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In a general setting, the transfinite interpolation problem requires constructing a single function f(x) that takes on the prescribed values and/or derivatives on some collection of point sets. The sets of points may contain isolated points, bounded or unbounded curves, as well as surfaces and regions of arbitrary topology. All such closed semi-analytic sets may be represented implicitly by real valued functions with guaranteed differential properties. Furthermore, such functions may be constructed automatically using the theory of R-functions. We show that such implicit representations may be used to solve the general transfinite interpolation problem using a generalization of the classical inverse distance weighting interpolation for scattered data. The constructed interpolants may be used to approximate boundary value and smoothing problems in a meshfree manner.

General structures of solutions (GSS) of mathematical physics boundary-value problems in domains containing thin cuts, small cracks, hollows, etc. are considered. It is shown that in such cases, the classic L. V. Kantorovich GSS leads to incompleteness of structure formulas. New complete structure formulas are proposed for the domains.

The R-Function Method (RFM) solution structure is a functional expression that satisfies all given boundary conditions exactly and contains some undetermined functional components. It is complete if there exists a choice of undetermined component that transform the solution structure into an exact solution. Such a structure was used by Kantorovich (Kantorovich and Krylov, 1958) and his students to solve boundary value problems with homogeneous boundary conditions on geometrically simple domains. RFM is based on the theory of R-functions (Rvachev, 1982) that allows construction of a set of functions vanishing on the boundary and can be applied to problems with arbitrarily complex domains and boundary conditions. The resulting solution method is essentially meshfree, in the sense that the spatial discretization no longer needs to conform to the geometry of the domain, and can be completely automated. This paper summarizes the main principles of RFM, proves its completeness, and presents numerical results for several simple test problems.

Application of recently developed non-Archimedean algebra to a flat and finite universe of total mass M0 and radius R0 is described. In this universe, mass m of a body and distance R between two points are bounded from above, i.e., 0≤m≤M0, 0≤R≤R0. The universe is characterized by an event horizon at R0 (there is nothing beyond it, not even space). The radial distance metric is compressed toward horizon, which is shown to cause the phenomenon of red shift. The corresponding modified Minkowski's metric and Lorentz transforms are obtained. Applications to Newtonian gravity shows a weakening at large scales (R→R0) and a regular behavior as R→0.

This paper is devoted to developing constructive methods of the theory of R-functions. It gives the first discussion of methods of constructing the equations of loci with symmetry of translation type for duplicated domains not separated by lines and with point symmetry of cyclic type for regions both possessing and lacking axial symmetry. The possible applications are illustrated by numerous examples carried out using the POLE system, including solutions of boundary-value problems involving the influence of cyclically located circular and star-shaped insulators on an electric field.

We present a computational model of the process of solidification of metal in molds of complicated configuration. The model is based on the theory of R-functions, which makes it possible to construct the normal equations of drawings and introduce additional parameters into these equations, by means of which one can assure the required behavior of the solution corresponding to the technological implementation of the process. Computational experiments were carried out using thePOLE system.

Solidification of metal castings can be modeled by an implicit real-valued function whose behavior is determined by physical parameters prescribed on the boundary of a casting. We show how to construct such functions using theory of R-functions for two-dimensional castings represented by their boundaries. The parameterized form of the constructed functions is convenient for studying, controlling, and optimizing their behavior in terms of the physical parameters specified on the boundary of the casting. The proposed approach can also be used for modeling multiple cavities in a same sand mold, generalizes to three-dimensional castings, and is applicable to other physical phenomena that may be suitable for analysis based on empirical knowledge.

The problem of calculating thermoelastic fields in the blading of aircraft engines is investigated. The problem is solved by the method of R functions (named after the author, V. L. Rvachev).

In blanking and stamping industrial practice, axially symmetric industrial equipment is used in forming conical tubular blanks by transmission media which has external circumferential ribbing. This reduces the metal content of the stamp while simultaneously ensuring sufficient strength. In this case, an external load in the form of a uniformly distributed pressure is applied to the internal surface of an axially symmetric conical ribbed thick shell. In the general case, the problem of determining the stressstrain state of an elastic body is formulated as follows [5]. Within the region occupied by the elastic body, we should determine the solution of the Lain6 equation

Described are the concepts and applications of the R-functions theory in continuum mechanics boundary value problems which model fields of different physical natures. With R-functions there appears the possibility of creating a constructive mathematical tool which incorporates the capabilities of classical continuous analysis and logic algebra. This allows one to overcome the main obstacle which hinders the use of variational methods when solving boundary value problems in domains of complex shape with complex boundary conditions, this obstacle being connected with the construction of so-called coordinate sequences. In contrast to widely used methods of the network type (finite difference, finite and boundary elements), in the R-functions method all the geometric information present in the boundary value problem statement is reduced to analytical form, which allows one to search for a solution in the form of formulae called solution structures containing some indefinite functional components. A method of constructing solution structures satisfying the required conditions of completeness has been developed. The structural formulae include the left-hand sides of the normalized equations of the boundaries of the domains or their regions being considered, thus allowing one to change the solution structure expeditiously when changing the geometric shape. Given in the work is a definition of the basic class of R-functions, solution with their help of the inverse problem of analytical geometry (construction of equations of specified configurations); generalization of the Taylor-Hermite formulae for functional spaces in which points are represented by lines and surfaces; and construction of solution structures of some types of boundary value problems. Shown are the solutions of a number of concrete problems in these application fields with the use of the RL language and POLYE system.

The development of the theory of R-functions has stimulated the emergence of efficient numerical-analytical methods for a wide class of problems associated with physico-mechanical fields. R-functions are the basis for computer-aided level source language, which is sufficiently close to an ordinary mathematical language. Despite the powerful capabilities of the POLE family systems and the extensive support for the analysis of computation results, these systems are still not widely used in practice, as they assume a user with special mathematical qualifications. It is therefore necessary to develop special-purpose systems that do not require the user to be proficient in mathematical physics, the theory of R-functions, variational methods, etc. In other words, we need systems that are oriented toward technical-profile engineers specializing in design and calculations. In this article, we propose some principles for the construction of a special-purpose interactive programming system POLE-PLAST capable of solving problems in the mechanics of deformable solids of complex shape. As the specific application domain for the description of the system we have chosen problems that deal with bending, stability, and vibration of elements of thin-walled constructions modeled by thin plates.

We propose a method of determining the structure of a field on a finite interval of the real line. Using the field operations we construct an analog of the differential calculus having the property that the result of every operation is bounded. We propose using this approach to develop mathematical models that are free of divergence.

A method for the approximate solution of mixed problems of elasticity theory for nonhomogeneous bodies of revolution af finite size is proposed. Based on this method the stress-formed state of a cylinder wtih a rigid inclusion is determined.

We consider the stationary flow in a cavity with a moving lid, and as a mathematical model we use a fourth-order partial differential equation in the current function with boundary conditions of Neumann and Dirichlet type. The solution is constructed using the apparatus of R-functions.

The effect of the ratio of characteristic diode base dimensions to its height for a constant volume is determined. The effect of the shape of the diode side surface on its resistance is investigated. Practical recommendations are proposed on the application of the obtained results to design optimization of solid state components used in radiometric devices.

The skin-effect resistance in Schottky-barrier diodes with a complex form of the boundaries is calculated using the theory of R functions and the Ritz variational method. The minimum resistance due to the skin effect is determined by the choice of the diode geometry. A numerical experiment and aphysical analysis enabled the proposed method of calculation to be recommended to designers of both individual solid-state components and radiometer systems.

Modern radio physics has in recent years been characterized by rapid development of theory and engineering practice in the millimeter and submillimeter wave ranges. The dimensions of the systems, which are usually quite arbitrary in shape, are commensurate with or substantially in excess of the wavelength. This necessitates determination of the eigenvalues and eigenfunctions of regions with boundaries of complex form (waveguides, microstrip structures, resonators, etc.). One important problem is to lay the electrodynamic foundation for computer-aided design of microwave devices over a broad wavelength range [i]. A procedure for analytic description of the complex geometric shapes of real objects is especially important for development of the appropriate algorithm and their numerical implementation in computers. The theory of R functions, which is utilized in solving interior and exterior electrodynamic boundary value problems [2-11], is promising for this purpose.

The temperature field was studied in a steel pressure chamber during tests of an asynchronous electric motor. A method based on the theory of R functions is proposed for calculation of the temperature fields in enclosed volumes. The numerical results are compared with the test data.

A calculation is presented on the three-dimensional temperature distribution in the cylinder block of an axial piston pump; the heat-conduction problem is solved by the joint use of finite integral transformations, the structural method, and the Bubnov-Galerkin method. The results are compared with experiment.