Motailo A. Geometric modeling of scalar and vector fields on the lattices of tetrahedral-octahedral structure

To study three-dimensional models, both analytical and numerical methods of solution are used, in particular, the finite element method. At realization of the given method by means of universal program systems of the finite element analysis for construction of settlement area with known parameters and characteristics use methods of geometrical modeling. Sampling of the three-dimensional region is based on the use of basic elements of form, including hexahedron, tetrahedron, pyramid and prism. The influence of a new basic element of the form  octahedron, or its generalizations  bipyramid  in solid-state modeling on its approximation qualities of numerical calculations by the finite element method with the use of alternative grids is studied in this paper. The features of approximation of the functions of scalar and vector fields by finite element method on the lattices of tetrahedral-octahedral structure in comparison with tetrahedral lattices are investigated. In this work it is first a number of different systems of nodal coordinate functions for the finite element in the form of an octahedron with 6 interpolation nodes have been constructed, which allows us to reconcile the choice of the basis with the peculiarities of the solvable problem. It was established that finite element in the form of a six-node octahedron with polynomial and piecewise linear basic functions is bestowed with better local interpolation qualities than the bases of the seven-node octahedron, which creates positive predictive conditions for their use in field-restoration tasks by the finite element method on lattices with cells in the form of the octahedron. The multiplicity of polynomial bases of the second degree of a finite element in the form of an octahedron, which are harmonic in Laplace, is shown, which allows solving an additional problem of optimization of the choice of the basis of the octahedron by the chosen local characteristic in solving boundary problems for the Laplace equation. It was established that the degrees of approximation of the octahedron with piecewise linear and quadratic functions are equal to 1 and coincide with the degree of approximation of a linear tetrahedron, whose nodal coordinate functions are the nodal coordinates of its vertices. It allows to reduce the number of nodes during the discretization of the three-dimensional region by the lattice of the tetrahedral-octahedral structure compared with the tetrahedral, while preserving the accuracy of the resulting solutions. It is proved that the piecewise linear and quadratic approximations by the finite element method of the displacement field functions on the lattice of the tetrahedral-octahedral structure converge to the exact solution in problems with a differential operator of the second order, which allows to reduce the dimension of the global stiffness matrix when solving the FEM boundary problems for the restoration of the functions of the vector field on the lattices of the tetrahedral-octahedral structure in comparison with the tetrahedral ones. It is proved that piecewise linear approximation by finite elements method of a temperature field with a differential operator of the second order is an admissible function in an area that is discrete by a tetrahedral-octahedral lattice, which allows to reduce the dimension of the global stiffness matrix when solving the ITE boundary problems for the restoration of the functions of the scalar field on the lattices of the tetrahedral-octahedral structure in comparison with the tetrahedral ones. An interpolation formula for numerical integration in the region of a finite element in the form of an octahedron is obtained, which allows precisely finding the elements of a stiffness matrix of the 6-node octahedron with piecewise linear basic functions ten times faster than the analytical integration methods embedded in the system of computer mathematics Maple. A finite element was found in the form of a bipyramid, for which the best accuracy of the calculations in the mean-square sense is predicted in the absence of the correct geometric form of a finite element, which allows it to be used as a cell of a spatial lattice in a group with a linear tetrahedron that is adapted to the calculated areas of complex geometry, when solving the finite element method of boundary problems of mathematical physics. The method of constructing the basic functions of convex n-dashboards is further developed. In the dissertation for the first time it is convincingly constructed that for n = 8 it is possible to construct a polynomial basis of the finite element in the form of an octahedron with 6 interpolation nodes. The proposed enhancement of the method of the Wachspress with respect to the modeling of the basic functions of the octahedron has less cognitive and operational complexity than other methods of constructing these functions.