Keywords: Delaunay, grid, rock mechanics, discrete elements, boundary elements, numerical simulations, fracture.

Fracturing processes in rock are often a combination of new cracks in intact material and of the activation of pre-existing discontinuities (e.g. a cemented fracture) or defects (e.g. micro-fissures). In order to model explicitly the opening and sliding of cracks, a discontinuous model is required. This implies that grid elements have to be allowed to open and to undergo a shear displacement.

In the study related to this paper, a predefined grid of possible fracture paths is used. To be able to simulate a fracture path in a realistic way, one needs a dense grid of elements in which each element becomes a crack (`activated') if its associated failure criterion is exceeded [1,2]. In principle, a (linear) element in a mesh applied in a numerical simulation can represent different types and sizes of cracks, depending, among others, on the corresponding scale. The behaviour of the (background) material is governed by a network of elements in which the pre-existing discontinuities have to be included.

A grid generator is required that allows a good control over element properties as well as the possibility of integrating pre-existing structures (e.g discontinuities, defects). Instead of adapting an existing grid generator, it was preferred to develop a point generating program on which triangulation is performed using Matlab.

The program starts by translating the coordinates of the predefined structure and the outer boundary into a series of structural points. Next, the sample is divided in multiple concentric annuli. Subsequently, each annulus is divided in different sectors of a similar area. In each sector a field point is generated at random, or to the user's preference. Minimal element length controls are built in, in order to avoid areas with a too high grid density (i.e. clusters). The program allows a good overview and control over the different geometrical parameters that influence the resulting grid (i.e. different distance parameters). Adaptations can be made to control the grid generation if required. This can be for example a given element length distribution, based on data from the grain size distribution. Next, Delaunay triangulation is performed using Matlab on the collection of field points and structural points. The triangulation algorithm is based on the Quickhull algorithm for constructing convex hulls [3].

Finally, the grid data is adapted in order to suit the numerical code to be used. Each grid element is classified as 'intact' or 'structural'. The latter includes the outer boundary of the structure and additional features that are modelled (e.g. fractures, layering, grain clusters). This allows the numerical code to recognize different types of elements that may be assigned different strength parameters.

Three examples of grids generated by the grid generator are discussed: a coal sample with a cleat system, a sample with one central discontinuity and a borehole in layered shale. For the first two examples element size distribution and mesh quality parameters (minimum angle per triangle, triangle aspect ratio and triangle quality factor) are presented. The mesh quality parameters are very similar for both models. This is logical, given a similar distribution of field points, and given that the triangulation algorithm remains unchanged. The (variable) structural points have only a limited influence on the triangle parameters. The grids are well-structured and show no extreme deformations (e.g. clustering, skinny triangles). The examples discussed have a circular form, but the program could easily be changed for other forms.

The generated grids are intended for the simulation of fracture growth in discrete element and displacement discontinuity boundary element models as applied in reference [4].

1

J.A.L Napier, T. Dede, "A comparison between random mesh schemes and explicit growth rules for rock fracture simulation", International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts, 34, 356, 1997. doi:10.1016/S1365-1609(97)00257-8

2

B. Van de Steen, A. Vervoort, J.A.L. Napier, "Numerical modelling of fracture initiation and propagation in biaxial tests on rock samples", International Journal of Fracture, 108, 165-191, 2001. doi:10.1023/A:1007697120530

3

C.B. Barber, D.P Dobkin, H. Huhdanpaa, "The Quickhull Algorithm for Convex Hulls", ACM Transactions on Mathematical Software, 22, 4, 469-483, 1996. doi:10.1145/235815.235821

4

B. Debecker, A. Vervoort, J.A.L. Napier, "Fracturing in and around a natural discontinuity in rock: a comparison between boundary discontinuity and discrete element models" in B.H.V. Topping, G. Montero and R. Montenegro, (Editors), "Proceedings of the Fifth International Conference on Engineering Computational Technology", Civil-Comp Press, Stirling, UK, 2006. doi:10.4203/ccp.84.168

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