Botvinovska S. Theoretical basis of shape formation in discrete modeling of objects in architecture and designing

The thesis is devoted to the development of theoretical foundations for the shape-formation of discrete surface basis by generalizing S.N. Kovalev’s static-geometric method (SGM) for solving a wide range of shaping problems in technical designing and architecture. The replacement of the coordinates of the discrete surface frame nodes being modeled by certain functions from these coordinates essentially extends the shapeforming capabilities of the SGM. This approach lets us to take into account not only the geometric and static, but also aesthetic requirements for the simulated surfaces of architecture, structures and designed objects. The representation of the external load forming the discrete surface frame in SGM as functions of the topological and metric parameters of the discrete frame allowed to describe the various given properties of the modeled surface in the form of a single generalized equation of equilibrium for the nodes of a discrete point frame of the surface on one hand, and on the other hand it greatly expanded the list of aesthetic, constructive and physical properties that can be taken into account in modeling. Additional possibilities in controlling the shape of discrete frame surfaces of designed objects that were SGM constructed are opened by using geometric transformations, as well as by setting the discrete analogues of conical points or flattening points on the simulated surfaces. The present research proposes the method for transferring an image conceived by a designer onto a simulated surface if an arbitrary supporting contour or layer where a surface frame is formed is defined. The possibilities of forming the curvilinear surfaces in architecture were expanded: in order to increase the rigidity of the coating constructions in architecture, a method of giving a momentless surface the features of waviness, folds or cyclicity was proposed; generalized SGM of forming a discrete frame by uneven excessive internal or external pressure allowed to determine point frames of momentless surfaces of reservoirs for liquid and structures enclosing underwater constructions. The method for a discrete surface frame formed by unevenly distributed internal and external pressure was developed. It allows for determining the point frames of momentless reservoirs shells for fluids or momentless membranes of structures of enclosing underwater constructions. The scientific novelty of the research is the creation of a generalized method of modeling discrete frame surfaces with the assigned properties by using geometric transformations and establishing interconnections between the parameters of the external shaping load and the ones of a discretely determined surface for creating objects in architecture and designing. The present research is first to: propose new algorithms of the formation of discrete frame surfaces using geometric transformations, and this became the first direction of generalization of the static-geometric method; develop the algorithms for the distribution of the external forming load between the nodes of the discrete grid in order to provide the conditions of geometric, physical and aesthetic character, and this became the second direction of generalization of the static-geometric method; develop, in the framework of generalization of static-geometric a method of formation of discrete carcasses of surfaces within the layer defined by continuous surfaces; provide, in the framework of the static-geometric method, a possibility of forming the non-momentary shells by uneven internal or external loss-making pressure; create a mathematical device for transferring the image with assigned differential geometric properties of the surface-prototype conceived by a designer onto the surface with arbitrary boundary conditions in the process of its modeling by a static-geometric method. The research improved: a method of determining individual differential characteristics of discretely represented surfaces by using spherical mapping in order to determine the zones of positive and negative Gaussian curvature on a discrete surface; The further development was given to: a parametric analysis of grids with arbitrary initial and boundary conditions in order to match the number of equations equilibrium to the number of unknown coordinates; the relationship between the topological characteristics and the metric parameters of the grids through the example of the formation of discrete frames of different order paraboloids; analysis of convergence of iterative processes in nonlinear problems of static-geometric method of modeling discrete frames of curves or surfaces. Key words: geometric modeling, discrete surface modeling, architectural shaping, surface properties, discrete surface representation, static-geometric method, discrete frame, external load, topological grid outline, geometric transformations.