ALGORITHM OF SEARCHING FOR SPECIAL POINTS OF THE GEOMETRIC REGION, AS SET BY THE IMPLICIT FUNCTION - Al-Atаmneh, B. G. M. (Computer Science and Applied Mathematics, 2017)
Numerical analysis of various classes of boundary value problems requires the generation of grid (discrete) models of geometric regions of complex shape. For example, the use of the finite element method is based on assuming the existence of a finite element model of the investigated region. The problem of constructing grid models can be divided into two independent tasks: 1) formalization of the description of the initial geometric domain (construction of its model in a form suitable for subsequent computer processing); and 2) construction of a grid model from the previously obtained formal description of the initial geometric domain. Classical approaches to solving the first problem are the use of framework, surface and solid geometric modeling. The main disadvantage of these approaches is the complexity of describing the geometric domain of non-standard form. In the event that there is no element in the library of geometric primitives, then the construction of a complex geometric model becomes problematic. One possible effective way to solve this problem is to describe geometric regions by means of functional relationships in the form of a certain real function that takes negative values outside the region. This approach to the description of geometric models is called Function Representation (FREP) and is a generalization of the theory of R-functions of V. Rvachev. Practical application of FREP involves a number of significant difficulties, as it is necessary to solve the following tasks: 1) automate the search for nodes on the boundary of the implicitly defined FREP-model of the original geometric domain; 2) reconstruction of the boundary of the grid model of the original region from the existing set of support nodes on its boundary; 3) visualization of the FREP-model. Therefore, the solution of the above problems is currently an actual problem. In this paper, an algorithm for constructing the boundary of an implicitly defined geometric domain described in the form of a FREP-model is described. The implemented approaches to optimizing the resulting grid model are described by two criteria: the ratio of the lengths (areas) of neighboring boundary segments and the magnitude of the angle between them. Examples of the construction of grid models for several planar geometric regions are given.