INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 40, 1957–1975 (1997) ASPECTS OF 2-D DELAUNAY MESH GENERATION HOUMAN BOROUCHAKI AND PAUL LOUIS GEORGE INRIA; Domaine de Voluceau; Rocquencourt; BP 105; 78153 Le Chesnay Cedex; France SUMMARY This paper aims to outline the dierent phases necessary to implement a Delaunay-type automatic mesh generator. First, it summarizes this method and then describes a variant which is numerically robust by mentioning at the same time the problems to solve and the dierent solutions possible. The Delaunay insertion process by itself, the boundary integrity problem, the way to create the eld points as well as the optimization procedures are discussed. The two-dimensional situation is described fully and possible extensions to the three-dimensional case are briey indicated. ? 1997 by John Wiley & Sons, Ltd. KEY WORDS: Delaunay triangulation; constrained triangulation; 2-D-mesh generator 1. INTRODUCTION There exist a large number of papers dealing with 2-D-mesh generation using the Delaunay method (see Reference 1). Nevertheless, we propose a new scheme with ambition to optimize, if possible, the dierent steps used in the practical application of the algorithm. We have tried to develop for each step an optimal solution. The method has been implemented and results show the performance of the algorithm. So we can construct a well-shaped mesh constituted by a million of triangles in a few minutes on a HP 735 workstation. The 2-D case is fully detailed for clarity but most of the results can be extended without diculty in three dimensions. The problem to be solved is to construct a consistent mesh of a domain essentially from its boundary data segments. The resulting mesh will be the support of a nite element computation which can indicate if it is adapted or must be adapted by any method (it is not the goal of this paper). Nevertheless, it is important to begin with a well-shaped mesh knowing that this request can only be based on geometric considerations, since the sole known information is of geometric nature. Thus, the shape and the size of the elements must be consistent with these data. This paper is organized as follows. In Section 2, we recall some tools and denitions which are used afterwards. Section 3 describes, on the one hand, the classical point insertion algorithm and, on the other, a robust version of the same algorithm. In Section 4 we propose some acceleration procedures for practical implementation of the previous procedure. In Section 5 we give a simple solution to the boundary integrity problem to construct an initial mesh of the domain. Section 6 deals with the most important part of a mesh generator in two dimensions, i.e. the eld point generation. Section 7 is a brief summary of mesh-improvement procedures that we can apply if necessary. In Section 8 we describe the mesh-generator scheme and give some numerical results. Finally, in Section 9, we propose the possible extensions to three dimensions. CCC 0029–5981/97/111957–19$17.50 Received 5 January 1996 ? 1997 by John Wiley & Sons, Ltd. Revised 25 June 1996