Determination of load distribution in a given medium according to the values of the loads at certain points
Authors: Mostovenko Oleksandr , Kovalov Serhii , Botvinovska Svitlana 
Published: July 2021

Taking into account force, temperature and other loads, the stress and strain state calculations methods of spatial structures involve determining the distribution of the loads in the three-dimensional body of the structure [1, 2].

In many cases the output data for this distribution can be the values of loadings in separate points of the structure. The problem of load distribution in the body of the structure can be solved by three-dimensional discrete interpolation in four-dimensional space based on the method of finite differences, which has been widely used in solving various engineering problems in different fields. A discrete conception of the load distribution at points in the body or in the environment is also required for solving problems by the finite elements method [3-7].

From a geometrical point of view, the result of three-dimensional interpolation is a multivariate of the four-dimensional space [8], where the three dimensions are the coordinates of a three-dimensional body point, and the fourth is the loading at this point. Such interpolation provides for setting of the three coordinates of the point and determining the load at that point. The simplest three-dimensional grid in the three-dimensional space is the grid based on a single sided hypercube. The coordinates of the nodes of such a grid correspond to the numbering of nodes along the coordinate axes.

Discrete interpolation of points by the finite difference method is directly related to the numerical solutions of differential equations with given boundary conditions and also requires the setting of boundary conditions.

If we consider a three-dimensional grid included into a parallelepiped, the boundary conditions are divided into three types: 1) zero-dimensional (loads at points), where the three edges of the grid converge; 2) one-dimensional (loads at points of lines), where the four edges of the grid converge; 3) two-dimensional (loads at the points of faces), where the five edges of the grid converge. The zero-dimensional conditions are boundary conditions for one-dimensional interpolation of the one-dimensional conditions, which, in turn, are boundary conditions for two-dimensional conditions, and the two-dimensional conditions are boundary conditions for determining the load on the inner points of the grid.

If a load is specified only at certain points of boundary conditions, then the interpolation problem is divided into three stages: one-dimensional load interpolation onto the line nodes, two-dimensional load interpolation onto the surface nodes and three-dimensional load interpolation onto internal grid nodes.

The proposed method of discrete three-dimensional interpolation allows, according to the specified values of force, temperature or other loads at individual points of the three-dimensional body, to interpolate such loads on all nodes of a given regular three-dimensional grid with cubic cells.

As a result of interpolation, a discrete point framework of the multivariate is obtained, which is a geometric model of the distribution of physical characteristics in a given medium according to the values of these characteristics at individual points.