# ADAPTIVE DISCRETE MODELS OF FUNCTIONALLY REPRESENTED OBJECT SHAPES

 DOI https://doi.org/10.15407/pmach2018.04.049 Journal Journal of Mechanical Engineering – Problemy Mashynobuduvannia Publisher A. Podgorny Institute for Mechanical Engineering Problems National Academy of Science of Ukraine ISSN 0131-2928 (Print), 2411-0779 (Online) Issue Vol. 21, no. 4, 2018 (December) Pages 49-56 Cited by J. of Mech. Eng., 2018, vol. 21, no. 4, pp. 49-56

Author

Serhii VChoporov, Zaporizhzhia National University (6, Zhukovskyi Str., Zaporizhzhia, 69600, Ukraine), e-mail: s.choporoff@znu.edu.ua, ORCID: 0000-0001-5932-952X

Abstract

Designers often use a numerical analysis of mechanical engineering product models. The analysis is based on partial differential equations. One of the most used numerical methods is the finite element method, in which the continuous object model is replaced by a discrete one. As a result, the first stage of modeling is the construction of a discrete object shape model as the final union of simple shapes. The distribution of elements in a discrete object shape model has a significant impact on the accuracy of numerical analysis. One of the most universal approaches to the computer modeling of object shapes is functional representation. This approach is based on using implicit functions to determine the set of points that corresponds to the object shape. Moreover, implicit functions for complex objects can be created constructively using combinations of simpler functions. For this, one can apply the real functions that are proposed in the R-functions theory and correspond to logical operations. Although functional representation makes it possible to check whether a point belongs to a set, it requires that methods for constructing discrete models be developed. In this paper, a method is proposed for constructing adaptive discrete models of object shapes represented functionally. This method uses an estimate of the accuracy of the finite element analysis to determine the areas where nodes and elements are refined. In the process of refinement, the refinement templates of elements are used that are proposed for the most common elements (triangles, quadrangles, tetrahedra and hexagons), with reprojection on the domain boundary of boundary nodes. Examples of constructing adaptive discrete models for solving two- and three-dimensional problems of studying stress-strain state are shown.

Keywords: discrete model, object shape, implicit function, R-function, finite element method.

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