Bicherakhova O.S. Construction of Triangular Meshes of Multiply Connected Domains Based on Delaunay Triangulation
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https://doi.org/10.15688/mpcm.jvolsu.2025.3.4
Olga S. Bicherakhova
Assistant Lecturer, Department of Mathematical Analysis and Function Theory, Volgograd State University
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https://orcid.org/0009-0000-3089-6029
Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation
Abstract. When numerically solving boundary value problems of mathematical physics in areas of complex shape, an important issue that must be studied separately is the construction of computational grids. The most suitable for this are grids with triangular cells, since triangles (tetrahedra in the spatial case) are the simplest linear figures, the geometric characteristics of which are quite easy to calculate. This article considers the problem of constructing triangulation of multiply connected areas. Existing methods for constructing triangular grids are either designed for convex areas or are very difficult to implement in software. In this article, a method for constructing triangular computational grids of multiply connected regions is proposed, based on the classical Delaunay triangulation of a finite system of points. A distinctive feature of the proposed method is that the algorithm returns a triangulation of a multi-connected region that satisfies the Delaunay condition. The idea to use Delaunay triangulation as the basis for the algorithm arose due to its property of maximizing the minimum angle in triangles, which leads to a reduction in too sharp angles in the grid, which negatively affect the accuracy of calculations. Thus, the algorithm returns a triangulation of sufficiently good quality, suitable for further mathematical calculations. A more detailed study of the quality of the resulting grid is a separate interesting task, the study of which is planned in future articles. The algorithm for constructing a triangulation of a multi-connected region presented in this article is simple to implement and is suitable for triangulating regions defined both mathematically and using an .obj file. The article provides code in the Python programming language that implements the presented algorithm, and also shows examples of triangulation of various flat multi-connected regions that are defined in different ways.
Key words: triangular mesh, Delaunay triangulation, multiply connected domain, non-convex triangulation, Delaunay condition.
Construction of Triangular Meshes of Multiply Connected Domains Based on Delaunay Triangulation by Bicherakhova O.S. is licensed under a Creative Commons Attribution 4.0 International License.
Citation in English: Mathematical Physics and Computer Simulation. Vol. 28 No. 3 2025, pp. 37-49
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