Automatic Mesh Generation and Finite Element Computation by P.L. George

By P.L. George

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A < Other patterns slightly different in two dimensions, as well as suitable patterns in three dimensions, can be defined to make the mesh generation process efficient. , ,,;,, i"""'" _- '"-" -· ---- -_ two elements are created one element is created - I---· - -·-- FIG. 6. The three retained patterns and associated constructions. 120 PL. George CHAPTER I in two dimensions and, * remove the faces belonging to the tetrahedron now formed, * add those which are not shared by two elements existing now, in three dimensions.

Finding an optimal location With regard to the manner in which the current front is analyzed and the nature of the departure zone, front items are dealt with in a predefined order. Assume that we examine a front item (edge or face according to the space dimension) obtained by one method or another, then we are able to define an optimal point very easily. The process is as follows (for simplicity, we first examine the twodimensional case and denote by AB the front edge under consideration (Fig. 2), after which the three-dimensional case will be discussed briefly): * point C is constructed so that triangle ABC is equilateral, * hloc(C) is computed using the control space: according to hloc,, and the current distances from C to A and B, point C is relocated on the mediatrice of AB in such a way that these three values become equal, * (a) points of the front other than C, falling in some neighborhood, are examined in order to decide if they can be connected with AB in such a way that triangles ABPi, where Pi denote these points, are valid.

In this case segments may produce, not only quadrilaterals, but triangles, triangles may produce degenerated pentahedra and quadrilaterals may produce degenerated hexahedra. Depending on the situation, the degenerate elements are valid, or not, in the usual finite element context. In this respect, for example, hexahedra may degenerate into pentahedra, which are admissible elements, or into non-admissible elements (see Fig. 2), while pentahedra may lead to the creation of tetrahedra or non-admissible elements.