SKOCHKO V. Methods of interpretive geometric modeling of mesh structures and their application

The dissertation on competition of a scientific degree of the doctor of technical sciences on a specialty 05.01.01 – applied geometry, engineering graphics. – Kyiv National University of Construction and Architecture. – Kyiv, 2021. The dissertation is devoted to the development of the tool base of methods of discrete geometric modeling of multicomponent objects, phenomena and processes that can be interpreted by mesh structures. Thus the nature of the appropriate objects, phenomena and processes can be described by both differential and other functional regularities including scalar and vector fields. In terms of system analysis all interpretative model mesh structures are composed of typical elements and have common features. In particular, in the simplest form, the constituent elements of the mesh structures are free and fixed nodes, as well as rectilinear links that connect them and express the degree of interaction between the respective nodes. The most striking example of mesh structures are rod architectural and building structures with a hinged connection of links, which in ideal condition can be formed and change the values of the components of the stress-strain state as a result of external functional loads. The process of interaction of mesh structures as complex systems with the environment lies in perception, reallocation and links to further internal efforts to transfer basis through bearing (fixed) nodes. Based on this idea of the work of mesh structures, a number of scientific and practical problems related to the formation and adjustment of the parameters of their state are set and solved. As the parameters of the condition of links of models it is offered to accept the density of internal efforts in them. Depending on the method of interpretation of the physical or abstract value of these parameters and external influences, different methods of forming and adjusting of mesh structures are proposed. The corresponding methods are developed on the basis of the generalized form of static-geometric method of applied discrete geometry, the equilibrium equation of which was supplemented by differential regularities between geometric and physical parameters of mesh structures, scalar and vector fields that balance or gear the interpretive models. These regularities represent the equations of state of free nodes and links of models that have been generalized and adapted to solve both static and dynamic application problems in design spaces of arbitrary dimension. On the one hand, there are developed the methodical system management of parameters of a condition of system structures for tasks, which provide impossibility of influence on the form of models due to the change of external nodal loading. On the other hand, there are created the methods of forming discrete geometric images (in the form of mesh structures), which is based on the transformation of parametric equations of state of model nodes into functions in Lagrangian form, which contain additional unknown variation parameters. The presence of appropriate parameters allows to impose additional modeling conditions on the objective of shaping and to provide discrete images with certain differential and metric properties, turning the modeling process itself into a problem of finding conditional optimums.