International Journal of Computational Methods

This paper presents a review on the numerical manifold method (NMM), which covers the basic theories of the NMM, such as NMM components, NMM displacement approximation, formulations of the discrete system of equations, integration scheme, imposition of the boundary conditions, treatment of contact problems involved in the NMM, and also the recent developments and applications of the NMM. Modeling the strong discontinuities within the framework of the NMM is specially emphasized. Several examples demonstrating the capability of the NMM in modeling discrete block system, strong discontinuities, as well as weak discontinuities are given. The similarities and distinctions of the NMM with various other numerical methods such as the finite element method (FEM), the extended finite element method (XFEM), the generalized finite element method (GFEM), the discontinuous deformation analysis (DDA), and the distinct element method (DEM) are investigated. Further developments on the NMM are suggested.

In order to deal with the kinematic and dynamic boundary conditions in the Eulerian framework, we develop an Eulerian finite cover method (FCM) for large deformation solid dynamics by incorporating the approximation strategy of the FCM into the existing Eulerian explicit finite element method. The operator split method is employed to solve Eulerian solid dynamics problems, and the resulting numerical algorithm consists of two steps. One is the nonadvective step, in which the standard Lagrangian FC analysis is carried out with the explicit time integration scheme, and the other is the advective step, in which the CIVA method is applied to project the solution obtained in the nonadvective step to the Eulerian mesh. Two representative numerical examples are presented to validate the proposed Eulerian FCM and demonstrate its capability especially in appropriately treating the kinematic and dynamic boundary conditions.

Based on the complex variable moving least-squares (CVMLS) approximation and the finite cover theory, the complex variable meshless manifold method (CVMMM) for fracture problems is presented in this paper. The CVMMM employs two cover systems which are the mathematical cover system and the physical cover system. The shape function in the CVMMM is derived with the CVMLS approximation and the finite cover theory. The finite cover theory is used to model cracks which lead to interior discontinuous displacements. At the tip of a crack of a problem, we use the analytical solution near the tip of a crack to extend the trial function of the CVMMM, then the corresponding approximation function is obtained. From the minimum potential energy principle, the corresponding formulae of the CVMMM for fracture problems are presented. Some numerical examples are presented to demonstrate the method in this paper.

This paper presents an overview of the numerical manifold method (NMM) as a unifying framework for modeling fracturing phenomena in quasi-brittle materials, and particularly in concrete. NMM is attractive due to its potential to provide an efficient framework for modeling the entire transition between continuum to discontinuum, from a continuum point of view, without remeshing. To achieve this, the following characteristics are employed: (1) discontinuities are introduced in a discrete manner, without remeshing; (2) the approximation is improved locally, for any arbitrary level, without remeshing. Furthermore, essential boundary conditions are enforced using projection matrices and higher-order boundary issues are identified.

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 NMM to the three-dimensional domain based on the 2D fundamentals. The general framework of the 3D NMM is introduced, including the cover geometry patterns (GP) with division structure from hexahedron to tetrahedron, and general formulations based on the virtual work principle. The block cutting process to generate discrete blocks are discussed through the topological structure description of blocks, and the operation rule of blocks is explained. The proposed 3D block generation algorithm allows for any arbitrary discrete structure or block system. Three numerical examples are presented to demonstrate that the developed 3D numerical manifold code is effective and applicable to 3D continuum solids. Further developments aim to incorporate contact models to simulate complicated discrete block system.

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.

A numerical manifold method (NMM) for plane micropolar elasticity is proposed in this paper to consider the microstructure influence of materials. First, the basic formulations of plane micropolar elasticity are briefly introduced. Then, the equations of stiffness matrix, boundary conditions, and equivalent force in micropolar NMM are derived based on energy minimization principle. Two manifold elements for micropolar elasticity are provided later. From geometry analysis, the proposed SQ-ME model will have about 40% less degree of freedoms than the corresponding finite element method model. Patch tests are used to validate these two manifold elements and the results show these elements could pass patch tests exactly. Furthermore, the micropolar NMM is used to predict the stress concentration of a micropolar membrane containing a small hole. The results are in agreement with the micropolar analytical solution. Finally, convergence analysis is performed and conclusions are derived.

As one type of rock slope failures, topping failure can be accurately simulated only when several aspects are correctly calculated such as deformation and stress, contacts between blocks, contact stress, movement of blocks, open/close of contacts between blocks, development of failure plane, and crack generation and propagation. Current numerical methods encounter many difficulties in simulating toppling failure, especially for rock slope with lots of rock-bridges. Numerical manifold method (NMM) can deal with these highly discontinuous problems and be used to model the toppling failure of rock slopes. This paper first introduces the fundamental principles, modeling of contacts, calculation of contact force and stress, and modeling of failure in NMM. Then, several case studies are conducted to testify the accuracy and convergence of method; comparisons with method, based on limit equilibrium principle, which was proposed by Goodman and Bray (G–B method) and centrifuge test are conducted. Finally, the topping failure of left bank of one high dam is simulated. Results show that the NMM can be used to correctly calculate the toppling safety factor, simulate the failure process of slope toppling, and accurately model the whole failure process of rock slopes with many rock-bridges.

This paper presents a general formulation for simulating the singular stress field at the vicinity of the crack-tip for linear fracture mechanics problems, based on the edge-based smoothed finite element method (ES-FEM) settings. This novel "singular ES-FEM" makes use of the unique feature offered by the ES-FEM that only the assumed displacement values (not the derivatives) are required to compute the stiffness matrix of the discretized system. The present singular ES-FEM method uses a basic mesh of linear triangular elements and a layer of novel "five-noded crack-tip elements" sharing the crack-tip node. The five-noded crack-tip element has one additional node on each of the edges connected to the crack-tip, and the locations of the "edge-node" can be arbitrary. A number of examples are analyzed and the results demonstrate that the present singular ES-FEM is generally softer and much more accurate than the existing FEM. The stress intensity factors obtained using the singular ES-FEM are very stable for different area-integration paths designed around the crack-tip. The present singular ES-FEM is found an excellent alternative to the standard FEM for fracture problems.