Thesis for the degree of Doctor of Technical Sciences in the specialty 05.01.01 - Applied Geometry, Engineering Graphics. Kiev National University of Construction and Architecture, Kiev, Ukraine, 2018.
The thesis is devoted to the solution of the scientific problem of modeling multi-factor systems and processes with any predetermined number of input factors of different physical nature. For this purpose, a composite method for geometric modeling (CMGM) has been developed, which is a further development of the principles and tools of the Baluba-Naidish point calculus (BN-calculus). With the help of CMD, the quality and validity of managerial decisions are improved in order to increase the efficiency of business entities in various sectors of the economy.
As a result of theoretical studies, an algorithm for the formation of the Baluba-Naidish coordinates (BN-coordinates) for interpolating the initial points of the parametric Baluba curve (B-curve) was developed and substantiated. One of the main tools of KMGM is the Balyuba-Naidish geometric matrix (BN-matrix). The use of BN matrices allowed us to generalize the geometric formalization of input factors and simplify problem solving.
The key method in CMGM is the method of forming segments of one-parameter B-curves of the Euclidean n-space, based on the transition from the global coordinate system to solving the problem in local simplexes. The transition from the original geometric figure to the B-curve with BN-coordinates in the local coordinate system ensures a decrease in the number of parameters when parameterizing the initial data of the model. A technique has been developed for the formation of continuous BN coordinates for a surface segment in the form of a parametric BN matrix, interpolates the points of the original geometric figure. At the same time, the solution of a complex n-dimensional problem is carried out by solving n simple single-factor problems and then combining the results. This allows you to avoid additional transformations in the event of a change in the position of the original geometric shape relative to the global coordinate system. The developed method of forming k-dimensional projections for n-dimensional B-figures allows for a detailed study of the influence of a group of heterogeneous factors on the behavior of the system.
An important feature of KMGM is the separation of the parametric and geometric parts of the original geometric figure. Due to this, the parametric part of the model (which determines all the properties and connections of the object of modeling) remains unchanged when the initial points of the geometric figure change. This allows for a convenient analysis of the system activity over time by simply changing the coordinates of the source points.
Using the above research results, a compositional method of geometric modeling (KMGM) is developed, the difference of which from existing methods is that it combines the factors at the final stage of formation of a composite geometric model. Compositional geometric modeling is implemented according to the modular principle, in which heterogeneous factors are involved sequentially according to the levels of decomposition of the system, object. The specified method of successive integration of factors allows to add an unlimited number of heterogeneous factors to the composite geometric model and create multilayer multi-factor geometric models. The principles of compilation of descriptions and creation of the structure of modules for composite geometric models are formulated, the schemes of their construction and the methods of geometrical formalization in the BN-matrix form are proposed for them. The given gives the opportunity to take into account a sufficiently large number of heterogeneous factors for modeling systems and objects, which promotes the adoption of more motivated management decisions, the introduction of which will increase the economic effect of the results of the system, the object. The monofactor principle of constructing composite geometric models in the BN-matrix form provides an opportunity for a deeper study of the influence of a separate factor on the work of the system, the object, which results in changes in the structure of the system, in order to improve the functioning of the system.
