BASES OF BIPYRAMID

Abstract

Research is devoted to the study of functions interpolation, applying the method of finite elements in the area of quadrilateral b bipyramid. Main task of the study is to obtain general formula of finite-element basis of bipyramid and analysis of approximation properties of basis functions on this polyhedron.

In the given study bipyramid is considered as a finite element, obtained from the regular octahedron as a result of linear deformations of its semi-axes. Depending on the number of directions of deformation bipyramid may have one/two/three moving nodes. Such method of bipyramid formation is the result of adaptation of gratings of tetrahedrally-octahedral structure to the boundaries of  areas, which have complex geometric form.

The paper contains general formulas for the bases of bipyramid with six and seven interpolation nodes, which are parametric functions of three variables. Values of parameters correspond to the elongation compression factors of the  semi-axes of bipyramid and weight coefficient for six-node basis functions.

For the functions of bipyramid bases with seven and six interpolation nodes geometric properties of functions which are associated with the vertices of a polyhedron, are studied. The given functions are presented by the surfaces of zero level. Dependence of the type of surface on the value of parameters of bases functions of bipyramid is determined.

Approximation properties of seven node base of bipyramid functions are investigated in the paper. Quality criterion of approximation minimal trace of the stiffness matrix on a polyhedron is chosen. Critical values of elongation /compression parameters of bipyramid semi-axes which deliver minimum functionality to the trace of the stiffness matrix of bipyramid with seven nodes of interpolation are found. Additionally interval assessments for the parameters of seven node base function of bipyramid are obtained, they determine the boundaries of finite element deviation in the form of bipyramid from regular octahedron.

Perspective of further studies is mathematical substantiation of bipyramids application in the method of finite elements, determination of quality indices of bipyramid, used in the systems of finite-element analysis for asymmetric 3D-elements, obtaining of cubature formulas.