Abstract
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.
Similar content being viewed by others
References
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.
K. Hollig, U. Reif and J. Wipper, Weighted extended B-spline approximation of dirichlet problems, SIAM J. Numer. Anal., 2002, 39(2): 442–462.
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.
G. Pelosi, The Finite-Element Method, Part I: R. L. Courant: Historical Corner, Antennas and Propagation Magazine, IEEE, 2007.
J. Oden, Finite elements: An introduction, Handbook of Numerical Analysis, 1991, (II): 3–15.
G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Pub. Co, Amsterdam, 1978.
D. H. Norrie and G. de Vries, The Finite Element Method, Academic Press, New York, 1973.
N. Zabaras, Introduction to the finite element method for elliptic problems, http://mpdc.mae.cornell.edu/Courses/MAEFEM/Lecture1.pdf, 2008, Online; Accessed 19–July-2008.
C. De Boor, A Practical Guide to Splines, Springer-Verlag, New York, 1978.
G. Farin, J. Hoschek, and M. S. Kim, Handbook of Computer Aided Geometric Design, Elsevier, 2002.
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.
S. A. Coons, Surfaces for Computer-Aided Design for Space Forms, Technical Report TR-41, MIT, 1967.
G. Farin, Curves and Surfaces for Computer-Aided Geometric Design: A Practical Guide, Academic Press, San Diego, 1997.
G. Farin and D. Hansford, Discrete coons patches, Computer Aided Geometric Design, 1999, (16): 692–700.
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.
Z. Quan and S. H. Xiang, A GMRES based polynomial preconditioning algorithm, Mathematica Numerica Sinica (in Chinese), 2006, 28(4): 365–376.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research is partially supported by NKBRSF under Grant No. 2011CB302404, NSFC under Grant Nos. 10871195, 10925105, 60821002, and 50875027.
This paper was recommended for publication by Editor Xiao-Shan GAO.
Rights and permissions
About this article
Cite this article
Liu, Y., Li, H. B-spline patches and transfinite interpolation method for PDE controlled simulation. J Syst Sci Complex 25, 348–361 (2012). https://doi.org/10.1007/s11424-012-1148-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11424-012-1148-4