In the field of practical rock engineering, there are two independent computations: continuous computation and limit equilibrium computation. Limit equilibrium is still the fundamental method for global stability analysis. For any numerical method, reaching limit equilibrium requires large displacements, discontinuous contacts, precise friction law, multi-step computation and stabilized time-step dynamic computation. Therefore three convergences are unavoidable: convergence of equilibrium equations, convergence of open-close iterations for contacts and convergence of the contact forces of dynamic computations. This paper focuses mainly on applications of two or three dimensional discontinuous deformation analysis (DDA) and two dimensional manifold method (MM). The applications show DDA and MM have the ability to reach limit equilibrium of block systems. This paper presents dam foundation damage computation, where the block sliding is a main issue. This paper also presents underground chamber rock stability and underground chamber bolting computations.
1. DISCONTINUOUS DEFORMATION ANALYSIS (DDA)
DDA works on block systems. Each block has linear displacements and constant stresses and strains. The current version of 2d-DDA bas 6 unknowns per block: x direction movement, y direction movement, rotation, x direction strain, y direction strain, shear strain. DDA uses multi-time steps. Both static and dynamic cases use dynamic computation. Static computation is the stabilized dynamic computation in the natural way. Therefore DDA can perform discontinuous and large deformation computation for both static and dynamic cases. For each time steps, DDA usually has several open-close iterations. DDA readjust open, close or sliding modes until every contact position has the same contact mode before and after the equation solving then going to next time step. Here, for each open-close iteration of each time step, DDA solves global equilibrium equations. The friction law is ensured in DDA computation. The friction law is the principle law of stability which is inequality equations in mathematics. Every single block of 2-d DDA can be a generally shaped convex or concave two dimensional polygon. Each block can have any number of edges. Based on simplex integration, the stiffness matrices, the inertia matrices and all other matrices of DDA are analytical solutions. DDA has complete linear contact modes. If the time step is small enough and the total step number is large enough, DDA can simulate any possible complex movements of simply deformable block systems. DDA serves as a bridge between FEM and limit equilibrium method. DDA has strict equilibrium at each time step. After some time steps, the DDA reaches dynamic or static limit equilibrium for whole simply deformable block systems. DDA also served as implicit version of DEM method. DDA has all advantages of dynamic relaxation yet the convergence is strict and the result is accurate. More important, DDA is a very well examined method by analytical solutions, physical model tests and large engineering projects.
2. MANIFOLD METHOD (MM)
The "manifold" in this paper is a generalization of the "manifold" of Mathematics. Based on finite cover systems, the newly developed "manifold method" has the potential to meet more engineering requirements. The term "manifold" here comes from the topological manifold and differential manifold, which is the main subject of differential geometry, algebraic topology, differential topology and modern algebra of mathematics.