Analytical R-functions-based design of complex constructions
Description
The work is based on fundamental
research in the R-functions theory developed by V.L. Rvachev, Academician of
the National Academy of Sciences of Ukraine. The R-functions theory has
allowed solving the inverse problem of analytical geometry formulated first
by Descartes: given a geometrical object, it is necessary to write down its
equation. The R-functions theory allows describing geometric objects with the
help of continuously differentiable functions whose values define the boundary,
the interior part and other properties of the simulated object. These
functions allow accounting for geometrical information analytically, and they
are convenient for executing different integro-differential computations.
Innovative aspects and main
advantages
New technologies have been
developed for synthesis and analysis of complex constructions in analytical
design systems with computer-based building of multiparameter normalized
equations in 2D and 3D space for mechanical engineering objects. Methods have
been developed for building normalized equations of boundaries of geometrical
objects possessing translational, cyclic and helical-type symmetry. Methods
of translation for finite intervals have been developed. It is first proposed
to construct equations of complex geometrical objects using standard
primitives. This is a technological basis for automating the process of
constructing these equations. Representing a geometric object as an
analytical formula dramatically reduces the volume of data describing the
type and spatial orientation of the object. To construct a complex geometric
object, one needs but parameters that describe its constructional elements
instead of huge FEM data arrays. Collections of elements (primitives) can
represent a wide spectrum of geometric forms that depend on the subject
domain.
Areas
of application
Mathematicians and
scientists in many countries of the world use the theory of R-functions for
solving problems in geometrical design, solid-state modeling, pattern
recognition and research in physico-mechanical fields in complex-shape
objects. It can be used for building solution structures, which precisely
satisfy all boundary conditions in a boundary- value problem. Such structures
can be combined further with many numerical methods for finding approximate
solutions of a problem without dividing fields into grids.
Stage of development
Computer implementation of fluid
dynamics, electromagnetic, thermal, strain and other field problems is done
using the POLYE system developed at the Podgorny Institute for Mechanical Engineering
Problems of NASU. Representation of geometrical and physical information with
alphabetic parameters allows for multivariant computations. 3D visualization
of constructed equations of geometrical objects is done with the RANOK system
developed at the Zaporozhye National University.
Contact details
Prof. Sheyko T.I.
Podgorny Institute for Mechanical Engineering Problems
Tel. +38(0572)942774
E-mail: sheyko@ipmach.kharkov.ua
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