R-Functions in Boundary Value Problems in Mechanics

Abstract

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.

Comparative Analysis of Ck- and C0-GFEM Applied to Two-dimensional Problems of Confined Plasticity

“…On the contrary, in the case of non-convex clouds, it is necessary to replace the edge functions of a pair of non-convex edges with a single new edge function obtained through a boolean product (Rvachev and Sheiko, 1995) before performing the weighting functions computation, as proposed by Duarte et al (2006). Therefore, the resulting PoU is at least k-times continuously differentiable, and the resulting approximation functions of the product of Shepard PoU with the enrichment functions will have the same continuity, since the enrichments are at least also C k .…”

Introduction to the Web-method and its applications

“…[19] for details). This algorithmic approach combines particularly well with simulations for constructive solid geometry models, hence covers many typical engineering applications.…”

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Mathematically Reduced Chemical Reaction Mechanism Using Neural Networks

“…Zhang and Karunamuni, (2000). The other approach for correcting for boundary effects, is to use the new approach of R-Functions, Rvachev and Sheiko (1995). This is an elegant approach that will be adopted in this work.…”