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To the Matter of Triangulation Method Selection of Three-Dimensional Bodies with Reference to Internal Electrodynamic Problem Solution


V.Y. Aronov, I.Y. Kolchugin

To solve present-day electrodynamic problems and an internal electrodynamic problem particularly it is possible to use the finite elements method (FEM). This method, in comparison with other numerical methods, has an advantage of lesser dependency on geometric forms of objects to work with. At the first stage of internal electrodynamic problem solution by FEM a mesh of finite elements modeling internal space of object is built. Methods that build a mesh of tetrahedrons (triangulation methods) are most developed and widely-spread. It also has to be noticed that FEM doest not require any kind of geometric or topological regularity of a mesh, that makes easier a triangulation problem. There are two classes of triangulation methods exist – direct methods where a mesh is built in one step and iterative ones where a mesh is built in a sequence of steps: one or more elements are added at an each step. To triangulate a finite area, most suitable method can be used. Particularly, for areas of a simple geometry shape (parallelepipeds, spheres, cylinders, prisms) direct methods can be used – application of more resource-exacting methods is not practical in this case. Iterative methods, unlike direct ones, can be used for areas of a fairly random shape triangulation. Among iterative triangulation methods, it is possible to specify three approaches to building a mesh: boundary correction methods, methods based on the Delaunay criteria and methods of exhaustion. Boundary correction methods are the fastest of iterative methods but have some limitations. Mesh is built in two stages in these methods. In the first stage a certain simple area that includes a given area is triangulated. In the second stage nodes of an obtained mesh are projected to a surface of a border, and nodes situated outside of a given area are deleted. Methods based on the Delaunay criteria are often called simply Delaunay methods. Nodes are placed in the given area and links are established among them according to Delaunay criteria then. The main point of the methods of exhaustion consists in consecutive excision of tetrahedron-shaped fragments from given area until an area exhausted. A triangulation of an unexhausted area border (called front) is a basic data for each iteration. At each iteration one tetrahedron or a whole layer of tetrahedrons is excised. After that, front is renewed, and then a new iteration starts. The conclusion, based on the analysis carried out on this three groups of methods, is that it is most practical to use exhaustion methods for geometrically complex areas triangulation as boundary correction methods can not be implemented for discretization of an areas with given border triangulation, and Delaunay methods have a range of problems when used in the three-dimensional space. Result of border triangulation giving a necessary base data (initial front) for exhaustion methods is a considerable advantage over other methods for them too.
May 29, 2020

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