CONSTRUCTION OF BIPYRAMID BASIS WITH TWO MOVABLE NODES

Abstract

This paper is devoted to the study of the possibilities of interpolation by the method of finite elements of functions of three independent variables in the region, which has the shape of a bipyramid. The main task of this study is to improve the approximation qualities of the lattice of the tetrahedral-octahedral structure by including cells in the form of bipyramids with two movable nodes. In this article, a bipyramid is considered as a finite element formed by elongation/compression of two semi-axes of an octahedron. Similar linear deformations occur in elements located in the boundary layer of the calculated region, when some nodes have to be removed to the boundary of the region. In cases where only one octahedron node is carried to the boundary of the region, a bipyramid is also formed. In this case, the node that was transferred to the border of the region is considered movable. In the author’s previous works, the bases of the bipyramid with seven and six interpolation nodes were built. The approximation qualities of the constructed bases are investigated theoretically. A positive forecast of their use in finite element calculations is obtained. The results were verified by the problem of thermal conductivity for timber. In the work by geometric and condensation method, two polynomial bases of the bipyramid with seven and six interpolation nodes are constructed. The geometric and interpolation qualities of the basic functions of a bipyramid with two movable nodes are investigated. The constructed bases have, respectively, two and three indefinite parameters that make it possible to give the basic functions of the bipyramid properties expedient in the finite element method. In this paper, the interpolation quality criterion is the trace of the bipyramid stiffness matrix. The coefficients of linear deformation of two semi-axes of the octahedron at which the trace of the bipyramid stiffness matrix is minimal are determined. The limits of permissible linear deformations of the semi-axes of the octahedron, which turn it into a bipyramid with two movable nodes, are analyzed in the article. The basis is the asymmetry index of Skewness, which is used in ANSYS. Estimations of elongation/compression parameters of octahedron semi-axes are obtained, which guarantee high and sufficient accuracy of finite-element calculations when using bipyramids with two movable nodes. The revealed dependences of the coefficients of linear deformation of the semi-axes of the octahedron correspond to the condition of minimality of the trace of the bipyramid stiffness matrix, which indicates a positive forecast of the interpolation qualities of the lattice with cells in the form of bipyramids with two movable nodes. The prospect of further research is the construction of numerical integration formulas on this polyhedron in order to include it in the algorithm of the finite element method in solving applied problems of mathematical physics.