A method for the automatic generation of triangular elements on a surface

Abstract

SUMMARYAn algorithm is given to discretize a polygon or curved surface into triangular arrays. The method first involves the manual division of the array into quadrilaterals with specified numbers on rows and columns. The individual points are then filled in automatically and elements are automatically generated. A point of particular importance is that the number of columns or rows may be made to change. A subroutine is given to perform this task.Edgeberg' and Zienkiewicz and Phillips' proposed a method for generating two-dimensional grid structures by first manually partitioning the domain into quadrilaterals (the 'piece and side' method). In each of these quadrilaterals the number of points on each side was restricted to the same number as those on the opposite side. While the method has proved quite useful in partitioning many types of structure, it often requires complex and inconvenient structures involving triangles (a special case of the generalized quadrilateral) in order to change the number of points in one direction. THE METHODOur method of discretizing a quadrilateral region is based on a method of storing its elemental corner points. They are arranged in rows and columns with no more columns at the top than at the bottom. Such a set of consecutively numbered points is shown in Figure 1.Note that there are two types of regions in this structure: one in which the number of columns is constant, and the other in which the columns decrease with descending rows. The top array may be treated by the method of Zienkiewicz and Phillips, while the bottom array must be treated differently. The elements are connected as shown in Figure 2. The method of connecting the points in the bottom rows is unique. The actual location of the points can be specified by a procedure similar to a transfinite i n t e r p~l a t i o n~.~ or a Coons patch.5 The location (x, y) of each point is given bywhere s = -1 + 2 ( i -l)/(n, -l), and t = -1 + 2 ( j -l)/(nc -1). The number of the row from the top is i, and n, is the total number of rows. The number of the column from the left is j, and n, is the total number of columns for that particular point. The vector location of the kth corner of the quadrilateral piece has been given as (x, y), and the number of these corners starts in the lower left corner and increases counterclockwise. Examples of the row and column blending functions, f and g, are given in Table I.

Element free formulation used for connecting domain boundaries

“…In recent years, many research results have been related to the development of automatic mesh generation. In a two dimensional plane problem, a triangle is the simplest form of element shape used in automatic mesh generation (Cavendish, 1974;Cohen, 1980;Haber et al, 1981;Brown, 1981;Ho-Le, 1988). However, convergence behavior of a triangle element is inferior to that of a quadrilateral element.…”

“…General software packages for commercial use which have automatic mesh generation are largely based on the mapping method (Cohen, 1980;Haber et al, 1981;Brown, 1981;Ho-Le, 1988;Zienkiewicz and Phillips, 1971;Gordon and Hall, 1973;Kikuchi, 1985;Chinnaswamy et al, 1991;Cheng and Li, 1996). This method is fast, simple and easily controls geometry and mesh density as well as being applicable to highly changeable geometric boundaries and mesh shapes.…”

Mesh generation over curved surfaces with explicit control on discretization error

“…Today, general‐purpose, user‐friendly, interactive mesh generators for two‐dimensional domains are available commercially, and there is also considerable success in the three‐dimensional mesh generation[4, 5, 6]. However, it seems that there is relatively little effort being spent on the development of high quality surface mesh generators[7, 8, 9, 10] which are essential for the shell structure analysis widely used in the automobile and airplane industries. A general surface mesh generator also serves as the starting point of many 3D mesh generators especially those based on the advancing front technique[11, 12].…”