B-splines[J].Computer Aided Geometric Design,2009,26:174-191.
    [4]
    SPELEERS H,MANNI C,FRANCESCA P,et al.Isogeometric analysis with Powell-Sabin splines for advection-diffusion-reaction problems[J].Comput Methods Appl Mech Engrg,2012,221-222: 132-148.
    [5]
    WANG R.Multivariate Splines and Its Application[M].Beijing：Science Publishers,1994.
    [6]
    KANG H M,CHEN F L,DENG J S.Hierarchical B-splines on regular triangular partition[J].Graphical Models,2014,76(5): 289-300.
    [7]
    冯玉瑜,曾芳玲,邓建松.样条函数与逼近论[M].合肥：中国科学技术大学出版社，2013.
    [8]
    QU K.Multivariate splines and some application[D].Dalian: Dalian University of Technology,2010.
    [9]
    KANG H M.Splines suitable for analysis and modeling[D].Hefei: University of Science and Technology of China,2014.)
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