Development of Coupled Discontinuous Deformation Analysis and Numerical Manifold Method (Nmm–dda)

Abstract

Discontinuous Deformation Analysis (DDA) and Numerical Manifold Method (NMM) have been widely used for the analyses of discontinuous rock masses. Recently, these discontinuum-based numerical methods have been applied to the simulations for slope failure due to earthquakes, where one of the key issues is the estimation of traveling velocities and distances for the collapsed rock blocks. For the dynamic response analysis of rock slopes, it is necessary to consider the local variation of seismic forces, especially when the slope size is large and/or the slope geometry becomes complicated. In DDA, there is difficulty to consider the local displacements and stress condition of the single block for the basement because of mathematical principle (in DDA, the displacement function is defined at the gravity center of the blocks and the strain in the block is uniform). On the other hand, NMM can simulate both continuous and discontinuous deformation of the block systems. However, the rigid body rotation of blocks cannot be treated properly because NMM does not deal with the rigid body rotation in explicit form. According to the above-mentioned features and drawbacks, it is reasonable to combine DDA and NMM from practical point of view. In this paper, the formulation for the coupled NMM and DDA (NMM–DDA) was presented. For the formulation, NMM and DDA can be easily combined by choosing displacements of the DDA blocks and NMM cover nodes as unknowns, because the processes to establish the equilibrium equations (minimizing total potential energy) and kinematics for block system are same between DDA and NMM. In this paper, some applications of the NMM–DDA to both dynamic and static problems were also presented and the validity and applicability of newly developed DDA–MM were discussed.

Modeling rock failure using the numerical manifold method followed by the discontinuous deformation analysis

“…The problem of material domain area change thus will bring about unreliable numerical results. With linear displacement function, an unnatural deformation phenomenon due to rotation in the NMM modelling was also reported by Miki et al [18], as illustrated in Fig. 5b, which can lead to extremely considerable errors to the modelling results due to its influence on the contact identification and treatment across discontinuous surfaces.…”

“…However, this phenomenon does not appear in our modelling, which may be related to the mesh adopted. Miki et al [18] used a FEM manner triangular mesh, and a structured triangular mesh is adopted in this study.…”

“…In the NMM, the above material area change problem and the unnatural deformation problem can not be solved easily because the rigid body rotation is not represented in an explicit form [18]. One possible approach to avoid these problems is to select a sufficiently small step time.…”

“…This method, however, does not solve the problem essentially and restricts the computational efficiency. The rotation-induced area change problem in the NMM can not be easily solved because the rigid body rotation is not represented in an explicit form [18]. Comparatively, in the DDA, which models the behaviour of discrete block systems and can be regarded as a special case of the NMM [14], however, the block rigid body rotation is represented explicitly and the block area change in rotation can be removed conveniently by various methods [19][20][21][22].…”

Numerical study on the bearing capacity of masonry platform with different number of layers

“…The details about discretization and matrices can be seen in the references (Shi and Goodman, 1988;Shi, 1991;Miki et al, 2010). Using Newmark's β method, the acceleration term in Eq.…”