The numerical manifold method (NMM) is a combination of the finite element method (FEM) and discontinuous deformation analysis (DDA) method. It provides a robust numerical solution to a solid medium with dense discontinuities. This paper extends the numerical manifold method to the three dimensional domain from the fundamental concepts. It explains the three essential entities, i.e, the mathematical cover, the physical cover and the manifold element in the 3-D version under the framework of the finite cover system. Furthermore, it tests the robustness and accuracy of the proposed algorithm through a tetrahedral wedge sliding simulation in a rock slope.
Recently, high frequent and prevalent geologic hazards cause catastrophic damage and human death. The increased consciousness regarding safety and economy has led the engineers to seek more rational solutions to the geotechnical problems related to civil and underground engineering. It is more and more urgent to have a proper analysis tool for strengthening the understanding to hazard mechanism and designing efficiency to reduce disaster damage. Following the characteristics of geotechnical applications, the 2-D framework of NMM was proposed in 1991 , which is one of most suitable numerical methods in rock engineering. NMM could be considered as the combination of FEM and DDA. Although the 2-D NMM has already been widely proven by various applications, the real problems in engineering practices are always in the three dimensional domain. It has been a long time challenge to extend the 2-D NMM to 3-D due to the complexity in geometry description and the absence of a reliable 3-D contact algorithm. In this paper, we firstly introduce basic concepts for 3-D NMM. It later utilizes simple tetrahedral wedge sliding in the rock slope to test the accuracy of the proposed algorithm
The numerical manifold method is proposed by Shi  as a combination of the mathematical domain and physical domain. The Target Physical Objects (TPOs) in physical domain includes the boundaries of the material volume, joints, and the interfaces of different material zones. The TPOs represents material conditions which cannot be chosen artificially. The mathematical mesh defines the fine or rough approximation of unknown functions. This mesh is chosen by the user according to the problem geometry, solution accuracy requirement, and the physical property zoning. The mathematical mesh is used to build mathematical covers that represent small regions of the whole field and can be of any shape and size. They can overlap each other and do not need to coincide with the physical cover as long as they are large enough to cover the physical blocks. The 3-D NMM is also based on the three important concepts, i.e, the mathematical cover (MC), the physical cover (PC) and the manifold element (ME). MCs are user-defined overlapping patches. One significant advantage is that arbitrary geometric shapes (e.g. polyhedron in 3-D) can be the basic MC of the mathematical domain. Each different shaped MC has its own mathematical description inside.