We consider conjugate boundary-value problems of heat exchange for the cases where a viscous incompressible liquid moves through channels with noncanonical cross section flowing around a bundle of rods. The influence of the type of packing on the velocity and temperature distributions is investigated. For the solution, we use the R -functions method in combination with the Ritz variational method. We consider the cases of cyclic, staggered, and rectangular packing of fuel elements.

In this paper we investigated the possibilities and proposed methods of functional representation of a geometric object in 3D for information on the equation of the boundary sections of the object being restored. The article describes the various methods postreniya geometry equations according to their section. The functional representation of a geometrical object defines it as a unit by one real continuous function of several variables. In 3D on the basis of the theory of R-functions developed by V. L. Rvachev works are devoted to the solution of the inverse problem of analytical geometry. The technique of creation of the equations of the composite geometrical objects described in them is based on operations with the known equations of three-dimensional primitives. At the same time set-theoretic operations are defined in an analytical view with the help of R-functions. However often there is a need of the functional representation of a geometrical object for 3D, being based not on the known equations of three-dimensional primitives, and according to information on the equations of borders of sections of the restored object. Constructed geometric objects using the apparatus of the theory of R-functions and its supporting software. This method of constructing geometric objects is a universal means of modeling and visualization. Analysing method for constructing geometric objects using R-functions, it should be noted that the function is positive inside the body is equal to zero on its surface and it is negative. Using literal parameters significantly expands the design possibilities of the implementation of the simulation geometry. Stored in the computer's memory model allows the researcher using the software three-dimensional interactive computer graphics to manipulate spatial images obtained by varying the value of literal parameters. Construction of mathematical models of geometric objects are their analytic identity, as evidenced by visualization of the derived equations.

In this article, was developed methodologies and constructed equations of different heat transfer surfaces of the fins, including Chevron with the help of R-functions theory. The resulting equations of surfaces have been implemented on a 3D printer. Ribbing not only increases the heat transfer surface, but also has a great influence on the hydrodynamics of the flow, and thus on the heat transfer coefficient. In experiments with various methods of finning of Fuel rod claddings were developed more favorable shape of the fins, the so-called Chevron and multi-zone. With the Chevron ribbing the entire surface of the shell is divided into four, six or eight sectors and adjacent sectors of the spiral are located symmetrically relative to the longitudinal axis. Because of the complexity and high cost of manufacturing Chevron transmission is used less frequently than helical, i.e. only in those cases when it is required to transmit high power and high speed, and the axial load of junk. On special machines V-wheels are made all of one piece. A disadvantage of the Chevron ribbing is the high complexity and cost of manufacturing. Thanks to the technology of 3D printing, these deficiencies can be remedied, because the benefits of using 3D printers is reducing the cost of production, reduction of terms of its appearance on the market, the modeling of objects of any shape and complexity, rapidity and high precision manufacturing, the use of different materials, including concrete, hydrogel, wood, metal, plastics, chocolate and even living cells. The article examines the technology of 3D printing. The analysis of the major representation schemes of models of continuous bodies, which revealed significant shortcomings. From the point of view of universality, one of the most promising functional representation, which is based on the use of language implicit mathematical functions with the structural features of the R-functions theory, developed by academician V. L. Rvachev. The analytical description of the designed objects enables to use symbolic geometrical parameters, complicated superposition of functions consequently allowing to change the design elements of these objects. The positivity feature of the built functions in the mid points of the object is convenient to 3D-printing implementing

In Amsterdam the team of architects (DUS Architects) works above the project, called to master one of the most important directions of development of a three-dimensional printing - the construction of buildings. In China (Yingchuang New Materials) the printer which is capable to out small one- and two-storeyed houses has appeared. At the University of Southern California professor Behrokh Khoshnevis has developed the project of the three-dimensional printer, which is capable to print the two-storeyed house within only 24 hours. The one of methods of the decision of a problem of the printing information task is the R-functions theory application. The creation on the R-functions theory basis of the mathematical and computer model of a small country house as a whole is the purpose of the given paper. The most simple system R 0 is used for the required equations construction. In case of presence of translation along direct symmetry or cyclic type dot symmetry the superposition with corresponding periodic functions is used. Stage-by-stage construction of the all constructive building elements' equations is carried out: the base of walls, the partitions, the window and doorways, the roof including versatile one. Besides for the decoration of the house elevation the equations of various ornaments and columns are constructed for their subsequent realization on the 3D-printer. The analytical identification of the projected objects has enabled to use alphabetic geometrical parameters, that, in turn, has allowed to change operatively constructive elements of a house, that also is illustrated in this paper.

The stage-by-stage modeling of the automobile body by the multiparameter equations with alphabetic parameters for geometrical characteristics and a technique of the surfaces equations of construction with continuous curvature function with the help of R-functions is considered in this work. The work of new high-speed system of the equations of geometrical objects surfaces visualization in 3D is illustrated.

Mathematical models of heat exchange in the motion of an incompressible viscous liquid through channels with a helical type of symmetry, including polyzonal finning of fuel elements, in curvilinear nonorthogonal coordinates are proposed. For a laminar flow in the domain of thermal stabilization, we reduce the three-dimensional problem to the two-dimensional problem and investigate the influence of a twist parameter on the distribution of a temperature field. Numerical solutions of model multiparameter problems of calculation of stationary temperature fields by the R-function method (RFM) are considered, which enables us to choose appropriate structural means of the RFM for the subsequent solution of real problems.

Normalized equations for regular no-gons translated nd times in a regular nb-gon are set up based on coordinate transformations that allow translating geometric objects on segments. Such approaches are especially important in solving automation problems in solid modeling, geometric design, and boundary-value problems of mathematical physics.

Based on R-functions, a general algorithm is proposed for automatic construction of predicate and analytical functions of a geometrical object consisting of a given list of standard primitives. Normalized functions of standard primitives with alphabetic parameters are constructed that determine the position, sizes, and orientation of the primitives. The capabilities of automation of constructing equations of geometrical objects from standard primitives are illustrated with the help of the system POLYE.