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In rock engineering, block theory is a fundamental theory that aims to analyze the finiteness, removability, and mechanical stability of convex blocks under different engineering conditions. In practice, the possible combinations of the fractures and joint sets that may generate key blocks can be identified by stereographic projection graphs of block theory. However, classic key block theory does not provide solutions for nonconvex blocks, which are very common in civil projects, such as those with underground edges, corners, and portals. To enhance the availability of block theory, a general algorithm that can analyze the removability and stability of blocks of arbitrary shapes is proposed in this paper. In the proposed algorithm, the joint pyramid for blocks of arbitrary shapes can be computed, and the faces of the blocks are grouped according to their normal vectors such that parallel or nonadjacent sliding faces with the same normal vector can be immediately identified when the sliding mode is determined. With this algorithm, blocks of arbitrary shapes can be analyzed, and users do not need to have experience interpreting graphs of block theory to take advantage of its accuracy and effectiveness. The proposed algorithm was verified by several benchmarking examples, and it was further applied to investigate the stability of the left bank rock slope of a dam. The results showed that the proposed algorithm is correct, effective, and feasible for use in the design and support of excavation in complex rock masses.

Due to the complexity of geometry and the difficulty of mesh discretization of 3D (three-dimensional) blocks cut by complexly distributed fractures, explicitly considering arbitrary fracture network in fractured porous medium (FPM) flow analysis is very challenging for various numerical methods. In this study, we developed a FPM flow model by taking full advantage of numerical manifold method (NMM) with independent covers. With the independent covers, arbitrarily-shaped 3D blocks identified by block-cutting analysis can be directly used as basic computational elements. Along the boundaries of the divided blocks, fractures elements are generated according to the fractures’ apertures. Therefore, it is able to handle very complicated fracture network in 3D flow analysis without need to subdivide 3D blocks into computational meshes. In order to refine the meshes, we introduced artificial fractures with same material properties as surrounding rock into a fracture network, without need to coordinate with the shapes of the blocks. We demonstrated our new model on different 2D examples. At last, we applied our model to 2D and 3D examples with complexly distributed fractures, and achieved reasonable results. The results show that our model is very powerful to analyze fluid flow in arbitrarily and complexly fractured rock mass in 3D.

Geological solid modeling has been widely studied and applied in geological analysis relevant to rock engineering, mining, geophysics, geo-hazard assessment, and geographic information systems (GIS). The discontinuities and other faces that may be planes or surfaces need to be fully considered in an attempt to yield solids formed by the mutual intersection and cutting between faces in practical geological modeling. As one kind of geological solid modeling approach, 3D block cutting analysis can simultaneously identify tens of thousands of solids (blocks) from a complicated 3D discontinuity network without user intervention. However, this analysis can only address planes and build flat blocks. In this paper, we propose a method of 3D curved block cutting analysis. The corresponding algorithm is presented completely, with some key processes of analysis thoroughly accounted for, such as solving the surface-to-surface intersection, analyzing the composition of vertices, and analyzing relevant loops. Our proposed method fully exploits the advantages of topology and computational geometry. The 3D curved blocks can be accurately constructed and represented with minor computation and memory. Analysis examples showed that several hundreds of curved blocks can be automatically constructed within one minute using a microcomputer. This method is of significance for the development of geological solid modeling and can provide a reference for related studies.

Summary Finite element generation of complicated fracture networks is the core issue and source of technical difficulty in three-dimensional (3-D) discrete fracture network (DFN) flow models. Due to the randomness and uncertainty in the configuration of a DFN, the intersection lines (traces) are arbitrarily distributed in each face (fracture and other surfaces). Hence, subdivision of the fractures is an issue relating to subdivision of two-dimensional (2-D) domains with arbitrarily-distributed constraints. When the DFN configuration is very complicated, the well-known approaches (e.g. Voronoi Delaunay-based methods and advancing-front techniques) cannot operate properly. This paper proposes an algorithm to implement end-to-end connection between traces to subdivide 2-D domains into closed loops. The compositions of the vertices in the common edges between adjacent loops (which may belong to a single fracture or two connected fractures) are thus ensured to be topologically identical. The paper then proposes an approach for triangulating arbitrary loops which does not add any nodes to ensure consistency of the meshes at the common edges. In addition, several techniques relating to tolerance control and improving code robustness are discussed. Finally, the equivalent permeability of the rock mass is calculated for some very complicated DFNs (the DFN may contain 1272 fractures, 633 connected fractures, and 16,270 closed loops). The results are compared with other approaches to demonstrate the veracity and efficiency of the approach proposed in this paper.

Summary It is necessary to consider a great number of arbitrarily developed fractures in applications involving realistic 3-d fracture network flow models of rock masses. In order to model the fluid flow in a complicated arrangement of discrete fracture networks (DFNs), two core issues have to be solved. Firstly, how does one identify the connection relationships between fractures in the 3-d arbitrary fracture network? Secondly, how can one calculate numerically the fluid flow in arbitrarily-shaped 2-d domains? This paper first proposes that the boundaries of all enclosed blocks form flow pathways. All enclosed blocks can be identified using a 3-d block cutting method. The boundaries of the blocks are composed of loops which are formed by intersecting lines between all faces (fractures and other surfaces). Therefore, the connection relationships between fractures can be determined according to the linkages of the loops. Accordingly, the linkages of the finite element nodes between different faces can also be determined. On the other hand, the fluid flow occurring in the fractures can be treated as a 2-d continuous flow within the arbitrarily shaped loops in the fracture’s local coordinates. We propose that these loops can be meshed into triangles using a method of simplex triangulation of the arbitrarily shaped domain. Then, according to the linkages between nodes, the global conductivity matrix can be assembled and the solution of the equations governing flow can be derived. Three cases are used to validate the model. Finally, an analysis is made of a practical engineering example (with 8914 lines of intersection and 4188 loops) which shows that the proposed flow model is practicable and can deal with complicated DFNs.

Computer-assisted instruction (CAI) is one modern information technology of the core content and important techniques that applying in education. This article aims at basketball theory teaching and brings CAI technology into the practice courses. This is in favor of improving teaching approach and system. Moreover, it can develop students’ learning initialization, intensify learning pertinence, increase teachers’ teaching level, and accelerate sports knowledge upgrade.