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The numerical manifold method (NMM) has been used to solve geotechnical problems with valuable results. In this paper, a numerical model based on the NMM is proposed to investigate two primary concerns in slope stability analyses, i.e., critical slip surface (CSS) determination and safety factor (Fs) calculation. In our proposed numerical model, the optimization algorithm of MAX-MIN ant colony (MMACOA), which is considered to be an algorithm with the best performance for such optimization problems, will be used to determine CSS. In the process of determining the CSS, the trial safety factor for a trial slip surface is calculated with the vector sum method (VSM) according to the stress field produced by an NMM analysis. Accuracy of the numerical model is validated with three typical examples. The numerical model has great potential in the evaluation of stability of real slopes.

In the present paper, the digital image processing (DIP) technology is employed to establish the structure model of a real SRM (soil-rock-mixture) slope. Based on the structure model, the most recently developed strength reduction numerical manifold method (SRNMM) is adopted to investigate the stability of the SRM slope. Based on the proposed numerical model, the testing results show that: 1) the DIP technology can accurately build the structure model of the SRM slope; 2) the SRNMM can obtain the slopes’ FOSs (factors of safety) with satisfactory accuracy, as well as obtain the main failure mode of the slopes; 3) the presence of rock blocks will improve the stability of the soil slopes.

A high-order(HO) three dimensional numerical manifold method (HO3DNMM) is proposed. In the present numerical model, HO global displacement function can be built without using extra nodal DOFs/nodes. Additionally, continuous stresses/strains among inter-element faces can be obtained with the HO3DNMM without using any stress smoothing technique. A number of numerical examples have been adopted to investigate the properties of the proposed HO3DNMM. Numerical results assessed from the HO3DNMM indicate that 1) HO3DNMM doesn't suffer from the LD (linear dependence) issue; 2) For discretized model with regular meshes, accuracy of the HO3DNMM is much better than that of the T3DNMM (traditional 3DNMM); 3) HO3DNMM is immune from the mesh distortion, which is a very attractive property as compared to the T3DNMM.

An improved NMM (Numerical Manifold Method) with mathematical cover refinement (RNMM) is adopted to investigate the stability of a SRM (soil-rock-mixture) slope. In the RNMM, refined mathematical meshes are deployed at specified domains where stress concentration may occur. Compared to the TNMM (Traditional NMM), the computational efficiency of the RNMM is better for slope stability problems. Besides, the recently proposed scheme called MLC (mathematical cover systems with multiple layers) is adopted to further improve the computational efficiency of the RNMM. Based on the proposed RNMM+MLC (RNMM with MLC), stability analyses about two typical examples including a soil slope and a SRM slope are investigated. The numerical results demonstrated that: 1) The FOS (factor of safety) based on the proposed RNMM is better than that based on the TNMM; 2) The FOS based on the RNMM + MLC is almost the same to that based on the RNMM, but the computational efficiency of the RNMM + MLC is better than that of the RNMM; 3) The distribution mode of plastic zones in a SRM slope differs a lot from a soil slope.

A mixed three-node triangular element with continuous nodal stresses (mixed T3-MINI-CNS) is presented for dynamics of porous media based on the three-variable u-w-p Biot model and the numerical manifold method (NMM). The displacement and velocity approximations are derived using the constrained and orthonormalized least-square (CO-LS) scheme, which possess continuous derivatives and Delta property at nodes. The pressure approximation takes linear piecewise interpolation. Relative to higher-order element, the mixed quadratic six-node triangle, mixed T3-MINI-CNS can model dynamic response of porous media more precisely with fewer unknowns, especially the short-term transient response. Moreover, mixed T3-MINI-CNS is immune from locking in both undrained and rigid skeleton limits and achieves more accurate pressure results than other characteristic locking-free elements. Based on NMM and u-w-p formulation, the most versatile three-node triangular mesh can always be used, avoiding difficulties in mesh generation. As fluid acceleration is involved, mixed T3-MINI-CNS is capable of totally predicting dynamic response of porous mixture, especially under rapid loading condition. In addition, reliability and precision of the time integration are assessed in terms of the energy balance condition. Through calculating benchmark problems, convergence, accuracy, and reliability of the mixed T3-MINI-CNS are thoroughly investigated and validated.

By using the MC system (mathematical cover system) and the PC system (physical cover system), the NMM (numerical manifold method) is able to cope with continuum and discontinuum problems in the same framework. In this study, a HM (hydro-mechanical) coupling model which is based on the “cubic law” is incorporated into the NNM for the purpose of modeling multiple crack propagation problems driven by water flow. A series of numerical examples have been conducted to evaluate the performance of the proposed numerical model. The results show that the proposed numerical model can not only simulate the processes of multiple crack propagation driven by fluid flow, but also can carry out hydraulic analysis of fissured rock networks with high accuracy.

Recently, fruitful results of geotechnical problems solved by the numerical manifold method (NMM) have been gained. In the present work, the shear strength reduction technique (SSRT) together with the ϕ–υ inequality is implemented into the NMM to carry out stability analysis of soil-rock-mixture (SRM) slopes. Several examples are used to validate the correctness and effectiveness of the developed numerical model. The results from these examples indicate that the proposed model can predict the factor of safety (FOS) of slopes with high accuracy, and capture the failure patterns of slopes. In addition, the influence of rock blocks on the stability of SRM slopes can be captured.

Recently, fruitful results of geotechnical problems solved by the NMM (abbreviated form for the numerical manifold method) have been gained. In the present work, a free surface updating strategy is implemented into the NMM to simulate unconfined seepage flow in the soil-rock mixtures (SRMs). Several examples are employed to validate the effectiveness and correctness of the proposed numerical model. Results from the numerical examples indicate that the proposed model can predict the location of free surface with high accuracy, as well as capture the main feature of the unconfined seepage flow. In addition, the influence of rock blocks on the seepage flow in the SRMs can be captured.

In the dynamic analysis with explicit time integration scheme, a lumped mass matrix (LMM) is always preferable than the consistent mass matrix (CMM), since the solving of large scale simultaneous algebraic equations can be avoided if a LMM is employed. In this study, a mass lumping scheme which is suitable for the 10-node tetrahedral element is proposed. A series of numerical examples show that the natural frequencies assessed from the proposed LMM are very close to those from CMM. In addition, the dynamic responses of structure predicted by the proposed LMM are in very good agreement with those from CMM if an implicit time integration scheme is adopted. More importantly, the proposed LMM is less sensitive to mesh distortion than the CMM. The proposed LMM can supersede the CMM at any cases.

Since the advent of finite element methods, the dynamic response analysis of solids and structures follows such a route without exception. Firstly the spatial discretization is carried out and the system of second order ordinary differential equations with the degrees of freedom as the unknown functions of time is derived, which is called the semi-discrete scheme. Then the temporal discretization is performed to the system of ordinary differential equations and the system of algebraic equations, referred to as the fully-discrete scheme, is obtained. This route has been working well for most problems, where, the meshes deform continuously and, in all the time steps, all the degrees of freedom are valid and the number of them keeps invariant. In the simulation of crack propagation, however, even the number of degrees of freedom varies with crack propagation and those degrees of freedom associated with crack tips become meaningless after the crack tips move away. While this causes no difficulties in linear static solutions, it is not readily handled in time-dependent solutions, leading to the transfer issue of degrees of freedom. Opposite to the conventional order of discretization, in this study the temporal discretization is put prior to the spatial discretization. In this way, all the degrees of freedom are valid only within the current time step. The transfer issue of degrees of freedom is accordingly resolved elegantly. The implementation of the proposed procedure is in the framework of the numerical manifold method, illustrated by some typical examples, where compressed and sheared cracks are involved with frictional contact.

In this study, a zero-thickness cohesive element-based Numerical Manifold Method (NMM) combined with detailed micro-scale characterization is proposed for modeling the rock mechanical response and failure process of rock. To represent the rock micro-structure, a Voronoi tessellation technique is adopted to generate the random polygonal grains. Since the contact fracture model not only requires input micro-parameters which are difficult to be obtained directly from laboratory tests, but also becomes very time consuming due to extensive contact searching and judging processes, a zero-thickness cohesive element is inserted between the rock grains to more efficiently and accurately capture the interaction between the rock grains before failure. To more efficiently model the interaction between the rock grains after failure, the original contact searching technique is improved. To validate and illustrate the efficiency of the developed method, a series of numerical tests are performed using the developed method and their results are compared with those obtained from laboratory tests and original NMM predictions. Since the NMM adopts an implicit scheme to solve the problem, a large time step is allowed, which makes it possible to realistically model the static loading condition without need of raising up the loading rate required by other explicit-based methods.

As a partition of unity method (PUM), the numerical manifold method (NMM) is capable of constructing global approximation by simply multiplying PU function with local approximation. In order to enhance accuracy, high order polynomials can be specified as local approximation. This, however, will hinder the engineering application of NMM by its ill conditioning of the global stiffness matrix. In this study, an improved NMM (iNMM) without extra degree of freedoms (DOFs) is developed. Without the extra DOFs, the resulting global stiffness becomes linear independent. In addition, the stresses are continuous at all nodes. Numerical studies show the iNMM's excellent accuracy.

A hybrid ‘FE-Meshfree’ four-node quadrilateral element with continuous nodal stress using radial-polynomial basis functions (Quad4-RPIMcns), was recently proposed for static analysis. The Quad4-RPIMcns element can be considered as a development of the previous partition-of-unity (PU) based ‘FE-Meshfree’ QUAD4 element (Quad4-RPIM) which uses FE shape functions to construct the PU and radial-polynomial basis functions to construct the local approximation (LA), so as to synergize the individual strengths of finite element and meshfree methods. As a result, high order global approximations in Quad4-RPIMcns element could be easily constructed without adding extra nodes and DOFs, thereby achieving high accuracy and convergence rate. In this paper, the element is further applied to conduct free vibration, forced vibration and geometric nonlinear analyses of two-dimensional solids. Several numerical test problems are solved and the performance of the element is compared with that of the three-node triangular element (Trig3) and four-node isoparametric quadrilateral element (Quad4). Numerical results show that Quad4-RPIMcns element has higher tolerance to mesh distortion and gives more accurate solution as compared to Trig3 and Quad4 elements.

By introducing the concept of mathematical cover and physical cover, the numerical manifold method (NMM) is able to solve continuous and discontinuous problems in a unified way. In this paper, the NMM is developed to analyze three dimensional (3D) fracture propagation. The maximum tensile stress criterion is implemented to determine whether the fracture will propagate and the direction of fracture propagation. Three benchmark problems are analyzed to validate the present algorithm and program. The numerical results replicate available experimental results and existing numerical results. The present algorithm and 3D NMM code are promising for 3D fracture propagation. They deserve to be further developed for the analysis of rock mechanic problems in which the initiation and propagation of multiple fractures, tensile and shear fractures, and fracture propagation under compressive loading are taken into account.

The recently published four-node quadrilateral element with continuous nodal stress (Quad4-CNS) is extended to free and forced vibration analyses of two-dimensional solids. The Quad4-CNS element can be regarded as a partition-of-unity (PU) based ‘FE-Meshfree’ element which inherits better accuracy, higher convergence rate, and high tolerance to mesh distortion from the meshfree methods, while preserving the Kronecker-delta property of the finite element method (FEM). Moreover, the Quad4-CNS element is free from the linear dependence problem which otherwise cripples many of the PU based finite elements. Several free and forced vibration problems are solved and the performance of the element is compared with that of the four-node isoparametric quadrilateral element (Quad4) and eight-node isoparametric quadrilateral element (Quad8). The results show that, for regular meshes, the performance of the element is superior to that of Quad4 element, and comparable to that of Quad8 element. For distorted meshes, the present element has better mesh-distortion tolerance than Quad4 and Quad8 elements.

A new ‘FE-Meshfree’ three-node triangular element (Trig3-RPIM) is developed based on the partition of unity (PU) concept. The Trig3-RPIM element employs the shape function of classical three-node triangular element (Trig3) to construct the PU and the radial-polynomial basis function which is free from the possible singularity of the moment matrix to construct the nodal approximation. The Trig3-RPIM element synergizes the individual strengths of finite element method and meshfree method. Moreover, it is free from the linear dependence problem which otherwise cripples many of the PU based finite elements. Several linear, nonlinear and free vibration test problems are solved and the performance of the element is compared with those of the well-known three-node triangular element (Trig3) and four-node iso-parametric quadrilateral element (Quad4). The results show that, for regular meshes, the performance of the element is superior to those of Trig3 and Quad4 elements. For distorted meshes, the present element has better mesh-distortion tolerance than Trig3 and Quad4 elements.

The present work uses mean value coordinates to construct the shape functions of a hybrid ‘FE-Meshfree’ quadrilateral element, which is named as Quad4-MVC. This Quad4-MVC can be regarded as the development of the ‘FE-Meshfree’ quadrilateral element with radial-polynomial point interpolation (Quad4-RPIM). Similar to Quad4-RPIM, Quad4-MVC has Kronecker delta property on the boundaries of computational domain, so essential boundary conditions can be enforced as conveniently as in the finite element method (FEM). The novelty of the present work is to construct nodal approximations using mean value coordinates, instead of radial basis functions which are used in Quad4-RPIM. Compared to the radial basis functions, mean value coordinates does not utilize any uncertain parameters, which enhances stability of numerical results. Numerical tests in this paper show that the performance of Quad4-RPIM becomes even worse than four-node iso-parametric element (Quad4) when the parameters of radial basis functions are not chosen properly. However, the performance of Quad4-MVC is stably better than Quad4.

In the present work, a novel hybrid FE-Meshless quadrilateral element with continuous nodal stress is developed using radial-polynomial basis functions, named as Quad4-RPIMcns. Quad4-RPIMcns can be regarded as the development of the previous FE-Meshless quadrilateral element with radial-polynomial basis functions (Quad4-RPIM) and quadrilateral element with continuous nodal stress (Quad4-CNS). Similar to Quad4-RPIM, radial-polynomial basis functions are used to construct nodal approximations of Quad4-RPIMcns in the context of partition of unity, which avoids the possible singularity problem of constructing nodal approximations. The derivative of Quad4-RPIMcns shape function is continuous at nodes. Therefore, nodal stress can be obtained without any extra operation. Quad4-RPIMcns possesses Kronecker-delta property which is a very important property to impose essential boundary conditions directly as in the FEM. The numerical tests in this paper demonstrate that Quad4-RPIMcns gives better accuracy and higher convergence rate as compared to four-node iso-parametric quadrilateral element (Quad4). Additionally, Quad4-RPIMcns seems to have higher tolerance to mesh distortion than Quad4.