A new rock slicing method based on linear programming

One of the important pre-processing stages in the analysis of jointed rock masses is the identification of rock blocks from discontinuities in the field. In 3D, the identification of polyhedral blocks usually involve tedious housekeeping algorithms, because one needs to establish their vertices, edges and faces, together with a hierarchical data structure: edges by pairs of vertices, faces by bounding edges, polyhedron by bounding faces. In this paper, we present a novel rock slicing method, based on the subdivision approach and linear programming optimisation, which requires only a single level of data structure rather than the current 2 or 3 levels presented in the literature. This method exploits the novel mathematical framework for contact detection introduced in Boon et al. (2012). In the proposed method, it is not necessary to calculate the intersections between a discontinuity and the block faces, because information on the block vertices and edges is not needed. The use of a simpler data structure presents obvious advantages in terms of code development, robustness and ease of maintenance. Non-persistent joints are also introduced in a novel way within the framework of linear programming. Advantages and disadvantages of the proposed modelling of non-persistent joints are discussed in this paper. Concave blocks are generated using established methods in the sequential subdivision approach, i.e. through fictitious joints

Geometric Rounding and Feature Separation in Meshes

Geometric rounding of a mesh is the task of approximating its vertex coordinates by floating point numbers while preserving mesh structure. Geometric rounding allows algorithms of computational geometry to interface with numerical algorithms. We present a practical geometric rounding algorithm for 3D triangle meshes that preserves the topology of the mesh. The basis of the algorithm is a novel strategy: 1) modify the mesh to achieve a feature separation that prevents topology changes when the coordinates change by the rounding unit; and 2) round each vertex coordinate to the closest floating point number. Feature separation is also useful on its own, for example for satisfying minimum separation rules in CAD models. We demonstrate a robust, accurate implementation

Inner and Outer Rounding of Boolean Operations on Lattice Polygonal Regions

Robustness problems due to the substitution of the exact computation on real numbers by the rounded floating point arithmetic are often an obstacle to obtain practical implementation of geometric algorithms. If the adoption of the --exact computation paradigm--[Yap et Dube] gives a satisfactory solution to this kind of problems for purely combinatorial algorithms, this solution does not allow to solve in practice the case of algorithms that cascade the construction of new geometric objects. In this report, we consider the problem of rounding the intersection of two polygonal regions onto the integer lattice with inclusion properties. Namely, given two polygonal regions A and B having their vertices on the integer lattice, the inner and outer rounding modes construct two polygonal regions with integer vertices which respectively is included and contains the true intersection. We also prove interesting results on the Hausdorff distance, the size and the convexity of these polygonal regions

Mesh generation by domain bisection

The research reported in this dissertation was undertaken to investigate efficient computational methods of automatically generating three dimensional unstructured tetrahedral meshes. The work on two dimensional triangular unstructured grid generation by Lewis and Robinson [LeR76] is first examined, in which a recursive bisection technique of computational order nlog(n) was implemented. This technique is then extended to incorporate new methods of geometry input and the automatic handling of multiconnected regions. The method of two dimensional recursive mesh bisection is then further modified to incorporate an improved strategy for the selection of bisections. This enables an automatic nodal placement technique to be implemented in conjunction with the grid generator. The dissertation then investigates methods of generating triangular grids over parametric surfaces. This includes a new definition of surface Delaunay triangulation with the extension of grid improvement techniques to surfaces. Based on the assumption that all surface grids of objects form polyhedral domains, a three dimensional mesh generation technique is derived. This technique is a hybrid of recursive domain bisection coupled with a min-max heuristic triangulation algorithm. This is done to achieve a computationlly efficient and reliable algorithm coupled with a fast nodal placement technique. The algorithm generates three dimensional unstructured tetrahedral grids over polyhedral domains with multi-connected regions in an average computational order of less than nlog(n)

Embedding Stacked Polytopes on a Polynomial-Size Grid

A stacking operation adds a $d$-simplex on top of a facet of a simplicial $d$-polytope while maintaining the convexity of the polytope. A stacked $d$-polytope is a polytope that is obtained from a $d$-simplex and a series of stacking operations. We show that for a fixed $d$ every stacked $d$-polytope with $n$ vertices can be realized with nonnegative integer coordinates. The coordinates are bounded by $O(n^{2\log(2d)})$, except for one axis, where the coordinates are bounded by $O(n^{3\log(2d)})$. The described realization can be computed with an easy algorithm. The realization of the polytopes is obtained with a lifting technique which produces an embedding on a large grid. We establish a rounding scheme that places the vertices on a sparser grid, while maintaining the convexity of the embedding.Comment: 22 pages, 10 Figure