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Abstract. This paper describes a method for generating tetrahedral meshes. The algorithm, based on the Delaunay triangulation, can treat objects of essentially arbitrary complexity. In order to preserve the surface triangulation of solid objects, it is necessary to override the Delaunay property and redefine the triangulation when points are introduced that are close to solid boundaries. Details of .this constrained Delaunay algorithm are presented and an efficient implementation of the triangulation method is described. Techniques for controlling the distribution of mesh points and tetrahedron quality are also discussed.

SUMMARYA two-dimensional numerical model of microstructural e ects in brittle fracture is presented, with an aim towards the understanding of toughening mechanisms in polycrystalline materials such as ceramics. Quasi-static crack propagation is modelled using the extended ÿnite element method (X-FEM) and microstructures are simulated within the framework of the Potts model for grain growth. In the X-FEM, a discontinuous function and the two-dimensional asymptotic crack-tip displacement ÿelds are added to the ÿnite element approximation to account for the crack using the notion of partition of unity. This enables the domain to be modelled by ÿnite elements with no explicit meshing of the crack surfaces. Hence, crack propagation can be simulated without any user-intervention or the need to remesh as the crack advances. The microstructural calculations are carried out on a regular lattice using a kinetic Monte Carlo algorithm for grain growth. We present a novel constrained Delaunay triangulation algorithm with grain boundary smoothing to create a ÿnite element mesh of the microstructure. The fracture properties of the microstructure are characterized by assuming that the critical fracture energy of the grain boundary (G gb c ) is di erent from that of the grain interior (G i c ). Numerical crack propagation simulations for varying toughness ratios G gb c =G i c are presented, to study the transition from the intergranular to the transgranular mode of crack growth. This study has demonstrated the capability of modelling crack propagation through a material microstructure within a ÿnite element framework, which opens-up exciting possibilities for the fracture analysis of functionally graded material systems.