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.

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The R-Function Method (RFM) solution structure is a functional expression that satis®es 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.

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The traditional mathematical models of fields of various physical nature are boundary-value problems for partial differential equations. The usual statement of a boundary-value problem has the following form:where ffZ is the region in which the solution u is being sought; OfZi, i = 1, 2 ..... m, is a covering of the boundary t?~ of the region f) (the parts c~f~ i are not necessarily different and may be equal to Of)); f and (Pi are known functions (perturbing the field), vector-valued functions, tensors, or possibly elements of some other type. In the Rfunction method developed in the present article geometric information is taken into account by functions co, co i, chosen as a rule from among the elementary functions, so that 0~2 and 0f~i are the sets of their zeros, and the solution u is sought in the pencilHere r is a known function; B is an operator depending on the shape of the boundary Og2 and its parts OO i, i = 1, 2 ..... m, formed so that for any choice of an undetermined component ~ from a set M formula (3) satisfies the boundary conditions (2) exactly. If this formula has the property of completeness relative to the boundary-value problem (1)-(2), that is, it is possible to choose an undetermined component 9 that converts (3) into a solution of the problem with any prescribed precision (in some sense), it is called a general structure solution of the problem in question. -We recall first how the R-function method solves the problem of constructing the functions co and col that occur in formulas of the form (3). A complicated locus (also called a geometric object or a drawing) which the region f2, and hence also its boundary 0f~, may be, can be described by a logical formula connecting simple loci (primitives) after which transition to the usual equations (resp. inequalities) of the form to(x, y)= 0 (resp. co(x, y)> 0) recognized in analytic geometry is accomplished using R-functions. The technique of constructing them will be shown through examples below.In the majority of cases the following R-functions are used to construct the functions to, co/: