FORMATION OF A TRIANGULATE MESH UNDER THE ACTION OF A NORMAL EXTERNAL LOAD

This study is devoted to the development of the methodology of applied geometry and its tools in the direction of creating a geometric apparatus for forming discrete frames of surfaces formed under the action of an external load normal to the surface. The basis for the formation of such surfaces is the static-geometric method [1]. This method is a kind of visual interpretation of the finite difference method based on the description of the static equilibrium of the nodes of a discrete mesh formed under the action of an external load on the mesh nodes. The internal forces in the mesh links that balance the external load are considered proportional to the lengths of the corresponding links, which allows describing the balanced mesh by a system of linear equations. Designing structures of long-span coverings of architectural structures remains an urgent task in our time. Among such coverings, a special place is occupied by those whose shape cannot be set arbitrarily, but is formed under the action of a certain external load, in particular, the shape of low-pressure pneumatic shells is formed under the action of negative internal pressure. Such surfaces can be represented only in a discrete form. In this study, the direction of the force of the normal external load acting on an arbitrary node of a discretely represented surface (DSP) is defined as perpendicular to the plane tangent to the discretely represented surface at this node. The work presents an algorithm for forming a discretely represented surface under the action of a normal load. The components of this algorithm look as follows: the initial parameters of the discretely represented surface are given; the distribution of normal forces between the nodes of the grid is given; the coordinates of all nodes of the initial approximation of the grid are given or determined; the coefficients A, B, C of two planes connecting the corresponding nodes of the grid star are determined for all internal nodes of the grid; the parameters of the normal load forces at each internal grid node are determined; the coordinate components of the force vector applied to the grid node are determined; a system of equilibrium equations of the grid nodes of the next approximation is constructed and solved, where the coordinates of the internal grid nodes and the forces are unknown; the determined coordinates are compared with the coordinates of the previous approximation. The iterative process ends if the difference between the coordinates of the current and previous approximations does not exceed the permissible error. Otherwise, the iterative process continues, starting from point 4.