A. P. Motailo, A. N. Khomchenko, G. Ya. Tuluchenko


The bipyramid for the first time is considered as a 6-knots finite element (FE) in the article. For the construction of her biquadratic base
two different approaches are used: matrix method and internal condensation method for the bipyramid’s base as a 7-knots Lagrange FE. The first approach allows to investigate the fundamentally possible amount of bases for FE, and second approach does not give such possibility, but it is more economical. It is shown that after satisfaction of the traditional requirements to the basic functions in FEM in the biquadratic basic functions of bipyramid as a 6-knots FE, which are built by means of the named before approaches, there are a different amount of the indefinite coefficients. These coefficients are used future for the giving of the special properties to the basic functions, which adapt them to the solving of a boundary problems with Laplace’s equation. The value of trace of stiffness matrix is chosen as a criterion for the prognostic evaluation of approximation properties of FE in bipyramid’s form. The minimization of the trace value of the stiffness matrix results in the construction of the same biquadratic base of the bipyramid at the both approaches. On the basis of the got base the borders of the possible deformations of the bipyramid’s geometrical form are analysed. It is first wellproven in theory, that FE exists, at using of that as a cell of finite-element mesh, the best accuracy is arrived at the deviation of FE geometrical form from a regular polyhedron, in this case from an octahedron. The critical value of the aspect ratio which provides the achievement of a minimum of trace of stiffness matrix for the bipyramid of the studied geometrical form is found. A calculable experiment the results of that confirm the theoretical prognosis for properties of bipyramid as FE is conducted. The found
regularities allow to suppose the expediency of application of bases with higher-order for FE in form bipyramid.


finite element method, bipyramid, trace of stiffness matrix, tetrahedral-octahedral mesh.


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