DETERMINING THE QUALITY METRICS OF A FINITE ELEMENT IN THE SHAPE OF A BIPYRAMID

Abstract

The article deals with the metric properties of a finite element in the shape of a bipyramid with one/two/three moving nodes. The main task of this study is to determine the conditions for using the bipyramid, which is not a regular polyhedron, in the constructing of a finite element mesh.Four main indices of bipyramid quality were obtained in this paper: Element Quality, Aspect Ratio, Skewness, and Orthogonal Quality. All calculations were made according to the rules used in ANSYS.Using the formula for 3D elements, an interval estimate of the element quality metric is found, which determines the limits of bipyramids application in the discretization of the domain by the Finite Element Method in problems of linear mechanics, elasticity and thermal conductivity.Based on the analysis of the Aspect Ratio of the bipyramid faces, a formula for calculating the quality index of a bipyramid with one/two/three moving nodes is obtained. The limits of application of this formula are determined by the rules that apply to the Aspect Ratio in ANSYS Mechanical, Elecromagnetics and CFX.In this paper, interval estimates are found for the asymmetry index of the bipyramid with one/two/three moving nodes, which allow satisfying the ANSYS mesh quality requirements at the good and excellent levels. The obtained estimates are constraints on the values of the internal angles of the bipyramid faces, which are functions of the parameters of elongation/compres- sion of the polyhedron semi-axes. The identified dependencies between the coordinates of the bipyramid vertices and the values of its angles, which are different from zero, simplify the algorithm for checking the asymmetry quality indicator.In this paper, Orthogonal Quality metric is found in the form of a function of the bipyramid elongation/compression parameters. Based on the analysis of the function’s set of values, it is established that, regardless of the bipyramid parameters, Orthogonal Quality indicator takes positive values. According to the requirements set forth in ANSYS, positivedefinite Orthogonal Quality is a sufficient condition for the use of irregular geometric shape elements in the Finite Element Mesh Generation. The obtained results contain formulas and interval estimates that can be used in the finite element method algorithmization. The prospect of further research is to develop an algorithm for 3D-discretizing with a tetrahedral-octahedral mesh and carry out computational experiments using the Finite Element Method.