Symbolic-Numerical Algorithm for Generating Interpolation Multivariate Hermite Polynomials of High-Accuracy Finite Element Method
A symbolic-numerical algorithm implemented in Maple for constructing Hermitian finite elements is presented. The basis functions of finite elements are high-order polynomials, determined from a specially constructed set of values of the polynomials themselves, their partial derivatives, and their derivatives along the directions of the normals to the boundaries of finite elements. Such a choice of the polynomials allows us to construct a piecewise polynomial basis continuous across the boundaries of elements together with the derivatives up to a given order, which is used to solve elliptic boundary value problems using the high-accuracy finite element method. The efficiency and the accuracy order of the finite element scheme, algorithm and program are demonstrated by the example of the exactly solvable boundary-value problem for a triangular membrane, depending on the number of finite elements of the partition of the domain and the number of piecewise polynomial basis functions.
KeywordsHermite interpolation polynomials Boundary-value problem High-accuracy finite element method
- 2.Argyris, J.H., Buck, K.E., Scharpf, D.W., Hilber, H.M., Mareczek, G.: Some new elements for the matrix displacement method. In: Proceedings of the Conference on Matrix Methods in Structural Mechanics (2nd), Wright-Patterson Air Force Base, Ohio, 15–17 October 1968Google Scholar
- 3.Bathe, K.J.: Finite Element Procedures in Engineering Analysis. Prentice Hall, Englewood Cliffs/New York (1982)Google Scholar
- 9.Gusev, A.A., Chuluunbaatar, O., Vinitsky, S.I., Derbov, V.L., Góźdź, A., Le Hai, L., Rostovtsev, V.A.: Symbolic-numerical solution of boundary-value problems with self-adjoint second-order differential equation using the finite element method with interpolation hermite polynomials. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2014. LNCS, vol. 8660, pp. 138–154. Springer, Cham (2014). doi: 10.1007/978-3-319-10515-4_11 Google Scholar
- 10.Gusev, A.A., Hai, L.L., Chuluunbaatar, O., Vinitsky, S.I.: KANTBP 4M: Program for Solving Boundary Problems of the System of Ordinary Second Order Differential Equations. http://wwwinfo.jinr.ru/programs/jinrlib/indexe.html
- 21.Zienkiewicz, O.C.: Finite elements. The background story. In: Whiteman, J.R. (ed.) The Mathematics of Finite Elements and Applications, p. 1. Academic Press, London (1973)Google Scholar