DELAUNAY TRIANGULATION .1 Introduction

At the middle of the nineteenth century, Dirichlet (1850) showed that for a given set of points in two dimensions, it is possible to partition the plane into convex cells based on a proximity criterion. At the turn of the twentieth century, Voronoi (1908) studied the algebraic forms related to this geometrical criterion, which could be extended to a three-dimensional space.

The concept of the Voronoi diagram was thus established, i.e. there is a one-to-one cor-respondence between the cells and the given points of the set in the partition of space into cells, such that any point in a cell is closest to a particular point of the set than any other points in the set. Delaunay in 1934 stated that it is possible to construct a triangulation from the Voronoi diagram by duality, linking up points associated with the cells with those of the neighbouring cells, as shown in Figure 3.18. The triangulation derived from the Voronoi diagram has since been named as DT of the point set.

p_i

V_i

Figure 3.18 Voronoi diagram and Delaunay triangulation.

3.5.1.1 The convex hull of a given point set

Let  ={ ,p_i i=1, }n be a set of n distinct node points in the Euclidean 2D plane ², and define the set of polygons V = {V_i, i = 1,n}, where

V_i=

{

}

and ||x − p|| denotes the Euclidean distance between points x and p.

V_i represents a region on the plane ² whose points are nearer to node point p_i than to any other node points in the given set. Hence, V_i is an open convex polygon (called the Voronoi polygon) whose boundaries are portions of the perpendicular bisectors of the lines joining node p_i to node p_j of neighbouring polygons V_i and V_j. The collection of Voronoi polygon V is called the Dirichlet tessellation of the point set .

In general, the vertex of a Voronoi polygon is shared by two more neighbouring poly-gons so that connecting the three points associated with such adjacent polypoly-gons forms a triangle, as shown in Figure 3.18. The set of triangles so created at each vertex is called the DT. This construction can be shown to be a triangulation of the convex hull for the set of nodal points . The Dirichlet tessellation is always unique for a given set of points, whereas the DT will be unique as long as the points are in the general positions, or no four points are cyclic. In case there are four points lying on a circle, Delaunay triangles can be formed by cutting across either diagonal of the cyclic quadrilateral, and the DT is only unique up to the division of the cyclic quadrilateral. In the event that there are more than four nodes lying on a circle, the DT is unique up to the division of the cyclic polygon into triangles. If there are more than one spot where points are not at the general positions, the number of DTs depends on the combination of how these cyclic polygons are subdivided into triangles.

3.5.2 Properties of DT

i. Among all triangulations of a set of points in 2D, the DT maximises the minimum angle and minimises the maximum circumcircle.

ii. If every triangle in a triangulation is non-obtuse, it is a DT.

iii. Any 2D triangulation can be transformed into a DT by locally flipping of diagonals of adjacent triangles in O(n²) time (Joe 1993).

In a paper by Rajan (1994), some optimality properties of DT in two or higher dimensions are discussed. It is suggested that the DT is the most compact one in the following sense, which justifies its wide applications in many areas.

i. The containment sphere is the smallest. It is the smallest sphere that contains the simplex, which may or may not be the circumsphere depending on whether the circumcentre is inside the simplex or not. In a DT, the largest containment sphere among all simplices is minimised.

ii. The union of the circumspheres of the simplices incident to an interior point is the smallest. That is, the union of circumspheres of simplices incident to an interior point in a DT lies inside the union of the circumspheres of simplices incident to the same point in any other triangulation of the point set.

iii. The weighted sum of the squares of the edge lengths, where the weight is the sum of volumes of the simplices incident on the edge, is the smallest for DT.

A detailed description of the Voronoi diagram, dual of DT, was given by Aurenhammer (1991), and other properties of DT such as the maximum vertex degree have been estimated by Bern et al. (1991).

3.5.3 Time complexity in the construction of DT

Based on a sound geometrical concept and the optimality properties, DT has important applications in many fields, including data visualisation, terrain modelling, FE MG, sur-face reconstruction and structural networking for arbitrary point sets. The popularity of DT is attributed to its nice geometric properties as a dual of Voronoi tessellation and the speed with which it can be constructed in two or higher dimensions. The existence, uniqueness and other properties of DT have been studied for a long time as formal math-ematical topics in computational geometry, and their computation issues and complexity have been interesting problems for computer scientists. In view of its diverse applica-tions, many strategies for its construction have been proposed. In a paper by Su and Drysdale (1997), numerical tests on several DT algorithms, namely, divide-and-conquer, sweep line algorithms, incremental algorithms, a fast incremental construction algorithm, gift-wrapping algorithms and convex hull–based algorithms, were carried out, and their performances were compared.

There are some estimates of the time complexity of DT in 2D and 3D, which could be taken as a reference, but these asymptotic results may not be directly applicable to FE MG as it only makes sense for very large n values, and the algorithm may not be available or could not be easily implemented for practical use; and there are two more important steps in FE MG relative to triangulation, namely, the generation of interior points and boundary recovery.

The complexity of DT is very sensitive to the point distribution and the order of how these points are processed. For instance, for DT in 3D, the estimated time complexity of triangulating n points is O(n^4/3) for the Bowyer’s algorithm (Bowyer 1981) and higher for the algorithms of Watson (1981) and Avis and Bhattacharya (1983). Joe (1989) presented an algorithm that makes use of local transformations to construct a DT of a set of 3D points. The empirical time complexity of the algorithm is O(n^4/3), and O(n²) in the worst case. Edelsbrunner et al. (1990) proposed a scheme in the worst case time of O(n²) for the construction of 3D DT by projecting the given 3D points onto a paraboloid in four dimen-sions. The convex hull of the 4D points on the paraboloid is constructed, and the 3D DT is then obtained from an appropriate portion of the convex hull. Nevertheless, for practical engineering applications with distance between points not greater than a ratio of, say, 1:10⁶, DTs can be constructed in a virtually linear time complexity, as described in Chapter 8.

3.5.4 FE meshing by DT

By far, the most popular triangular MG schemes are based on the concept of DT (Cavendish et al. 1985; Weatherill 1988; Lo 1989a; Joe 1991a,b,c; Wright and Jack 1994; Lewis et al.

1995; Escobar and Montenegro 1996; Krysl and Ortiz 2001; Nishioka et al. 2001; Quey et al. 2011). The Delaunay criterion, also known as the empty-circle property, states that any node must not be contained within the circumscribing circle of any triangle in the tri-angulation, as shown in Figure 3.19. Although the Delaunay criterion has been known for many years since the pioneer paper by Delaunay (1934), it was not until the work of Lawson (1977), Bowyer (1981) and Watson (1981) that the criterion was exploited for developing algorithms to form a convex hull of a given set of points. With the rapid development of the FEM in the 1980s, the DT algorithm was further extended to generate valid FE meshes for numerical analysis by Baker (1989a,b), George and Hermeline (1992), Ghosh and Mallett (1994), Weatherill and Hassan (1994), Peterson et al. (1999) and others. For a comprehen-sive view on the theoretical aspects of DT as well as its applications to FE MG, readers are referred to the book by George and Borouchaki (1998).

In the incremental algorithm of Bowyer and Watson, the points are processed one at a time. In a typical step of point insertion, the triangles whose circumcircle contains the inser-tion point are identified and deleted. New triangles are constructed in the cavity left behind by the triangles removed. Hence, the efficiency of the triangulation scheme depends on how fast one can identify the triangles to be removed and determine correctly the cavity for inser-tion and the speed with which the circumcentres, circumradii and adjacency relainser-tionship of the new triangles are computed.

The Delaunay criterion itself is not an algorithm for MG. It merely provides a rule to connect a set of existing points in space to form a triangulation. As a result, although the boundary of the domain is well specified, it is necessary to devise a scheme to determine the number and the locations of node points to be inserted within the domain of interest. A typi-cal approach is to first create a triangular mesh large enough to contain the entire domain.

The boundary nodes are then inserted and connected according to the Delaunay criterion, and this forms a triangulation of the boundary nodes. More nodes are then inserted pro-gressively into the coarse boundary mesh, refining the triangles as each new node is intro-duced, until a desirable number of elements are formed at appropriate positions (Borouchaki et al. 1996).

In FE applications, there is a requirement that the triangulation contains the boundary of the domain, so that the boundary integrity can be easily enforced by deleting all the trian-gles outside the given domain. In most DT processes, before interior nodes are introduced, a tessellation of the nodes on the domain boundary is produced. However, in this process, there is no guarantee that boundary segments will all be present in the triangulation. In many implementations, the approach is to tessellate the boundary nodes using a standard Delaunay point insertion algorithm with no regard to the integrity of the domain boundary.

A second step is then employed to force or recover the boundary segments. Of course, by doing so, the triangulation in general is not strictly Delaunay at least locally, hence the term boundary-constrained DT. In 2D, the edge recovery is straightforward simply by swapping diagonals (Weatherill 1990), and in 3D, it is less obvious how boundary integrity could be achieved in general, which will be further elaborated in Chapter 5.

There are a number of ways to create interior points according to the node spacing requirement, which in fact would lead to meshes of different characteristics. Hermeline Figure 3.19 Delaunay triangulation of 14 points.

(1980) proposed a scheme in which points are inserted at the barycentre of the triangles, and some researchers have suggested to insert points at the circumcentres of triangles (Shenton and Cendes 1985; Holmes and Snyder 1988). Borouchaki and Geroge (1997) advocated the insertion of points along the edges of triangles. Others made use of a set of points at pre-determined positions with the aid of a regular grid, a Quadtree network or some sort of spatial decomposition methods such as the kd-tree. A combined scheme with the AFT was also put forward in which points are inserted at strategic positions as determined in a frontal process, and element connections are modified based on the Delaunay criterion. A quality guaranteed Delaunay refinement of a conforming triangu-lar mesh was presented by Shewchuk (2002) in which the interior angle of the triangles is bounded by some threshold related to the boundary. Based on a hexagonal grid, Suβner and Greiner (2009) produced a DT for a set of 2D points with refinement satisfying a number of conditions.

Borouchaki et al. (1997a) made use of the concept of control space and length measure for the insertion of points according to non-Euclidean metrics to create adaptive meshes of variable element sizes and aspect ratios. The element size map can be explicitly specified as a continuous function over the entire domain or implicitly defined over the domain by means of a background mesh (Borouchaki and George 1996). The idea was later extended with the introduction of a general metric tensor to measure the distance between two points for the generation of anisotropic meshes in which not only element size can vary but also elements are subject to different size requirements along different directions (Borouchaki et al. 1997b).

In the construction of DT, the point insertion algorithm proposed by Bowyer (1981) and Watson (1981) is perhaps the simplest but is the most general in extending to higher dimensions. Moreover, it is also easy to describe as well as to implement in a few clear distinct steps (Borouchaki and Lo 1995). It is also one of the most efficient approaches in terms of serial processing and parallel processing (Lo 2012a,b), such that for evenly distributed randomly generated points, the time complex is basically linear. In the point insertion scheme, the fact that DTs are always constructed for the points already inserted is a nice feature guaranteeing the robustness of the procedure. When the next point is introduced, circumcircles of some triangles may contain the newly inserted points, and hence they become non-Delaunay. According to the lemma of Delaunay, non-Delaunay triangles locally form a patch (connected piece), which could be easily identified by the empty-circle criterion. Upon the removal of the patch of non-Delaunay triangles, and the re-establishment of the connection of the newly inserted points with the boundary edges of the patch (cavity), a DT containing the newly inserted point is thus created. When all the points are processed one by one, a DT of the given point set is constructed. Hence, the point insertion algorithm is very flexible and reliable in dealing with arbitrary point sets or their subsets.

3.5.4.1 Fundamentals and strategy

Before a detailed presentation of the point insertion algorithm, the formal definition and some basic strategies for the construction of DT are discussed. The DT of a set of points on a plane is defined to be a triangulation such that the circumcircle of every triangle in the triangulation contains no point from the set in its interior. Such a triangulation exists for a given set of points, and it is the dual of the Voronoi tessellation. The triangulation is unique if the points are in general position, i.e. no four points are cyclic. A triangle T is said to be Delaunay with respect to a point p if p does not lie inside the circumcircle of T. A triangle T in a triangulation of a set of points is called a Delaunay triangle if T is Delaunay with respect

to every point in the set. A triangulation of a set of points is called the DT of the point set if every triangle in the triangulation is a Delaunay triangle, as shown in Figure 3.19. The idea of DT is very general, which can be easily extended to higher dimensions. For instance, the DT in 3D is given by replacing the triangle by a tetrahedron, the circle by a sphere and the 2D plane by a 3D space. The following lemma provides the basis for many algorithms in the construction and verification of DT.