Algorithm 884: A Simple Matlab Implementation of the Argyris Element

ABSTRACT

In this work we propose a new algorithm to evaluate the basis functions of the Argyris finite element and their derivatives. The main novelty here is an efficient way to calculate the matrix which gives the change of coordinates between the bases of the Argyis element for the reference and for an arbitrary triangle. This matrix is factored as the product of two rectangular matrices with a strong block structure which makes their computation very easy. We show and comment on an implementation of this algorithm in Matlab. Two numerical experiments, an interpolation of a smooth function on a triangle and the finite-element solution of the Dirichlet problem for the biLaplacian, are presented in the last section to check the performance of our implementation.

REFERENCES

• 1: Argyris, J. H., Fried, I., and Scharpf, D.W. 1968. The TUBA family of plate elements for the matrix displacement method. Aero. J. Roy. Aero. Soc. 72, 701--709.
• 2: Arcangéli, R., Cruz Lopez de Silanes M., and Torrens, J. J. 2004. Multidimensional Minimizing Splines. Grenoble Sciences, Kluwer.
• 3: Bell, K. 1969. A refined triangular plate bending finite element. Int. J. Numer. Methods Eng. 1, 101--122.
• 4: Bernadou, M. 1997. Méthodes d'Éléments Finis pour les Problèmes de Coques Minces. Dunod.
• 5: Bogner, F. K., Fox, R. L., and Schmit, L. A. 1965. The generation of inter-element compatible stiffness and mass matrices by the use of interpolation formulas. In Proceedings of the Conference on Matrix Methods in Structural Mechanics, Wright-Patterson Air Force Base, OH, 397--444.
• 6: Braess, D. 2001. Finite Elements, 2^nd ed. Cambridge University Press, Cambridge, MA.
• 7: Brenner, S. C. and Scott, L. R. 2002. The Mathematical Theory of Finite Element Methods. Springer.
• 8: Philippe G. Ciarlet, Finite Element Method for Elliptic Problems, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2002
• 9: Clough, R. and Tocher, J. 1965. Finite element stiffness matrices for analysis of plates in bending. In Proceedings of the Conference on Matrix Methods in Structural Mechanics, Wright-Patterson Air Force Base, OH, 515--547.
• 10: Dunavant, D. 1985. High degree efficient symmetrical Gaussian quadrature rules for the triangle. Int. J. Numer. Methods Eng. 21, 1129--1148.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.4 MATHEMATICAL SOFTWARE
      Subjects: Documentation

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.8 Partial Differential Equations
          Subjects: Finite element methods

General Terms:
Algorithms, Documentation

Keywords:
Argyris element, Matlab, finite elements