Delaunay triangulation of non‐convex planar domains

This paper investigates the possibility of integrating the two currently most popular mesh generation techniques, namely the method of advancing front and the Delaunay triangulation algorithm. The merits of the resulting scheme are its simplicity, efficiency and versatility. With the introduction of ‘non-Delaunay’ line segments, the concept of using Delaunay triangulation as a means of mesh generation is clarified. An efficient algorithm is proposed for the construction of Delaunay triangulations over non-convex planar domains. Interior nodes are first generated within the planar domain. These interior nodes and the boundary nodes are then linked up together to produce a valid triangulation. In the mesh generation process, the Delaunay property of each triangle is ensured by selecting a node having the smallest associated circumcircle. In contrast to convex domains, intersection between the proposed triangle and the domain boundary has to be checked; this can be simply done by considering only the ‘non-Delaunay’ segments on the generation front. Through the study of numerous examples of various characteristics, it is found that high-quality triangular element meshes are obtained by the proposed algorithm, and the mesh generation time bears a linear relationship with the number of elements/nodes of the triangulation.

[1]  O. C. Zienkiewicz,et al.  An automatic mesh generation scheme for plane and curved surfaces by ‘isoparametric’ co‐ordinates , 1971 .

[2]  Robert D. Cook,et al.  Observations regarding assumed‐stress hybrid plate elements , 1974 .

[3]  K. R. Grice,et al.  Robust, geometrically based, automatic two‐dimensional mesh generation , 1987 .

[4]  Richard H. Gallagher,et al.  A general two‐dimensional, graphical finite element preprocessor utilizing discrete transfinite mappings , 1981 .

[5]  Nguyen-Van-Phai,et al.  Automatic mesh generation with tetrahedron elements , 1982 .

[6]  S. Lo.  A NEW MESH GENERATION SCHEME FOR ARBITRARY PLANAR DOMAINS , 1985 .

[7]  Mark S. Shephard,et al.  Automatic three‐dimensional mesh generation by the finite octree technique , 1984 .

[8]  William H. Frey,et al.  An apporach to automatic three‐dimensional finite element mesh generation , 1985 .

[9]  W. J. Gordon,et al.  Smooth interpolation in triangles , 1973 .

[10]  J. Cavendish.  Automatic triangulation of arbitrary planar domains for the finite element method , 1974 .

[11]  O. C. Zienkiewicz,et al.  Reduced integration technique in general analysis of plates and shells , 1971 .

[12]  Robin Sibson,et al.  Locally Equiangular Triangulations , 1978, Comput. J..

[13]  J. L Finney,et al.  A procedure for the construction of Voronoi polyhedra , 1979 .

[14]  W. C. Thacker,et al.  A brief review of techniques for generating irregular computational grids , 1980 .

[15]  Lawrence L. Durocher,et al.  A versatile two-dimensional mesh generator with automatic bandwidth reduction , 1979 .

[16]  R. Shaw,et al.  Modification to the Suhara‐Fukuda method of network generation , 1978 .

[17]  A. Bykat,et al.  Design of a recursive, shape controlling mesh generator , 1983 .

[18]  André Denayer.  Automatic generation of finite element meshes , 1978 .

[19]  R. B. Simpson,et al.  Triangular meshes for regions of complicated shape , 1986 .

[20]  Mark Yerry,et al.  A Modified Quadtree Approach To Finite Element Mesh Generation , 1983, IEEE Computer Graphics and Applications.

[21]  Sergio Pissanetzky.  Kubik: An automatic three‐dimensional finite element mesh generator , 1981 .

[22]  Mark S. Shephard,et al.  AUTOMATIC MESH GENERATION FOR THREE-DIMENSIONAL SOLIDS , 1985 .

[23]  D. F. Watson.  Computing the n-Dimensional Delaunay Tesselation with Application to Voronoi Polytopes , 1981, Comput. J..

[24]  W. Frey.  Selective refinement: A new strategy for automatic node placement in graded triangular meshes , 1987 .

[25]  Adrian Bowyer,et al.  Computing Dirichlet Tessellations , 1981, Comput. J..

[26]  Firooz Ghassemi,et al.  Automatic mesh generation scheme for a two- or three-dimensional triangular curved surface , 1982 .

[27]  W. J. Gordon,et al.  Construction of curvilinear co-ordinate systems and applications to mesh generation , 1973 .

[28]  B. Fox,et al.  Construction of Voronoi Polyhedra , 1978 .

[29]  W. A. Cook.  Body oriented (natural) co-ordinates for generating three-dimensional meshes , 1974 .

[30]  S. Park,et al.  Drag method as a finite element mesh generation scheme , 1979 .

[31]  Edward A. Sadek,et al.  A scheme for the automatic generation of triangular finite elements , 1980 .

[32]  A. O. Moscardini,et al.  AGTHOM-automatic generation of triangular and higher order meshes , 1983 .

[33]  Yoichi Kodera,et al.  A generalized automatic mesh generation scheme for finite element method , 1980 .