CONSTRUCTING OF BIPYRAMID BASIS WITH THREE MOVABLE NODES

Abstract

The article discusses the possibilities of approximation by the finite element method of the function of three variables in the region that has the shape of a quadrangular bipyramid. The main task of this study is to improve the approximation properties of lattices of the tetrahedral-octahedral structure by replenishing them with cells that are not regular polyhedra. In particular, the author studies finite elements, which are formed as a result of linear deformations of the octahedron. In this work, the object of research is a finite element in the form of a bipyramid with three movable nodes that can be moved along the semi-axes of the polyhedron. This property makes it possible to adjust a finite element that is not a regular polyhedron to the boundary of the computational domain better than a regular polyhedron. Two finite-element bases of a bipyramid with seven and six interpolation nodes are constructed in the paper. Two methods are used to construct the basic functions of the bipyramid: the geometric method and the condensation method. The obtained bases are polynomial functions that parametrically depend on the values of the three elongation/compression coefficients of the semi-axes of the bipyramid. The basis with six interpolation nodes contains an additional parameter that is a weight coefficient, which is a consequence of the application of the condensation procedure to the functions of the seven-node basis of the bipyramid. The availability of parameters allows improving the interpolation properties of the constructed bipyramid bases in accordance with the approximation quality criteria used in the finite element method. In this work, such a criterion is considered to be the value of the trace of the stiffness matrix. According to the selected criterion, the values of the linear deformation coefficients of the three semi-axes of the octahedron are found, at which the trace of the bipyramid stiffness matrix is minimal. In the article we have obtained interval estimates for the elongation/compression coefficients of the three semi-axes of the octahedron, which characterize the deviation of the geometric dimensions of the bipyramid from the regular polyhedron and lead to the loss of all types of symmetry. To determine the limits of the permissible values of the coefficients of linear deformations of the octahedron, the Skewness asymmetry index used in the ANSYS finite element analysis system was calculated. The obtained interval estimates for the elongation/compression coefficients of the octahedron semi-axes correspond to the criterion of minimality of the trace of the stiffness matrix of a bipyramid with three moving nodes, which indicates a positive prediction of the use of this finite element as a lattice cell of the tetrahedral-octahedral structure. Prospect for further research is the construction of cubature formulas on a finite element in the form of a bipyramid with three moving nodes in order to use it in the algorithmization of the finite element method.