The article discusses the geometric aspects of the optimization of manufacturing a parabolic-type solar radiation concentrator, which is the main part of a solar photovoltaic thermal module. The method of three-dimensional parquetting of the surface of a parabolic concentrator is considered. To ensure the specified accuracy and smoothness of the surface bypassing at the stages of designing and manufacturing the device, a method of fan-shaped geometric parquetting of the surface of a parabolic concentrator is proposed. As an implementation of the method, a manufactured concentrator of a solar photovoltaic thermal module and illumination of the photodetector at its focus are presented.

Optimization of Parquetting of the Concentrator of Photovoltaic Thermal Module

Geometric simulation and its software for estimating the efficiency of deployment of solar panels on spacecraft and solar concentrators on the ground are considered in this work. Both the physical and mathematical set up of the problem for estimating the energy efficiency of solar panels, taking into account their shading both by each other and by other elements of a space station has been described in this paper. It has been shown that the known methods for mechanization and automation of such calculations are focused on objects of relatively simple geometric shapes (such as buildings), and are inefficient for objects of complex and diverse geometric shape, characteristic both for spacecraft themselves and their solar panels. Therefore, to solve this problem, a receptor (voxel) geometric model digitizing the computational space has been chosen. The receptor model’s uniqueness is that comparing the values of receptor codes allows easy determine the intersection of objects. Has been described a developed receptor geometric model for estimating the effective area of solar panels, taking into account their shading when the object (for example, a spacecraft) is illuminated by a flow of solar energy from a given direction. The essential difference between the developed receptor geometric model and the classical one is that the former is multiform, i.e. uses not the 2-digit code (0 and 1), but the 4-digit one (0, 1, 2 and 3). Has been demonstrated a software implementation of the described geometric model in C#, and a graphical shell developed for this problem, allowing see the obtained results’ numerical values. Have been provided examples of its implementation in solving of practical problems. The results of verification for the described receptor geometric model have been demonstrated. All this allows speak about efficiency of using receptor geometric models both in singular computation calculations and for creating the appropriate algorithmic, mathematical support and software for the corresponding CAD systems.

Discrete Geometric Models for Estimating the Degree of Shading in Solar Energy

In this paper the question related to the use of receptor (voxel) method for geometric modeling to solve practical design problems has been considered. The use of receptor methods is effective in solving a certain class of problems, primarily the problems of automated layout. The complexity of this method’s practical use is due to the fact, that receptor geometric models are never the primary ones. They are formed based on parametric models specified by designer. Receptor models are the internal machine ones. The main problem that prevents the widespread use of the receptor method is the lack of universal methods for converting parametric models into the receptor ones. Available publications show that in solving practical problems various authors have developed their own methods for creating receptor models for objects of \"primitives\" and \"composition of primitives\" classes. Therefore, it is extremely urgent to solve the problem of developing a universal method of forming receptor models for objects of complex technical forms. The essence of the proposed method is the transformation of a solid-state model created in a CAD system into a receptor matrix. First in the physical one, in which the solid-state model is discretized into cubes with receptor sizes, and then in the mathematical one — a three-dimensional array with binary codes of zeros and ones. The creation of a physical receptor matrix is carried out by means of the CAD-system itself, allowing diagnose the belonging of a single receptor to a simulated object. The fact of intersection or non-intersection a given position by a single receptor is encoded by \"1\" and \"0\" respectively, and this information is transferred to a mathematical receptor model (3-dimensional binary array). This calculation procedure is programmed in the form of a macro, providing a given position of a single receptor and fixing the fact of its intersection with the solid-state model. Have been demonstrated examples for described method’s practical application, and has been carried out CPU time cost estimation for the construction of a physical receptor model depending on the receptor size and object geometric complexity. Actions on data transformation from a solid-state model to a receptor one have been implemented in the form of C# programs.

Construction of Receptor Geometric Models for Objects of Complex Technical Forms

Methods of formation of receptor (voxel) geometric models for automated layout tasks

One of the approaches to the construction of graphic images of the stress state for the force vector applied to a point is considered in this work. Has been proposed a geometric model for a continuous medium, formed by a bunch of projection planes for each point of the examined object’s space. This permits to obtain a model for a volume vector in the form of a distributed decomposition into stress components at each point specified by a bunch of projection planes. 
The building a model for a volume vector, defined as a set of specified laws of direction and length, in the context of modeling stress from an applied force vector to a selected point, is based on strength of materials’ classical laws for calculation the stress state values at an inclined section. Such approach allows use a voxel graphic structure for computer representation of the simulated stress, rather than a finite element mesh. In such a case, there is no obtained result’s error dependence on the spatial position of the mesh nodal points, which is often a problem in FEM calculations. 
The resulting functional-voxel computer model of the volume stress vector is a structural unit for modeling the distributed load on areas of complex configuration. In this case, the elementary summation of such vectors allows any uneven distribution of the load relative to each point on the specified area. 
The considered approach works well with geometric models initially represented analytically in the form of a function space (for example, models obtained by the R-functional modelling – RFM-method), and reduced to functional-voxel computer models. 
A method for deformation modeling based on obtained stresses by means of local transformations of the function space, describing the investigated geometric object, is demonstrated.

Geometric Modeling of Stress Visualization Tools Based on the Functional-Voxel Method

In this paper have been considered questions related to automation of the layout for products with high layout density (primarily means of transport). It has been demonstrated how an aircraft’s geometric shape is formed on the basis of infrastructural and layout constraints. Influences of aerodynamic and internal layouts on the aircraft’s geometric shape have been described, taking into account mass-inertia characteristics of units placed in it. The layout’s reverse problem (when a required layout space is initial data for the aircraft’s geometric shape under hard infrastructure restrictions) has been presented. A project task of finding the rational parameters for the aircraft’s geometric shape as the task of multi-criterion discrete optimization has been described in a generalized form. It has been demonstrated that this task can be formulated as a search for the vector of design parameters as a multitude of admissible variants for design-and-engineering solutions. In the paper has been described a physical task formulation for automated layout as a system of restrictions on objects allocation indoor (required orientation, mutual compatibility, serviceability etc.). Mathematical task formulation for automated layout as the optimization problem has been described too. Since the allocation task is a classical geometric problem, it is necessary to develop appropriate geometric models for its solving. It has been shown that this process’s complexity is due to the complexity related to computer representation of information about geometric shape for layout objects of modern transport, especially the aerospace one. In this paper it has been shown that the abundance of models used in modern applied geometry and allowing describe geometric shapes for objects of any complexity, does not provide any solution for automated layout tasks. Possibilities of modern software have been also shown, and the reasons not allowing its direct use in the tasks of automated layout have been proved. The layout task’s mathematical formulation has been described as an optimization problem, specifying its objective function, limitations and efficiency criteria. Has been justified an approach (receptor methods and apparatus of normal equations) allowing, while creating geometric models for the automated layout, go from exhaustive options for allocation of layout objects to intellectual algorithms for automated allocation As has been shown in the paper, fractal theory is a good mathematical tool for study of rigid bodies’ surface geometry and mechanisms influencing on the obtaining surface structure.

М. Куприков

Papers

Geometric Aspects of Aircraft Automated Layout Design