Journal of Systems Science and Complexity

, Volume 25, Issue 2, pp 348–361 | Cite as

B-spline patches and transfinite interpolation method for PDE controlled simulation

  • Yuanjie LiuEmail author
  • Hongbo Li


This paper is to discuss an approach which combines B-spline patches and transfinite interpolation to establish a linear algebraic system for solving partial differential equations and modify the WEB-spline method developed by Klaus Hollig to derive this new idea. First of all, the authors replace the R-function method with transfinite interpolation to build a function which vanishes on boundaries. Secondly, the authors simulate the partial differential equation by directly applying differential operators to basis functions, which is similar to the RBF method rather than Hollig’s method. These new strategies then make the constructing of bases and the linear system much more straightforward. And as the interpolation is brought in, the design of schemes for solving practical PDEs can be more flexible. This new method is easy to carry out and suitable for simulations in the fields such as graphics to achieve rapid rendering. Especially when the specified precision is not very high, this method performs much faster than WEB-spline method.

Key words

B-spline representation finite element method RBF transfinite interpolation WEB-spline 


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  1. [1]
    T. Belyschko, Y. Krongaus, D. Organ, M. Fleming, and P. Krysl, Meshless methods: An overview and recent developments, Comput. Methods Appl. Mech. Enrg., 1996, 139(12): 3–47.CrossRefGoogle Scholar
  2. [2]
    K. Hollig, U. Reif and J. Wipper, Weighted extended B-spline approximation of dirichlet problems, SIAM J. Numer. Anal., 2002, 39(2): 442–462.MathSciNetCrossRefGoogle Scholar
  3. [3]
    S. K. Lodha and R. Franke, Scattered data techniques for surfaces, Proceedings of Dagstuhl Conference on Scientific Visualization, IEEE Computer Society Press, Washington, DC, 1997.Google Scholar
  4. [4]
    G. Pelosi, The Finite-Element Method, Part I: R. L. Courant: Historical Corner, Antennas and Propagation Magazine, IEEE, 2007.Google Scholar
  5. [5]
    J. Oden, Finite elements: An introduction, Handbook of Numerical Analysis, 1991, (II): 3–15.Google Scholar
  6. [6]
    G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Pub. Co, Amsterdam, 1978.zbMATHGoogle Scholar
  7. [7]
    D. H. Norrie and G. de Vries, The Finite Element Method, Academic Press, New York, 1973.zbMATHGoogle Scholar
  8. [8]
    N. Zabaras, Introduction to the finite element method for elliptic problems,, 2008, Online; Accessed 19–July-2008.
  9. [9]
    C. De Boor, A Practical Guide to Splines, Springer-Verlag, New York, 1978.zbMATHCrossRefGoogle Scholar
  10. [10]
    G. Farin, J. Hoschek, and M. S. Kim, Handbook of Computer Aided Geometric Design, Elsevier, 2002.Google Scholar
  11. [11]
    T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs, Isogeometric analysis: CAD, finite Elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Engrg., 2005, 194: 4135–4195.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    S. A. Coons, Surfaces for Computer-Aided Design for Space Forms, Technical Report TR-41, MIT, 1967.Google Scholar
  13. [13]
    G. Farin, Curves and Surfaces for Computer-Aided Geometric Design: A Practical Guide, Academic Press, San Diego, 1997.zbMATHGoogle Scholar
  14. [14]
    G. Farin and D. Hansford, Discrete coons patches, Computer Aided Geometric Design, 1999, (16): 692–700.Google Scholar
  15. [15]
    R. Barret, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, Templates for The Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd Edition, SIAM, Philadelphia, 1994.CrossRefGoogle Scholar
  16. [16]
    Z. Quan and S. H. Xiang, A GMRES based polynomial preconditioning algorithm, Mathematica Numerica Sinica (in Chinese), 2006, 28(4): 365–376.MathSciNetGoogle Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institute of Systems Science, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  2. 2.Graduate School of the Chinese Academy of SciencesBeijingChina

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