Numerical manifold method (NMM) has been proposed to simulate discontinuous rock mass for many years. Due to the advantages of the algorithm in processing discontinuous problems, this method has also been applied to model fracture process for both continuous and discontinuous solids. NMM still has great potential for further development, especially on high-order manifold cover functions, fluid-structure interactions, multiple crack generation and propagation, three-dimensional manifold methods, etc. This paper presents three-dimensional extension of the numerical manifold method by developing 3-D mathematical and physical covers for blocky rock mass system. Three-dimensional blocky rocks are considered as physical covers, while the mathematical covers are arbitrarily constructed by using regular brick elements as the traditional finite element method. Cutting random shape of rock mass is a key issue in the 3-D manifold method too. A few case studies show that the developed three-dimensional cutting program can simulate more approximately complicated rock mass system, while the generation of the physical and mathematical covers makes it very convenient to numerically simulate 3-D discontinuous blocky rock system. Based on advantages of the existent 2-D simplex integration formulas, a scheme for 3-D version has been put forward.

Numerical Manifold Method
Finite cover system

The numerical manifold method (NMM) adopts a two-mesh problem description, with the two meshes referred as the physical mesh and the mathematical mesh. The physical mesh includes boundaries, discontinuities, and interfaces of different material zones. It represents the material condition which cannot be chosen arbitrarily. The physical mesh limits the integration zones. The mathematical mesh defines the fine or rough approximation of unknown functions. This mesh is chosen by users, and it could be a mesh of some regular pattern or a combination of some arbitrary figures. The mathematical mesh does not need to conform to the boundaries and discontinuities of the problem domain. However, it should be large enough to include the whole problem domain inside. The mathematical mesh is used to build mathematical covers that form the small regions of the whole field. The intersection of the mathematical cover and the physical mesh defines the region of physical covers. A common area of the overlapped physical covers corresponds to a manifold element. Separating all the manifold elements produces a complete cover of the whole field without overlapping. The above concept of the finite cover system is illustrated by the example shown in Figure 1 [1]. Figure 1(a) depicts the triangular domain with a discontinuity inside, where the discontinuity and the boundaries are referred as the physical mesh in the NMM. The rectangle and the circle are arbitrarily chosen as the mathematical mesh, which forms two mathematical covers and as shown in Figure 1(b).

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