An overview of particle-based numerical manifold method and its application to dynamic rock fracturing
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paper_title: A novel meshless numerical method for modeling progressive failure processes of slopes

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Abstract A novel meshless numerical method, known as general particle dynamics (GPD3D), is proposed to simulate the failure process of slopes as well as the failure process of 2D and 3D samples. An elasto-brittle damage model based on the Mohr–Coulomb criterion is applied to investigate the initiation, growth and coalescence of cracks as well as the macro-failure of slopes. The Weibull statistical approach is applied to model the heterogeneity of geomaterials. Five macro-fracture modes of samples subjected to uniaxial compressive loads are modeled due to the heterogeneity of geomaterials. The safety factor and slip surface of the slope are determined by using the GPD3D approach. The numerical results from the GPD approach are in good agreement with those from the limit equilibrium method and FEM. The present numerical method is demonstrated to predict the complex fracture processes of slopes.
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paper_title: A discrete numerical model for granular assemblies

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The distinct element method is a numerical model capable of describing the mechanical behaviour of assemblies of discs and spheres. The method is based on the use of an explicit numerical scheme in which the interaction of the particles is monitored contact by contact and the motion of the particles modelled particle by particle. The main features of the distinct element method are described. The method is validated by comparing force vector plots obtained from the computer program BALL with the corresponding plots obtained from a photoelastic analysis. The photoelastic analysis used for the comparison is the one applied to an assembly of discs by De Josselin de Jong and Verruijt (1969). The force vector diagrams obtained numerically closely resemble those obtained photoelastically. It is concluded from this comparison that the distinct element method and the program BALL are valid tools for research into the behaviour of granular assemblies. La methode des elements distincts est un modele numerique capab...
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paper_title: Considerations of the discontinuous deformation analysis on wave propagation problems

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In rock engineering, the damage criteria of the rock mass under dynamic loads are generally governed by the threshold values of wave amplitudes, such as the peak particle velocity and the peak particle acceleration. Therefore, the prediction of wave attenuation across fractured rock mass is important on assessing the stability and damage of rock mass under dynamic loads. This paper aims to investigate the applications of the discontinuous deformation analysis (DDA) for modeling wave propagation problems in rock mass. Parametric studies are carried out to obtain an insight into the influencing factors on the accuracy of wave propagations, in terms of the block size, the boundary condition and the incident wave frequency. The reflected and transmitted waves from the interface between two materials are also numerically simulated. To study the tensile failure induced by the reflected wave, the spalling phenomena are modeled under various loading frequencies. The numerical results show that the DDA is capable of modeling the wave propagation in jointed rock mass with a good accuracy. Copyright © 2009 John Wiley & Sons, Ltd.
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paper_title: Three-Dimensional Numerical Investigations of the Failure Mechanism of a Rock Disc with a Central or Eccentric Hole

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The diametrical compression of a circular disc (Brazilian test) or cylinder with a small eccentric hole is a simple but important test to determine the tensile strength of rocks. This paper studies the failure mechanism of circular disc with an eccentric hole by a 3D numerical model (RFPA3D). A feature of the code RFPA3D is that it can numerically simulate the evolution of cracks in three-dimensional space, as well as the heterogeneity of the rock mass. First, numerically simulated Brazilian tests are compared with experimental results. Special attention is given to the effect of the thickness to radius ratio on the failure modes and the peak stress of specimens. The effects of the compressive strength to tensile strength ratio (C/T), the loading arc angle (2α), and the homogeneity index (m) are also studied in the numerical simulations. Secondly, the failure process of a rock disc with a central hole is studied. The effects of the ratio of the internal hole radius (r) to the radius of the rock disc (R) on the failure mode and the peak stress are investigated. Thirdly, the influence of the vertical and horizontal eccentricity of an internal hole on the initiation and propagation of cracks inside a specimen are simulated. The effect of the radius of the eccentric hole and the homogeneity index (m) are also investigated.
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paper_title: Numerical Simulation of Granular Flows in a Large Flume Using Discontinuous Deformation Analysis

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A long run-out rock avalanche is defined as an extremely rapid, massive, flow-like motion of fragmented rock derived from a bed-rock failure (Hungr et al. 2001). Due to their extraordinary mobility, rock avalanches have been extremely costly in terms of human lives and of engineering developments, and have interested many researchers. Numerical method appears a promising approach for analyzing most of the physical aspects of dry rock avalanches and for predicting the consequence of real events as well (Cundall and Strack 1979; Gonzales et al. 2002; Tommasi et al. 2008). The distinct element method (DEM) and discontinuous deformation analysis (DDA) are the two most popular discrete element methods used to tackle problems dealing with discontinuous rock mass. Limitations of the explicit DEM include its very small step for numerical stability and the use of artificial damping to absorb the energy generated from the relaxation analysis to maintain equilibrium, although some researchers (Calvetti et al. 2000; Valentino et al. 2008) have devoted to appropriately calibrate numerical damping in simulating run-out of granular flows. Shi (1988) claimed that the energy-based DDA can overcome the two limitations of the force-based DEM. Based on its advantages, this paper selected DDA as the analysis method to investigate the behavior of granular flows. In order to acquire confidence in the numerical simulations, it is vital to corroborate them by direct observation and to prove their efficiency. Due to the difficulty of reproducing the complexity of a rock avalanche on a reduced scale, few researchers tackled the problem through an accurate experimental work on physical smallscale models (Valentino et al. 2008). In this spirit, we performed a series of tests to investigate the effects of some factors influencing mass-front velocity and deposit characteristics of granular flows in a large flume. This paper compares DDA simulations and the experimental results in order to validate the efficiency of this numerical method. Q. Yang (&) Faculty of Geosciences and Environmental Engineering, Southwest Jiaotong University, Chengdu 610031, China e-mail: yangqq_71@126.com
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paper_title: Polygon-based contact description for modeling arbitrary polyhedra in the Discrete Element Method

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Abstract Many real-life applications of the Discrete Element Method (DEM) require a particle description which accounts for irregular and arbitrary shapes. In this work, a novel method is presented for calculating contact force interactions between polyhedral particles. A contact between two polyhedra is decomposed as a set of contacts between individual polygonal facets. For each polygon–polygon contact, an individual contact force is obtained by integrating a linear pressure over the area of its intersection. Both convex as well as partially concave polyhedra can be accurately represented. The proposed algorithm is validated by comparing to previously published experimental and computational gravitational particle depositions of identical cubes. Finally, the model is demonstrated in simulations of gravitational packing of various other polyhedral shapes.
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paper_title: EXPERIMENTAL STUDY OF STRAIN-RATE DEPENDENCE AND PRESSURE DEPENDENCE OF FAILURE PROPERTIES OF GRANITE

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Effects of the strain rate and pressure on various brittle properties of granite under compressional loading are studied experimentally. Granite specimens were tested to failure under various constant strain rates at confining pressures of 0.1 to 200 MPa. The strain rates in these tests varied from 10-4 to 10-8s-1. The results show that the strength of granite decreases linearly as the logarithm of the strain rate decreases, and that the strain-rate dependence on the strength is enhanced at high confining pressures. The dilatant strain and elastic wave velocity variations with stress were found to be independent of the strain rate if the stress is normalized by the strength. The dilatant-strain-versus-stress curve is significantly affected by confining pressure but the confining pressure effect on the dilatant strain versus normalized stress is small. The acoustic emission rate is accelerated at a stress level closer to the failure strength as the strain rate decreases. Some of the timedependent properties are explained by a theory based on the concept of subcritical stress-corrosion cracking. But it is also clear that the stress-corrosion cracking theory does not provide reasonable explanations of the variation of acoustic emission rate with stress and the variation of strain with stress at the stage immediately before fracture.
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paper_title: A 3-D discrete-element method for dry granular flows of ellipsoidal particles

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Abstract We use 3-D discrete element method (DEM) to simulate dry granular flows of non-spherical particles. In our simulation code, non-spherical particles are represented by clusters of spheres for contact detection and for contact-force calculation. For particles in contact, the normal force—displacement (NFD) model is simulated by a partially latching spring system; whereas the tangential force—displacement (TFD) model is simulated by an incremental relation that corresponds to a simplification of the Mindlin and Deresiewicz (R.D. Mindlin, H. Deresiewicz, ASME J. Appl. Mech. 20 (1953) 327–344) contact-mechanics theory. To establish the validity of the simulation procedure, benchmark tests such as superball and superbean simulations were carried out. We present various results obtained from the simulation of ellipsoidal particles flowing down a chute having a bumpy bottom. These ellipsoidal particles had their physical properties obtained from experiments.
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paper_title: A method to model realistic particle shape and inertia in DEM

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A simple and fast original method to create irregular particle shapes for the discrete element method using overlapping spheres is described. The effects of its parameters on the resolution of the particle shape are discussed. Overlapping spheres induce a non-uniform density inside the particle leading to incorrect moments of inertia and therefore rotational behaviour. A simple method to reduce the error in the principal moments of inertia which acts on the individual densities of the spheres is also described. The pertinence of the density correction is illustrated by the case of free falling ballast particles forming a heap on a flat surface. In addition to improve behaviour, the correction reduces also computational time. The model is then used to analyse the interaction between ballast and geogrid by simulating pull-out tests. The pulling force results show that the model apprehends better the ballast geogrid interlocking than models with simple representation of the shape of the particles. It points out the importance of modelling accurately the shape of particles in discrete element simulations.
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paper_title: Numerical investigation of arching mechanism to underground excavation in jointed rock mass

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Abstract In this paper, the discontinuous deformation analysis (DDA) is applied to numerically investigate the formation of pressure arch and the stabilization mechanism of rock bolts for underground excavation in laminated rock mass. A global pressure arch with elevated stress is observed sustaining the weight of the overburden and providing the natural shield over the opening. Below the pressure arch is a loosened zone, where a group of local voussoir beam arches are formed to transfer the weight of each layer to the abutments. Detailed parametric studies regarding the in-situ stress ratio, joint friction angle, overburden depth and dip angle of strata are conducted and their influences on the formation of global pressure arch and local voussoir beam arches are identified and discussed. Based on the observed phenomena, the corresponding reinforcement scheme is proposed and the effectiveness and stabilization mechanism are examined.
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paper_title: UDEC modelling on wave propagation across fractured rock masses

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Reference LMR-ARTICLE-2008-011doi:10.1016/j.compgeo.2007.01.001View record in Web of Science Record created on 2008-12-02, modified on 2016-08-08
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paper_title: MODELING BRITTLE FRACTURE WITH THE NODAL-BASED DISCONTINUOUS DEFORMATION ANALYSIS

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The finite element method (FEM) has been successfully applied to solve continuum-based engineering problems, and the discontinuous deformation analysis (DDA) has emerged as a more attractive model than the continuum-based methods for rock engineering problems due to its intrinsic feature of block discontinuity at the contact boundaries. This paper presents a hybrid approach for two-dimensional problems which takes the advantage of both the FEM and the DDA in solving problems involving continuum and discontinuum domains. The hybrid FEM/DDA method, named as the nodal-based DDA (NDDA), significantly improves not only the accuracy of stress distribution inside the DDA block but also the computational efficiency. A cracking scheme based on failure time is introduced to the NDDA so that it can handle the crack propagation inside blocks. Two validating examples show that the newly developed NDDA can model the brittle fracture well.
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paper_title: Moving least-square reproducing kernel methods ( I ) Methodology and convergence

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This paper formulates the moving least-square interpolation scheme in a framework of the so-called moving least-square reproducing kernel (MLSRK) representation. In this study, the procedure of constructing moving least square interpolation function is facilitated by using the notion of reproducing kernel formulation, which, as a generalization of the early discrete approach, establishes a continuous basis for a partition of unity. This new formulation possesses the quality of simplicity, and it is easy to implement. Moreover, the reproducing kernel formula proposed is not only able to reproduce any mth order polynomial exactly on an irregular particle distribution, but also serves as a projection operator that can approximate any smooth function globally with an optimal accuracy. ::: ::: In this contribution, a generic m-consistency relation has been found, which is the essential property of the MLSRK approximation. An interpolation error estimate is given to assess the convergence rate of the approximation. It is shown that for sufficiently smooth function the interpolant expansion in terms of sampled values will converge to the original function in the Sobolev norms. As a meshless method, the convergence rate is measured by a new control variable—dilation parameter ρ of the window function, instead of the mesh size h as usually done in the finite element analysis. To illustrate the procedure, convergence has been shown for the numerical solution of the second-order elliptic differential equations in a Galerkin procedure invoked with this interpolant. In the numerical example, a two point boundary problem is solved by using the method, and an optimal convergence rate is observed with respect to various norms.
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paper_title: A coupled distinct lattice spring model for rock failure under dynamic loads

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It is necessary to take into account the micro discretization of natural rock when studying its macroscopic failure behavior. This requirement has resulted in renewed and increased interest in the discrete or framework/lattice numerical modeling techniques. However, to fully construct a numerical model for practical applications using a discrete numerical model is computationally difficult with current computing technologies. Hence, a coupled model has been developed to overcome this limitation by coupling the Distinct Lattice Spring Model (DLSM) and the Numerical Manifold Method (NMM). In the coupled model, the microscopic discrete model of the rock is represented by a system of discrete particles interacting via springs while the macroscopic level model is represented by the NMM. The proposed model bears a structure of three layers corresponding to the DLSM model, the NMM model, and a model for coupling, respectively. The coupling model is based on a newly developed Particle based Manifold Method (PMM) to bridge the DLSM with the NMM. The proposed coupled model can reduce the computational resources needed for the purely discrete particle based model. This study introduces theoretical aspects of the coupled model together with a few examples to demonstrate its correctness and feasibility. (C) 2011 Elsevier Ltd. All rights reserved.
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paper_title: Dual-horizon Peridynamics

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Summary ::: In this paper, we develop a dual-horizon peridynamics (DH-PD) formulation that naturally includes varying horizon sizes and completely solves the ‘ghost force’ issue. Therefore, the concept of dual horizon is introduced to consider the unbalanced interactions between the particles with different horizon sizes. The present formulation fulfills both the balances of linear momentum and angular momentum exactly. Neither the ‘partial stress tensor’ nor the ‘slice’ technique is needed to ameliorate the ghost force issue. We will show that the traditional peridynamics can be derived as a special case of the present DH-PD. All three peridynamic formulations, namely, bond-based, ordinary state-based, and non-ordinary state-based peridynamics, can be implemented within the DH-PD framework. Our DH-PD formulation allows for h-adaptivity and can be implemented in any existing peridynamics code with minimal changes. A simple adaptive refinement procedure is proposed, reducing the computational cost. Both two-dimensional and three-dimensional examples including the Kalthoff–Winkler experiment and plate with branching cracks are tested to demonstrate the capability of the method. Copyright © 2016 John Wiley & Sons, Ltd.
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paper_title: A Meshfree Method based on the Local Partition of Unity for Cohesive Cracks

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We will present a meshfree method based on the local partition of unity for cohesive cracks. The cracks are described by a jump in the displacement field for particles whose domain of influence is cut by the crack. Particles with partially cut domain of influence are enriched with branch functions. Crack propagation is governed by the material stability condition. Due to the smoothness and higher order continuity, the method is very accurate which is demonstrated for several quasi static and dynamic crack propagation examples.
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paper_title: Meshless methods: A review and computer implementation aspects

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The aim of this manuscript is to give a practical overview of meshless methods (for solid mechanics) based on global weak forms through a simple and well-structured MATLAB code, to illustrate our discourse. The source code is available for download on our website and should help students and researchers get started with some of the basic meshless methods; it includes intrinsic and extrinsic enrichment, point collocation methods, several boundary condition enforcement schemes and corresponding test cases. Several one and two-dimensional examples in elastostatics are given including weak and strong discontinuities and testing different ways of enforcing essential boundary conditions.
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paper_title: Meshless methods: An overview and recent developments

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Meshless approximations based on moving least-squares, kernels, and partitions of unity are examined. It is shown that the three methods are in most cases identical except for the important fact that partitions of unity enable p-adaptivity to be achieved. Methods for constructing discontinuous approximations and approximations with discontinuous derivatives are also described. Next, several issues in implementation are reviewed: discretization (collocation and Galerkin), quadrature in Galerkin and fast ways of constructing consistent moving least-square approximations. The paper concludes with some sample calculations.
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paper_title: A parallel meshless dynamic cloud method on graphic processing units for unsteady compressible flows past moving boundaries

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This paper presents an effort to implement a recently proposed meshless dynamic cloud method (Hong Wang et al., 2010) on modern high-performance graphic processing units (GPUs) with the compute unified device architecture (CUDA) programming model. Within the framework of the meshless method, clouds of points used as basic computational stencils are distributed in the whole flow domain. The spatial derivatives of the governing equations are discretised by the moving-least square scheme on every cloud of points. Roe’s approximate Riemann solver is adopted to compute the convective flux. A dual-time stepping approach, which iterates in physical and pseudo temporal spaces, is employed to obtain the time-accurate solution. Simulation of steady compressible flows over a fixed aerofoil is firstly carried out to verify the GPU implementation of the method. Then it is extended to compute unsteady flows past oscillatory aerofoils. Numerical outcomes are compared with experimental and/or other reference results to validate the method. Significant performance speedup of more than an order of magnitude is verified by the numerical results. Systematic analysis shows that GPU is more energy efficient than CPU for solving aerodynamic problems. This demonstrates the potential of the proposed method to solve fluid–structure interaction problems.
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paper_title: A peridynamics formulation for quasi-static fracture and contact in rock

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Abstract We present a dual-horizon peridynamics (DH-PD) formulation for fracture in granular and rock-like materials. In contrast to discrete crack methods such as XFEM, DH-PD does not require any representation of the crack surface and criteria to treat complex fracture patterns such as crack branching and coalescence. The crack path is the natural outcome of the simulation. In this manuscript, a new penalty method to model the contact for compressive fractures and constraining the penetration conditions is developed. The new method is applied to several benchmark problems in geomechanics including the four-point shear test, the indirect tensile (Brazilian) test of rock disks with one or multiple initial cracks. By using an appropriate damping coefficient, the quasi-static solution for rock failure is obtained when the dynamic formulation is used. A good agreement is obtained between the results given by DH-PD and those by the experiments.
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paper_title: Smoothed Particle Hydrodynamics (SPH): an Overview and Recent Developments

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Smoothed particle hydrodynamics (SPH) is a meshfree particle method based on Lagrangian formulation, and has been widely applied to different areas in engineering and science. This paper presents an overview on the SPH method and its recent developments, including (1) the need for meshfree particle methods, and advantages of SPH, (2) approximation schemes of the conventional SPH method and numerical techniques for deriving SPH formulations for partial differential equations such as the Navier-Stokes (N-S) equations, (3) the role of the smoothing kernel functions and a general approach to construct smoothing kernel functions, (4) kernel and particle consistency for the SPH method, and approaches for restoring particle consistency, (5) several important numerical aspects, and (6) some recent applications of SPH. The paper ends with some concluding remarks.
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paper_title: A simple and robust three-dimensional cracking-particle method without enrichment

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Abstract A new robust and efficient approach for modeling discrete cracks in meshfree methods is described. The method is motivated by the cracking-particle method (Rabczuk T., Belytschko T., International Journal for Numerical Methods in Engineering, 2004) where the crack is modeled by a set of cracked segments. However, in contrast to the above mentioned paper, we do not introduce additional unknowns in the variational formulation to capture the displacement discontinuity. Instead, the crack is modeled by splitting particles located on opposite sides of the associated crack segments and we make use of the visibility method in order to describe the crack kinematics. We apply this method to several two- and three-dimensional problems in statics and dynamics and show through several numerical examples that the method does not show any “mesh” orientation bias.
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paper_title: A numerical approach to the testing of the fission hypothesis

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A finite-size particle scheme for the numerical solution of two- and three-dimensional gas dynamical problems of astronomical interest is described and tested. The scheme is then applied to the fission problem for optically thick protostars. Results are given, showing the evolution of one such protostar from an initial state as a single, rotating star to a final state as a triple system whose components contain 60% of the original mass. The decisiveness of this numerical test of the fission hypothesis and its relevance to observed binaries are briefly discussed.
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paper_title: A new approach for modelling slip lines in geological materials with cohesive models

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A methodology to model slip lines as strong displacement discontinuities within a continuum mechanics context is presented. The loss of hyperbolicity of the IBVP is used as the criterion for switching from a classical continuum description of the constitutive behaviour to a traction-separation model acting at the discontinuity surface. A version of the element-free Galerkin (EFG) method is employed where the slip line is represented as a set of slipped particles. The representation of the slip line as set of cohesive segments promises to remove difficulties in the propagation of the slip line. Two-dimensional examples are studied using the Drucker–Prager material model. Copyright © 2006 John Wiley & Sons, Ltd.
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paper_title: DDA validation of the mobility of earthquake-induced landslides

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Abstract This paper studies the run-out of earthquake-induced landslides. There are more than 20 mechanical models to explain long run-out of rapid landslides including some earthquake-induced landslides. However, notably few of them considered the effect of seismic loading on the run-out of landslides. In a previous study, we have proposed a model, which is called the multiplex acceleration model (MAM), to interpret the long run-out mechanism, and a shaking-table test has been performed to verify the MAM. Because the previous MAM is a conceptual model and only a single stone has been examined under sine waves by the shaking-table test, the MAM needs to be extended and further verified. In this paper, the MAM was extended by introducing the movement change that was induced by the so-called trampoline effect of earthquake loading, and the latest practical numerical simulation program, discontinuous deformation analysis (DDA), was used to verify the mobility of earthquake-induced landslides. After a conceptual landslide model was used to verify the effect of seismic loading on the mobility, the Donghekou landslide, which was a typical long run-out landslide that was induced by the 2008 Wenchuan earthquake, was analyzed. The results show that: the seismic loading could be one of the factors that could eventually help increase the run-out of landslides.
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paper_title: DEM–SPH simulation of rock blasting

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A numerical model is proposed for the simulation of rock blasting. A bonded particle system is utilized to mimic the behavior of rock. The particles interact at the contact points through normal and shear springs to simulate rock elasticity. To withstand the deviatoric stresses, the particles are glued to each other. If the applied force exceeds the contact strength, local failure occurs and microcracks are developed in the synthetic rock. For simulation of gas flow, the smooth particle hydrodynamic method is implemented. The interaction of gas particles with the rock grains is assumed to follow a perfect plastic collision model in which the initial momentum of the colliding particles is preserved. A detailed examination of the interaction of gas with blast hole is investigated. It is shown that the proposed hybrid model is capable of simulating the induced shock waves in the gas together with wave propagation in the rock material. The model successfully mimics crack propagation in rock. In particular, the crushed zone around the borehole, radial cracks, and surface spalling are all captured successfully. The results of numerical analysis suggest that gas–rock interaction can, in fact, generate a few successive compressive waves in the rock specimen, causing further extension of radial cracks with time as the weaker secondary and tertiary waves interact with the crack tips.
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paper_title: Reformulation of elasticity theory for discontinuities and long-range forces

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Some materials may naturally form discontinuities such as cracks as a result of deformation. As an aid to the modeling of such materials, a new framework for the basic equations of continuum mechanics, called the "peridynamic" formulation, is proposed. The propagation of linear stress waves in the new theory is discussed, and wave dispersion relations are derived. Material stability and its connection with wave propagation is investigated. It is demonstrated by an example that the reformulated approach permits the solution of fracture problems using the same equations either on or off the crack surface or crack tip. This is an advantage for modeling problems in which the location of a crack is not known in advance.
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paper_title: 3D dynamic fragmentation with parallel dynamic insertion of cohesive elements

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Dynamic fragmentation is a rapid and catastrophic failure of a material. During this process, the nucleation, propagation, branching, and coalescence of cracks results in the formation of fragments. The numerical modeling of this phenomenon is challenging because it requires high-performance computational capabilities. For this purpose, the finite-element method with dynamic insertion of cohesive elements was chosen. This paper describes the parallel implementation of its fundamental algorithms in the C++ open-source library Akantu. Moreover, a numerical instability that can cause the loss of energy conservation and possible solutions to it are illustrated. Finally, the method is applied to the dynamic fragmentation of a hollow sphere subjected to uniform radial expansion. Copyright (c) 2016 John Wiley & Sons, Ltd.
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paper_title: Mechanical response of a jointed rock beam—numerical study of centrifuge models

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In this paper we present a comparison between a set of benchmark centrifuge models of a jointed beam and the predictions of two numerical models: fast Lagrangian analysis of continua (FLAC) and discontinuous deformation analysis (DDA). The primary objective of this paper is a comparison between the measured deformation profiles and thrust evolution to predictions of the numerical methods employed. A secondary objective is an attempt to clarify the issue of compressive arch geometry which is still in controversy among researchers. ::: ::: ::: ::: It is found that both FLAC and DDA result in insufficiently accurate predictions to the measured displacements. The mode of deformation is only partially captured and is dependent on the aspect ratio of the individual blocks which made up the beam. It is shown that the accuracy of the predicted displacements is a function of the assigned interface stiffness. The thrust predicted by both methods is found to be considerably lower than that measured in the model; however, the linear evolution of thrust and equilibrium conditions are correctly captured. ::: ::: ::: ::: The geometry of the compressive arch as predicted by FLAC compares extremely well with the data measured in the physical model. Based on the FLAC analysis it is found that for a beam composed of equidimensional blocks the thickness of the compressive arch varies from 0.8t at the abutment interface to the entire beam thickness (1t) at a distance of a half block width from the abutment face, extending across the interface separating the block and its neighbour, and attains a value of 0.5t at the beam mid span. Copyright © 2006 John Wiley & Sons, Ltd.
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paper_title: Element‐free Galerkin methods

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An element-free Galerkin method which is applicable to arbitrary shapes but requires only nodal data is applied to elasticity and heat conduction problems. In this method, moving least-squares interpolants are used to construct the trial and test functions for the variational principle (weak form); the dependent variable and its gradient are continuous in the entire domain. In contrast to an earlier formulation by Nayroles and coworkers, certain key differences are introduced in the implementation to increase its accuracy. The numerical examples in this paper show that with these modifications, the method does not exhibit any volumetric locking, the rate of convergence can exceed that of finite elements significantly and a high resolution of localized steep gradients can be achieved. The moving least-squares interpolants and the choices of the weight function are also discussed in this paper.
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paper_title: Implementation of a high order lattice spring model for elasticity

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Abstract In a lattice spring model (LSM), the material is discretised into particles linked by springs. However, LSMs always adopt linear springs, which results in a stiff approximation of the corresponding elastic solution. In this work, a high order LSM is proposed to overcome this limitation by introducing additional degrees of freedoms (DOFs) to the particles. Based on the energy minimisation principle and the local strain technique, equations for the stiffness matrices of high order LSM are derived. Relationships between micro spring parameters and macro material constants are derived from the Cauchy-born rules and the hyperelastic theory. Numerical examples show that the high order LSM can provide a better solution than that of the linear LSM and that the LSM is more suitable for modelling singularity and fracture problems.
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paper_title: Safety evaluation of an ancient masonry seawall structure with modified DDA method

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Abstract The ancient masonry seawall along China’s Qiantang River is an important structure for defending the prosperous estuarine area, including Hangzhou, Jiaxing, and part of Shanghai, against violent floods, tides and waves. Modified discontinuous deformation analysis (DDA) is used here to analyze the failure mechanisms and stability of the blocky seawall structure. The backfilling soil is properly integrated into the DDA model, by means of the proposed soil discretization approach, and with an accurate interface shear strength model that the authors presented. According to the simulation results by the modified DDA, the existing seawall is generally in good state. Nevertheless, routine inspection and early restoration are recommended to avoid progressive failures of the seawall.
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paper_title: Hyperelastic model of a material which microstructure is formed by “balls and springs”

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Abstract We develop a model of an incompressible material which microscopic structure is formed by flexible but incompressible balls connected mutually by linear springs. The model is motivated by the structure of smooth muscle tissues that exhibit perfect elastic (or visco-elastic) behavior in a large extent of deformations. Moreover, their bulk (macroscopic) stiffness may be very effectively controlled and changed from very low values to essentially higher ones by simply defined structural changes inside individual muscle cells. In the continuum limit, the “balls and springs” model gives a nontrivial, highly nonlinear hyperelastic material. The stored strain energy function has generally no analytical expression. However, we find an approximate analytic formula, that is suitable for describing certain mechanical effects coming from the special arrangement of smooth muscle cells.
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paper_title: THE NUMERICAL MANIFOLD METHOD: A REVIEW

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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.
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paper_title: A two-scale model of granular materials

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A concurrent two-scale and two-method approach for modeling the mechanical behavior of dry frictional non-cohesive granular materials is presented. In domains of large eventually discontinuous deformation the material is modeled on the grain scale using a three-dimensional discrete element method. In the remaining domain the material is considered continuous and modeled by the finite element method using an elasto–plastic constitutive equation whose parameters are fit to the particle model via a homogenization scheme. The discrete and finite element models are coupled at the interface by the Arlequin method. An overlapping domain is introduced where the overall virtual work is interpolated between both models and compatibility is assured by kinematic constraints. For this purpose the discrete particle displacements are split into a fine and coarse scale part and equality of the coarse scale part and the continuum solution is enforced through the penalty method.
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paper_title: Dynamic discrete analysis of an earthquake-induced large-scale landslide

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Abstract The Chiu-fen-erh-shan landslide, estimated to have a volume of 36 million cubic meters, was triggered by the M w 7.6 Taiwan Chi-Chi earthquake of 1999. The debris covered an area of 1.95 km 2 and claimed 36 lives. During the sliding, the landslide mass disintegrated with its debris acquiring high speeds to reach over 1 km of travel distance. The main objectives of the study were to investigate if it was possible to numerically model the landslide progression, including slope disintegration, and to reproduce the post-failure configuration. Toward these goals, the study employed a discrete element method called discontinuous deformation analysis for the dynamic analysis. By incorporating the spacing and orientation information of the joints and bedding planes, the slope was modeled as an assemblage of discrete deformable blocks. Horizontal and vertical ground motions were simultaneously applied. The near-field digital strong records were affected by the large uplift of the hanging wall of the reverse thrust fault and corrections were made to address this issue. The study showed that the methodology adopted was able to capture essential characteristics of the landslide and gave a post-failure configuration resembling that observed in the field.
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paper_title: hp-Meshless Cloud Method

paper_content:

A methodology to build discrete models of boundary-value problems (BVP) is presented. The method is applicable to arbitrary domains and employs only a scattered set of nodes to build approximate solutions to BVPs. A version of moving least-square interpolation and the collocation method are used to discretize BVP equations, which results in a truly meshless method (i.e. without a background mesh of integration points). h- and p-adaptive strategies are tested and very good convergence of the method was observed. The improvements in the methodology, in particular the introduction of spectral degrees of freedom, result in a fast and accurate method, significantly more efficient than the Finite Element Method or Element Free Galerkin Method. Several practical applications of the method to solve various engineering problems are presented.
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paper_title: Dynamic stability analysis of jointed rock slopes using the DDA method: King Herod's Palace, Masada, Israel

paper_content:

Abstract A dynamic, two-dimensional, stability analysis of a highly discontinuous rock slope is demonstrated in this paper. The studied rock slope is the upper terrace of King Herod's Palace in Masada, situated on the western margins of the seismically active Dead Sea Rift. The slope consists of sub-horizontally bedded and sub-vertically jointed, stiff, dolomite blocks. The dynamic deformation of the slope is calculated using a fully dynamic version of DDA in which time-dependent acceleration is used as input. The analytically determined failure modes of critical keyblocks in the jointed rock slope are clearly predicted by DDA at the end of the dynamic calculation. It is found however that for realistic displacement estimates some amount of energy dissipation must be introduced into the otherwise fully elastic, un-damped, DDA formulation. Comparison of predicted damage with actual slope performance over a historic time span of 2000 years allows us to conclude that introduction of 2% kinetic damping should suffice for realistic damage predictions. This conclusion is in agreement with recent results of Tsesarsky et al. (In: Y.H. Hatzor (Ed.), Stability of Rock Structures: Proceedings of the Fifth International Conference of Analysis of Discontinuous Deformation, Balkema Publishers, Lisse, 2002, pp. 195–203) who compared displacements of a single block on an inclined plane subjected to dynamic loading obtained by DDA and by shaking table experiments. Using dynamic DDA it is shown that introduction of a simple rock bolting pattern completely stabilizes the slope.
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paper_title: A bonded-particle model for rock

paper_content:

A numerical model for rock is proposed in which the rock is represented by a dense packing of non-uniform-sized circular or spherical particles that are bonded together at their contact points and whose mechanical behavior is simulated by the distinct-element method using the two- and three-dimensional discontinuum programs PFC2D and PFC3D. The microproperties consist of stiffness and strength parameters for the particles and the bonds. Damage is represented explicitly as broken bonds, which form and coalesce into macroscopic fractures when load is applied. The model reproduces many features of rock behavior, including elasticity, fracturing, acoustic emission, damage accumulation producing material anisotropy, hysteresis, dilation, post-peak softening and strength increase with confinement. These behaviors are emergent properties of the model that arise from a relatively simple set of microproperties. A material-genesis procedure and microproperties to represent Lac du Bonnet granite are presented. The behavior of this model is described for two- and three-dimensional biaxial, triaxial and Brazilian tests and for two-dimensional tunnel simulations in which breakout notches form in the region of maximum compressive stress. The sensitivity of the results to microproperties, including particle size, is investigated. Particle size is not a free parameter that only controls resolution; instead, it affects the fracture toughness and thereby influences damage processes (such as notch formation) in which damage localizes at macrofracture tips experiencing extensile loading.
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paper_title: A clumped particle model for rock

paper_content:

Abstract Discrete element (particle) modeling is now often used to simulate the behavior of rock. While the method is attractive because it does not require the formulation of complex constitutive models, it requires extensive calibration with measured macro-scale results to determine the particle–contact parameters that will predict the macro-scale response. This calibration is usually carried out using the laboratory uniaxial compression test. The behavior of intact rock in compression is complex, resulting in a nonlinear failure envelope with high frictional resistance and a ratio of tensile to uniaxial compression strength of approximately 0.05–0.1. An extensive series of calibrations were carried out using a particle code to determine if the micro-parameters determined from uniaxial compression tests could be used to predict the intact rock response regardless of stress path, i.e., tension to triaxial compression. It is determined that the particle code cannot predict the failure envelope measured on laboratory samples, unless significant modifications are made. This study shows that adjusting the particle parameters had little effect on the macro-scale properties of compression and tension tests, but using a clumped-particle geometry improves the predictive capabilities of the particle code significantly. For both Lac du Bonnet granite and a weak synthetic rock, the particle code calibrated to uniaxial tests using the clumped-particle geometry predicts both the stress–strain behavior and the complete nonlinear failure envelope.
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paper_title: Block fracturing analysis using nodal-based discontinuous deformation analysis with the double minimization procedure

paper_content:

SUMMARY ::: ::: As a hybrid method, the nodal-based discontinuous deformation analysis (NDDA) greatly improves the stress accuracy within each DDA block by coupling a well-defined finite element mesh inside the DDA block; at the same time, the NDDA inherits the unique block kinematics of the standard DDA method. Each finite element mesh line inside the DDA block is treated as a potential crack, which enables the transformation of the block material from continuum to discontinuum through the tensile and shear fracturing mechanism. This paper introduces a double minimization procedure into the NDDA method to further improve the accuracy of the stresses evaluated at the finite element mesh lines and thus to obtain a more realistic fracture model. Three numerical examples are employed to demonstrate the improved stress accuracy by the implemented double minimization procedure and the accuracy and capability of the enhanced NDDA method in capturing brittle fracturing process. Copyright © 2013 John Wiley & Sons, Ltd.
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paper_title: Dynamic failure numeric simulations of model concrete-faced rock-fill dam

paper_content:

Conventional numeric simulations of rock-fill dams are generally performed finite element method (FEM), in which the rock-fill body is treated as continuum material. But the rock-fill body possesses strong discontinuity and FEM based on continuum idealization cannot simulate its failure process. The discontinuous deformation analysis (DDA) method is just the right tool of solving this problem satisfactorily. In this paper, two kinds of model dams, i.e. homogeneous rock-fill dam and concrete-faced rock-fill dam, are simulated using DDA method, their characteristics of response and failure process are presented. The results from numerical simulations are consistent with those from the author's previous dynamic experiments.
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paper_title: On three-dimensional modelling of crack growth using partition of unity methods

paper_content:

This paper reviews different crack tracking techniques in three-dimensions applicable in the context of partition of unity methods, especially meshfree methods. Issues such as describing and tracking the crack surface are addressed. A crack tracking procedure is proposed in detail and implemented in the context of the extended element-free Galerkin method (XEFG). Several three-dimensional cracking examples are compared to other results from the literature or the experimental data and show good agreement.
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paper_title: Contact forces between viscoelastic ellipsoidal particles

paper_content:

Abstract The normal (NFD) and tangential contact forces (TFD) between viscoelastic ellipsoidal particles are studied based on the contact mechanics and finite element method. It is found that the NFD and TFD force models previously formulated for spheres are valid for ellipsoidal particles and non-spherical particles of smooth surface, provided that some variables in the models are properly considered. The applicability of the NFD and TFD models are demonstrated in the collisions of two viscoelastic ellipsoids and shown to agree well with the results calculated by means of the finite element method. As part of the study, the applicability of the Linear-Spring-Dashpot (LSD) model which is widely used in the discrete modeling is also examined, and its limitation is identified.
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paper_title: Investigating the Influence of Mechanical anisotropy on the Fracturing Behaviour of Brittle Clay Shales with Application to Deep Geological Repositories

paper_content:

Clay shales are currently being assessed as possible host rock formations for the deep geological disposal of radioactive waste. However, one main concern is that the favourable long-term isolation properties of the intact rock mass could be negatively affected by the formation of an excavation damaged zone (EDZ) around the underground openings. This thesis investigated the deformation and failure process of a clay shale, namely Opalinus Clay, with particular focus on the inﬂuence of anisotropy on the ::: short-term response of circular tunnels. To achieve this goal, a hybrid continuum-discontinuum numerical approach was used in combination with new ﬁeld measurements from the Mont Terri underground research laboratory. The response of Opalinus Clay during the excavation of a full-scale emplacement (FE) test tunnel was characterized by geodetic monitoring of wall displacements, radial extensometers ::: and longitudinal inclinometers. The deformation measurements indicated strong directionality induced by the combined effect of in situ stress ﬁeld and presence of bedding planes striking parallel to the ::: tunnel axis, with the most severe deformation occurring in the direction approximately perpendicular to the material layering. Computer simulations were conducted using a newly-extended combined ::: ﬁnite-discrete element method (FEM/DEM), a numerical technique which allows the explicit simulation of brittle fracturing and associated seismicity. The numerical experimentation ﬁrstly focused on the ::: laboratory-scale analysis of failure processes (e.g., acoustic activity) in brittle rocks, and on the role of ::: strength and modulus anisotropy in the failure behaviour of Opalinus Clay in tension and compression. ::: The fracturing behaviour of unsupported circular excavations in laminated rock masses was then analyzed under different in situ stress conditions. Lastly, the modelling methodology was applied to the ::: aforementioned FE tunnel to obtain original insights into the possible EDZ formation process around emplacement tunnels for nuclear waste. The calibrated numerical model suggested delamination along bedding planes and subsequent extensional fracturing as key mechanisms of the damage process potentially leading to buckling and spalling phenomena. Overall, the research ﬁndings may have a potential impact on the constructability and support design of an underground repository as well as implications for its long-term safety assessment procedure.%%%%PhD
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paper_title: Numerical analysis of block stone displacements in ancient masonry structures: A new method to estimate historic ground motions

paper_content:

An innovative approach is presented, in which the discontinuous deformation analysis (DDA) method is used to estimate historic ground motions by back analysis of unique structural failures in archaeological sites. Two archaeological sites in Israel are investigated using this new approach and results are presented in terms of displacement evolution of selected structural elements in the studied masonry structure. The response of the structure is studied up to the point of incipient failure, in a mechanism similar to the one observed in the field. Structural response is found to be very sensitive to dynamic parameters of the loading function such as amplitude and frequency. ::: ::: ::: ::: Prior to back analysis of case studies, two validations are presented. Both compare the performance of DDA with analytical solutions and present strong agreement between the two. ::: ::: ::: ::: Using comprehensive sensitivity analyses, the most likely peak ground acceleration (PGA) and frequency that must have driven the observed block displacements are found for the two case studies—the Nabatean city of Mamshit and the medieval fortress of Nimrod in southern and northern Israel, respectively. It is found that horizontal peak ground accelerations (HPGA) of 0.5g and 1g were required to generate the observed deformations in Mamshit and Nimrod, respectively. Although these might seem too high, considering structural and topographic amplifications it is concluded that the analyses suggest ground motions of 0.2g at a frequency of 1.5 Hz for Mamshit and up to 0.4g at a frequency of 1 Hz for Nimrod. These values provide constraints on the seismic risk associated with these regions as appears in the local building code using a completely independent approach. Copyright © 2007 John Wiley & Sons, Ltd.
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paper_title: A detailed investigation of block dynamic sliding by the discontinuous deformation analysis

paper_content:

SUMMARY ::: ::: An extensive examination of the discontinuous deformation analysis (DDA) in block dynamic sliding modeling is carried out in this paper. Theoretical solutions for a single block sliding on an arbitrarily inclined plane by applying the horizontal/vertical seismic loadings to the sliding block as acceleration time histories or to the base as constraint displacement time histories are derived. As compared with the theoretical solutions, for a single block sliding, the DDA predicts the sliding displacements and block interaction forces accurately under various base incline angles and friction angles under both the harmonic loadings and a real seismic loading. The vertical seismic component may influence the block sliding displacements to different extent, and the DDA can capture these phenomena successfully and give accurate results. For the calculation of the single block relative sliding, both the theoretical and the DDA solutions indicate that applying the seismic accelerations as constraint displacement time histories (derived by integrating the seismic accelerations twice) to the base is equivalent to applying the seismic accelerations as volume forces to the sliding block in the opposite directions. The DDA modeling also demonstrates that this conclusion still stands for the case of multi-block sliding. Copyright © 2012 John Wiley & Sons, Ltd.
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paper_title: Modelling the dynamic failure of brittle rocks using a hybrid continuum-discrete element method with a mixed-mode cohesive fracture model

paper_content:

Abstract A cohesive fracture model that combines tension, compression and shear material behaviour is implemented into the hybrid continuum-discrete element method, i.e. Universal Distinct Element Code (UDEC), to simulate fracturing process in rock dynamic tests. The fracture model considers both elastic and inelastic (decomposed to fracture and plastic) displacements, with the norm of the effective inelastic displacement used to control the fracture behaviour. Two numerical examples, including notched semi-circular bending and Brazilian disc tests, are conducted to verify the applicability of the model in simulating the dynamic failure of rocks. Results show that the proposed approach is capable of realistically simulating the load rate effects on the mechanical behaviour of rock materials subjected to dynamic loads.
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paper_title: Coupled hydro-mechanical model for fractured rock masses using the discontinuous deformation analysis

paper_content:

Abstract A coupling analysis model is proposed to study the hydro-mechanical response of the fluid flow in fractured rock mass with the method of discontinuous deformation analysis (DDA). The DDA coupled hydro-mechanical model is interpreted in details by expressing the fracture fluid flow equations, the coupling process and the global coupled equations. For the mechanical response, the hydraulic pressure is determined first, followed by the coupled motion equations expressed under the DDA framework, to study the interaction between the fluid flow along the fractures and the movement of the rock blocks. In the fluid flow analysis, the cubic law is applied to study the steady flow along the fractures using the finite difference method (FDM). A real case of cavern excavation is analyzed by the proposed DDA coupled 2D hydro-mechanical model, to study the influence of fluid flow on the rock cavern stability during the excavation phase. The results show that the DDA coupled hydro-mechanical model is suitable for the stability and seepage analysis of practical engineering problems.
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paper_title: Simulating the process of reservoir-impoundment-induced landslide using the extended DDA method

paper_content:

Abstract In this work, three extensions to the traditional discontinuous deformation analysis (DDA) method are tested to simulate the entire process describing the behavior of landslides induced by reservoir impoundment. These extensions include the discretization of the area of interest, block gluing and the HM (hydro-mechanical) coupling approach. Several hydraulic boundary conditions are introduced based on the water level during the dry and rainfall seasons. As verification, the entire processes for two practical landslides, the Qianjiangping and the Majiagou landslides, which occurred in the Three Gorges Reservoir (TGR) of China, are simulated. The simulation results agree well with the observed failure modes, indicating that the proposed methodology can intuitively reproduce the entire reservoir impoundment induced landslide process.
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paper_title: A contact detection algorithm for multi-sphere particles by means of two-level-grid-searching in DEM simulations

paper_content:

Summary ::: In discrete element method simulations, multi-sphere particle is extensively employed for modeling the geometry shape of non-spherical particle. A contact detection algorithm for multi-sphere particles has been developed through two-level-grid-searching. In the first-level-grid-searching, each multi-sphere particle is represented by a bounding sphere, and global space is partitioned into identical square or cubic cells of size D, the diameter of the greatest bounding sphere. The bounding spheres are mapped into the cells in global space. The candidate particles can be picked out by searching the bounding spheres in the neighbor cells of the bounding sphere for the target particle. In the second-level-grid-searching, a square or cubic local space of size (D + d) is partitioned into identical cells of size d, the diameter of the greatest element sphere. If two bounding spheres of two multi-sphere particles are overlapped, the contacts occurring between the element spheres in the target multi-sphere particle and in the candidate multi-sphere particle are checked. Theoretical analysis and numerical tests on the memory requirement and contact detection time of this algorithm have been performed to verify the efficiency of this algorithm. The results showed that this algorithm can effectively deal with the contact problem for multi-sphere particles. Copyright © 2015 John Wiley & Sons, Ltd.
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paper_title: Dual-horizon peridynamics: A stable solution to varying horizons

paper_content:

Abstract In this paper, we present a dual-horizon peridynamics formulation which allows for simulations with dual-horizon with minimal spurious wave reflection. We prove the general dual property for dual-horizon peridynamics, based on which the balance of momentum and angular momentum in PD are naturally satisfied. We also analyze the crack pattern of random point distribution and the multiple materials issue in peridynamics. For selected benchmark problems, we show that DH-PD is less sensitive to the spatial than the original PD formulation.
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paper_title: Meshfree and particle methods and their applications

paper_content:

Recent developments of meshfree and particle methods and their applications in applied mechanics are surveyed. Three major methodologies have been reviewed. First, smoothed particle hydrodynamics ~SPH! is discussed as a representative of a non-local kernel, strong form collocation approach. Second, mesh-free Galerkin methods, which have been an active research area in recent years, are reviewed. Third, some applications of molecular dynamics ~MD! in applied mechanics are discussed. The emphases of this survey are placed on simulations of finite deformations, fracture, strain localization of solids; incompressible as well as compressible flows; and applications of multiscale methods and nano-scale mechanics. This review article includes 397 references. @DOI: 10.1115/1.1431547#
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paper_title: XLME interpolants, a seamless bridge between XFEM and enriched meshless methods

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In this paper, we develop a method based on local maximum entropy shape functions together with enrichment functions used in partition of unity methods to discretize problems in linear elastic fracture mechanics. We obtain improved accuracy relative to the standard extended finite element method at a comparable computational cost. In addition, we keep the advantages of the LME shape functions, such as smoothness and non-negativity. We show numerically that optimal convergence (same as in FEM) for energy norm and stress intensity factors can be obtained through the use of geometric (fixed area) enrichment with no special treatment of the nodes near the crack such as blending or shifting.
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paper_title: A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics

paper_content:

A local symmetric weak form (LSWF) for linear potential problems is developed, and a truly meshless method, based on the LSWF and the moving least squares approximation, is presented for solving potential problems with high accuracy. The essential boundary conditions in the present formulation are imposed by a penalty method. The present method does not need a “finite element mesh”, either for purposes of interpolation of the solution variables, or for the integration of the “energy”. All integrals can be easily evaluated over regularly shaped domains (in general, spheres in three-dimensional problems) and their boundaries. No post-smoothing technique is required for computing the derivatives of the unknown variable, since the original solution, using the moving least squares approximation, is already smooth enough. Several numerical examples are presented in the paper. In the example problems dealing with Laplace & Poisson's equations, high rates of convergence with mesh refinement for the Sobolev norms ||·||0 and ||·||1 have been found, and the values of the unknown variable and its derivatives are quite accurate. In essence, the present meshless method based on the LSWF is found to be a simple, efficient, and attractive method with a great potential in engineering applications.
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paper_title: An integrated finite difference method for analyzing fluid flow in porous media

paper_content:

The theoretical basis for the integrated finite difference method (IFDM) is presented to describe a powerful numerical technique for solving problems of groundwater flow in porous media. The method combines the advantages of an integral formulation with the simplicity of finite difference gradients and is very convenient for handling multidimensional heterogeneous systems composed of isotropic materials. Three illustrative problems are solved to demonstrate that two- and three-dimensional problems are handled with equal ease. Comparison of IFDM with the well-known finite element method (FEM) indicates that both are conceptually similar and differ mainly in the procedure adopted for measuring spatial gradients. The IFDM includes a simple criterion for local stability and an efficient explicit-implicit iterative scheme for marching in the time domain. If such a scheme can be incorporated in a new version of FEM, it should be possible to develop an improved numerical technique that combines the inherent advantages of both methods.
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paper_title: A review of discrete modeling techniques for fracturing processes in discontinuous rock masses

paper_content:

Abstract The goal of this review paper is to provide a summary of selected discrete element and hybrid finite–discrete element modeling techniques that have emerged in the field of rock mechanics as simulation tools for fracturing processes in rocks and rock masses. The fundamental principles of each computer code are illustrated with particular emphasis on the approach specifically adopted to simulate fracture nucleation and propagation and to account for the presence of rock mass discontinuities. This description is accompanied by a brief review of application studies focusing on laboratory-scale models of rock failure processes and on the simulation of damage development around underground excavations.
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paper_title: Extensions of discontinuous deformation analysis for jointed rock masses

paper_content:

Beginning with the original work of Shi [Discontinuous deformation analysis: a new numerical method for the statics and dynamics of block systems. Ph.D. thesis, University of California, Berkeley (1988)], called the Discontinuous Deformation Analysis (DDA) method, a number of extensions to the method have been explored. The extensions consist of improving the contact algorithm, adding block fracturing and sub-blocking capabilities. Contacts between blocks have been modeled using an Augmented Lagrangian Method instead of the penalty method originally proposed by Shi. This allows block-to-block contacts to be enforced more precisely and block contact forces to be determined more accurately. A sub-blocking capability has been developed, whereby blocks are discretized into sub-blocks. The continuity of the sub-block contacts is preserved and the variation of stresses in each large block can be determined. The sub-blocking capability is done using a consistent formulation in which the same methodology is used for the sub-blocks as the original large blocks. This is different from other discrete block methods that imbed finite difference zones or finite elements inside larger blocks. Finally, two block fracturing algorithms have been implemented in the DDA method. Using a three-parameter (cohesion, friction, tensile strength) Mohr-Coulomb criterion, one algorithm allows intact rocks to be broken into smaller blocks. Fracturing can be in shear or tension. The second algorithm allows fractures to propagate in the sub-blocks either in Mode I (tensile fracturing) or Mode II (shear fracturing). All three extensions have been implemented into the original DDA program of Shi. With the three extensions, the DDA method is more applicable to a greater range of rock mechanics problems and other engineering problems involving blocky systems. Examples of application of the method, for plane stress condition, are presented with regard to rock fall, slope stability and underground excavation problems.
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paper_title: The importance of modelling ballast particle shape in the discrete element method

paper_content:

The discrete element method has been used to model railway ballast. Particles have been modelled using both spheres and clumps of spheres. A simple procedure has been developed to generate clumps which resemble real ballast particles much more so than spheres. The influence of clump shape on the heterogeneous stresses within an aggregate has been investigated, and it has been found that more angular clumps lead to a greater degree of homogeneity. A box test consisting of one cycle of sleeper load after compaction has been performed on an aggregate of spheres and also on an alternative aggregate of clumps. The interlocking provided by the clumps provides a much more realistic load- deformation response than the spheres and the clumps will be the basis for future work on ballast degradation under cyclic loading.
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paper_title: Validation of a three-dimensional Finite-Discrete Element Method using experimental results of the Split Hopkinson Pressure Bar test

paper_content:

Abstract A full-scale 3D analysis of a Split Hopkinson Pressure Bar experiment on granite material using a 3D combined Finite-Discrete Element Method (FDEM) is shown. Previous efforts to simulate Split Hopkinson Pressure Bar experiments using the 2D FDEM had obtained a very good match for the loading portion of the experiment. This work extends those efforts by modeling the entire 3D Split Hopkinson Pressure Bar experimental setup, and reproducing the softening behavior of the sample after breakage. This modeling effort introduces the effect of a compliant interface between the bars and the sample.
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paper_title: Energy-conserving contact interaction models for arbitrarily shaped discrete elements

paper_content:

Abstract This work aims to establish a unified theoretical framework for normal contact of arbitrarily shaped discrete elements in discrete element modelling solely based on the energy-conservation principle for elastic contact. It will show that the normal force must be a potential field vector associated with a potential. With the construction of an appropriate potential function, a complete normal contact model, including the magnitude and direction of the normal force and the normal contact line, can be unambiguously derived for a pair of any shaped bodies. The model in its final form is simple and has a clear geometric perspective for 2D bodies. It can also recover some well-known models for bodies with simple geometric shapes.
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paper_title: The effect of strain rate on rock strength

paper_content:

The effect of the strain rate on strength has been evaluated for two widely different rock types, a brittle limestone (Tyndallstone) and a ductile salt rock (Lanigan potash rock).
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paper_title: Smoothed Particle Hydrodynamics and Its Diverse Applications

paper_content:

This review focuses on the applications of smoothed particle hydrodynamics (SPH) to incompressible or nearly incompressible flow. In the past 17 years, the range of applications has increased as researchers have realized the ability of SPH algorithms to handle complex physical problems. These include the disruption of free surfaces when a wave hits a rocky beach, multifluid problems that may involve the motion of rigid and elastic bodies, non-Newtonian fluids, virtual surgery, and chemical precipitation from fluids moving through fractured media. SPH provides a fascinating tool that has some of the properties of molecular dynamics while retaining the attributes of the macroscopic equations of continuum mechanics.
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paper_title: Viscous boundary of DDA for modeling stress wave propagation in jointed rock

paper_content:

Note: Roches Reference LMR-ARTICLE-2007-010View record in Web of Science Record created on 2007-09-24, modified on 2016-10-14
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paper_title: Response of aluminum nitride (including a phase change) to large strains, high strain rates, and high pressures

paper_content:

This article contains a description of a computational constitutive model for brittle materials subjected to large strains, high strain rates, and high pressures. The focus of this model is to determine the response of aluminum nitride under high velocity impact conditions that produce large strains, high strain rates, and high pressures. The strength is expressed as a function of the pressure, strain rate, and accumulated damage; and it allows for strength of both intact and failed material. The pressure is primarily expressed as a function of the volumetric strain, but it also includes the effect of bulking for the failed material. For materials without a phase change this model is an extension of the previous Johnson–Holmquist models for brittle materials. The primary new feature of this model is the capability to include a phase change, and this is required for aluminum nitride. Computations are performed to illustrate the capabilities of the model, to compare computed results to experimental results, and to provide an understanding of the behavior of aluminum nitride under the conditions of high velocity impact.This article contains a description of a computational constitutive model for brittle materials subjected to large strains, high strain rates, and high pressures. The focus of this model is to determine the response of aluminum nitride under high velocity impact conditions that produce large strains, high strain rates, and high pressures. The strength is expressed as a function of the pressure, strain rate, and accumulated damage; and it allows for strength of both intact and failed material. The pressure is primarily expressed as a function of the volumetric strain, but it also includes the effect of bulking for the failed material. For materials without a phase change this model is an extension of the previous Johnson–Holmquist models for brittle materials. The primary new feature of this model is the capability to include a phase change, and this is required for aluminum nitride. Computations are performed to illustrate the capabilities of the model, to compare computed results to experimental results,...
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paper_title: Modeling complex crack problems using the numerical manifold method

paper_content:

In the numerical manifold method, there are two kinds of covers, namely mathematical cover and physical cover. Mathematical covers are independent of the physical domain of the problem, over which weight functions are defined. Physical covers are the intersection of the mathematical covers and the physical domain, over which cover functions with unknowns to be determined are defined. With these two kinds of covers, the method is quite suitable for modeling discontinuous problems. In this paper, complex crack problems such as multiple branched and intersecting cracks are studied to exhibit the advantageous features of the numerical manifold method. Complex displacement discontinuities across crack surfaces are modeled by different cover functions in a natural and straightforward manner. For the crack tip singularity, the asymptotic near tip field is incorporated to the cover function of the singular physical cover. By virtue of the domain form of the interaction integral, the mixed mode stress intensity factors are evaluated for three typical examples. The excellent results show that the numerical manifold method is prominent in modeling the complex crack problems.
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paper_title: Particle Manifold Method (PMM) for Multiscale Continuous-Discontinuous Analysis

paper_content:

Keywords: particle manifold method (PMM) ; continuous and discontinuous analysis ; wave propagation and transmission ; particle contact ; micro failure mechanism ; GPU parallelization ; CUDA ; multiscale analysis ; rock dynamics ; loading rate effect These Ecole polytechnique federale de Lausanne EPFL, n° 5515 (2012)Programme doctoral StructuresFaculte de l'environnement naturel, architectural et construitInstitut d'ingenierie civileLaboratoire de mecanique des rochesJury: J.-F. Molinari (president), H. Konietzky, L. Laloui, A. Munjiza Public defense: 2012-10-5 Reference doi:10.5075/epfl-thesis-5515Print copy in library catalog Record created on 2012-09-27, modified on 2016-08-09
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paper_title: NBS contact detection algorithm for bodies of similar size

paper_content:

Large-scale discrete element simulations, as well as a whole range of related problems, involve contact of a large number of separate bodies. In this context an efficient and robust contact detection algorithm is necessary. There has been a number of contact detection algorithms with total detection time (CPU time needed to detect all couples close to each other) proportional to Nln(N) (where N is the total number of separate bodies) reported in recent years. ::: ::: In this work a contact detection algorithm with total detection time proportional to N is reported. The algorithm is termed NBS, which stands for no binary search. In other words, the proposed algorithm involves no binary search at any stage. In addition the performance of the algorithm in terms of total detection time is not influenced by packing density, while memory requirements are insignificant. The only limitation of the algorithm is its applicability to the systems comprising bodies of similar size. © 1998 John Wiley & Sons, Ltd.
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paper_title: Modified contact model with rock joint constitutive in numerical manifold method

paper_content:

Abstract In the classical numerical manifold method (NMM), contact conditions are enforced by applying or releasing contact springs repeatedly. Penalty spring stiffness is constant during the calculation process. In this study, a modified contact model describing joint mechanical behavior is proposed using the NMM. Two rock joint constitutive models, namely the BB model and hyperbolic model, are used to modify normal and shear stiffness, respectively. In the modified contact model, stiffness varies with normal and shear inter-penetration distance in each iteration step. Therefore, stiffness varies for different contact points. With increasing inter-penetration distance, the normal stiffness increases and shear stiffness decreases. As a result, the force needed for trigger sliding along the joint increased. Finally, three example simulations are conducted to validate the effectiveness of the modified contact model.
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paper_title: Numerical simulation of blasting-induced rock fractures

paper_content:

Abstract In the present study, the Johnson–Holmquist (J–H) material model is implemented into the commercial software LS-DYNA through user-subroutines to simulate the blasting-induced rock fractures. The J–H model consists of strength models for both intact and fully fractured materials, a polynomial equation of state, and a damage model that represents the material from an intact state to a fully fractured state. Influences of the key parameters in smooth blasting, viz., loading rate, distance from a free face, earth stress, and pre-existing joint planes, etc., on fracture patterns are explored. According to the simulation results, the rock fracture pattern is significantly influenced by the loading rate. Fracture control techniques, namely, notched borehole and charge holder with slits are also simulated. Effectiveness of the fracture control techniques is demonstrated. The numerical simulation in the present study reproduces some of the well-known phenomena observed by other researchers. It has the potential to be applied in practical blast control and gas and hydraulic fracturing engineering.
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paper_title: Particle-based numerical manifold method to model dynamic fracture process in rock blasting

paper_content:

AbstractA particle-based numerical manifold method (PNMM) is developed to investigate rock fracturing behavior under dynamic loading. The basic idea of PNMM is to represent the microstructure of rock with a group of particles and to simulate the macroscopic behavior of rock masses through polygonal elements, which is performed by a dual-level discretization system. The first discretization is in the manner of FEM, which reduces the infinite degrees of freedom (DOF) of a continuum to the finite DOF of polygonal elements. However, the DOF of the model are not further affected by the latter discretization. The behaviors of internal particles are obtained from the mechanical fields of their governing elements. A dual-layer-cover system, i.e., the mathematical cover and the physical cover, is adopted in PNMM. Fracture initiation and propagation at a macroscopic level are represented as the coalescence of microcracks induced at the microscale level between particles. The Johnson-Holmquist-Beissel (JHB) model is...
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paper_title: Application of the numerical manifold method to model progressive failure in rock slopes

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Abstract The present study develops the numerical manifold method as a tool to investigate the progressive failure in rock slopes. The entire processes of the progressive slide surface development related to crack initiation, propagation, coalescence and degradation to eventual catastrophic failure are successfully captured. To overcome the limitation of the original NMM associated with an improper removal of the interface cohesion of the discontinuities, the displacement-dependent cohesion removal method is adopted. Simple examples of failure of rock slopes containing different arrangements of discontinuities are modeled and their results are compared with those based on the popular limit equilibrium method proposed by famous pioneers Jennings (1970) [1] and Jaeger (1971) [2]. The present results illustrate that the types of crack propagation and coalescence obtained by the developed NMM can help explain the relevant cracking processes in nature and experimental studies.
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paper_title: The numerical manifold method for 2D transient heat conduction problems in functionally graded materials

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Abstract Benefiting from the use of two cover systems, that is, the mathematical cover and the physical cover, the numerical manifold method (NMM) is capable of solving both continuous and discontinuous problems in the same platform. Presently, the NMM is further developed to tackle two-dimensional transient heat conduction problems in the functionally graded materials (FGMs). Firstly, the governing equation, the associated boundary conditions and the initial condition are presented. Then, the fundamentals of the NMM are briefly reviewed. Following, the NMM discrete formulations are derived based on the Galerkin-form weighted residual method and then solved with the backward difference scheme. Finally, for verification, three numerical examples with increasing complexity are tested on uniform mathematical covers composed of square mathematical elements, and our results well demonstrate the advantages of the proposed method in discretization and accuracy; besides, the effects of material gradient on the thermal behavior of FGMs are also examined.
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paper_title: Modeling complex crack problems using the numerical manifold method

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In the numerical manifold method, there are two kinds of covers, namely mathematical cover and physical cover. Mathematical covers are independent of the physical domain of the problem, over which weight functions are defined. Physical covers are the intersection of the mathematical covers and the physical domain, over which cover functions with unknowns to be determined are defined. With these two kinds of covers, the method is quite suitable for modeling discontinuous problems. In this paper, complex crack problems such as multiple branched and intersecting cracks are studied to exhibit the advantageous features of the numerical manifold method. Complex displacement discontinuities across crack surfaces are modeled by different cover functions in a natural and straightforward manner. For the crack tip singularity, the asymptotic near tip field is incorporated to the cover function of the singular physical cover. By virtue of the domain form of the interaction integral, the mixed mode stress intensity factors are evaluated for three typical examples. The excellent results show that the numerical manifold method is prominent in modeling the complex crack problems.
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paper_title: Three new boundary conditions for the seismic response analysis of geomechanics problems using the numerical manifold method

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Abstract The numerical manifold method (NMM) provides a unified approach to address continuum-discontinuum problems in geotechnical engineering. Owing to the dynamic nature of its governing equations, the NMM should be suitable for modelling dynamic problems such as those associated to earthquakes. However, due to the current limitations in far-field boundary conditions implemented in the original NMM formulation, NMM has not been used to carry out seismic response analysis. In the present study, three new boundary conditions have been developed to extend the capability of the NMM to conduct seismic response analysis: (1) the classical viscous boundary condition, which allows for the absorption of the seismic wave energy at the boundaries, based on the viscous boundaries, and the seismic motion input method is also proposed; (2) the free field boundary condition, which captures the free field motion and absorption of the reflected waves at the sides of the model. The algorithms to generate free field mesh and its coupling calculations with the main mesh are also presented; (3) the static-dynamic unified boundary, which models the transition from fixed boundary condition in static state to free field boundary condition in seismic state, thus ensuring the accuracy and consistency of the numerical simulation. Finally, five numerical examples are shown to validate the proposed methods. The numerical results indicate that the improved NMM can be successfully adopted for seismic response analysis.
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paper_title: Unified continuum/discontinuum modeling framework for slope stability assessment

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Abstract Current dominant methods for slope stability analysis are the limit equilibrium method and the strength reduction method. Both methods are based on the limit equilibrium conditions. However, the limit equilibrium methods are limited to the rigid body assumption, while the strength reduction method is computationally expensive and has convergence issues due to the non-linear iterative computations. In this paper, we propose a current stress-based search algorithm to directly obtain the critical slip surface and the safety factor. The numerical manifold method, which unifies the continuum and discontinuum analysis problems, is used for stress analysis to obtain the stress distribution of soil slopes or rock slopes cut by joints. Based on the stress results obtained, a graph theory is used to convert the solution of the critical slip surface to a shortest path problem, which can be directly solved by the Bellman–Ford algorithm. The proposed method couples the numerical manifold method and the graph theory allowing for stability analyses of both rock and soil slopes within the same framework. The method completely removes the computational effort needed for iterations in the strength reduction method as well as eliminating the rigid body assumptions in the limit equilibrium method.
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paper_title: A coupled distinct lattice spring model for rock failure under dynamic loads

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It is necessary to take into account the micro discretization of natural rock when studying its macroscopic failure behavior. This requirement has resulted in renewed and increased interest in the discrete or framework/lattice numerical modeling techniques. However, to fully construct a numerical model for practical applications using a discrete numerical model is computationally difficult with current computing technologies. Hence, a coupled model has been developed to overcome this limitation by coupling the Distinct Lattice Spring Model (DLSM) and the Numerical Manifold Method (NMM). In the coupled model, the microscopic discrete model of the rock is represented by a system of discrete particles interacting via springs while the macroscopic level model is represented by the NMM. The proposed model bears a structure of three layers corresponding to the DLSM model, the NMM model, and a model for coupling, respectively. The coupling model is based on a newly developed Particle based Manifold Method (PMM) to bridge the DLSM with the NMM. The proposed coupled model can reduce the computational resources needed for the purely discrete particle based model. This study introduces theoretical aspects of the coupled model together with a few examples to demonstrate its correctness and feasibility. (C) 2011 Elsevier Ltd. All rights reserved.
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paper_title: Development of three-dimensional numerical manifold method for jointed rock slope stability analysis

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Abstract This paper presents a three-dimensional numerical manifold method (NMM) for jointed rock slope stability analysis. The main barriers for realistic rock slope stability analysis are discussed, and the unique features of the NMM which make it stand out in slope analysis are emphasized. The framework of the 3-D NMM is established via a regularly-patterned tetrahedral mesh-based mathematical cover system. A contact detection algorithm is then customized to fit the 3-D NMM characteristics. The developed code is first calibrated by two simple cases: inclined plane sliding and tetrahedral wedge sliding, and then applied to a more complex slope scenario. The complete failure process involving large displacement and rotation of multiple interaction blocks is exhibited dynamically. Two typical stabilization/protection techniques are also investigated. The developed 3-D NMM could be potentially used to find the trigger/failure mechanism of jointed rock slopes, so as to optimize the slope stabilization or protection design.
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paper_title: Three-dimensional slope stability analysis using independent cover based numerical manifold and vector method

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Abstract A three-dimensional (3D) slope stability analysis method is presented in this paper based on three-dimensional independent cover based manifold method (ICMM3D) and vector sum method (VSM). ICMM3D is proposed in the framework of independent cover, which avoid the time-consuming and error-prone cover system generation of convention numerical manifold method (NMM). It is very suitable for the continuous/discontinuous deformation analysis in slope engineering. Then, with the stress field obtained by ICMM3D, VSM is employed to calculate the factor of safety, which is effective in computation and clear in physical meaning. In addition, a new strategy of discretization of the slip surface is proposed, which discretizes the slip surface into a set of calculating points in order to avoid errors caused by the special handling in the boundary columns in limit equilibrium method (LEM) and the tedious triangulation of the slip surface in VSM. Finally, genetic algorithm is employed to search for the critical slip surface. Numerical examples demonstrate the efficiency, accuracy and robustness of the proposed method.
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paper_title: Formulation of a Three-dimensional Numerical Manifold Method with Tetrahedron and Hexahedron Elements

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Numerical Manifold Method (NMM) is an extension of Discontinuous Deformation Analysis which contains and combines FEM and joint/block oriented DDA in a unified form. At present, application of NMM is mainly limited to two-dimensional analysis which is not realistic for many underground excavation works. In the present paper, three-dimensional NMM based on tetrahedron and hexahedron elements is proposed. The corresponding three-dimensional cover contact detection algorithms which have some fundamental properties different from the corresponding two-dimensional analysis are also developed. Based on the present formulations, engineering problems with arbitrarily oriented discontinuities can be modeled and realistic modeling of construction works becomes possible.
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paper_title: Cover refinement of numerical manifold method for crack propagation simulation

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Abstract A cover refinement method is proposed for the numerical manifold method (NMM) to simulate crack propagation in brittle materials. New mathematical covers are defined for manifold elements near a crack tip. The refinement is done for corresponding mathematical covers of the selected manifold elements. The updating process of mathematical cover with respect to different boundary conditions is introduced in detail. When a mathematical cover is updated, the corresponding physical covers and manifold elements are updated accordingly. Furthermore, the loops of the considered domain are updated as well. Three numerical examples are analyzed to validate the proposed cover refinement method. The numerical results are all in good agreement with those results in the existing studies. It is demonstrated that the proposed cover refinement method has higher accuracy for crack propagation simulation comparing to the traditional numerical manifold method which has a consistent mathematical cover system. The proposed cover refinement method also does not significantly change the manifold elements at the vicinity of the crack tips. A rock slope model with a bilinear failure mode is simulated and progressive failure process of the rock slope is obtained, which demonstrates the applicability of the proposed method in practical rock engineering.
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paper_title: The MLS-based numerical manifold method with applications to crack analysis

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In order to solve problems, from a continuum point of view and in a unified way, involving continuum and discontinuum deformation, and small deformation and large movement, the numerical manifold method (NMM) introduces two covers, namely the mathematical cover (MC) and the physical cover (PC). This study generates the MC with the influence domains of nodes in the moving least squares (MLS) interpolation instead of commonly-used finite element meshes, significantly simplifying the generation of PCs and the simulation of crack growth. Advantageous over the conventional meshfree method, the MLS-based NMM can naturally treat complex geometry without recourse to those complicated but contrived criteria or operations. Moreover, the treatment of large movement caused by cracking is much easier with the MLS-based NMM than with the FE-based NMM.
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paper_title: Development of high‐order manifold method

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The Manifold Method of Material Analysis (MM) with high-order displacement functions has been derived based on triangular meshes for the requirement of high accurate calculations from practical applications. The matrices of equilibrium equations for the second-order MM have been given in detail for program coding. The derivation of the method is made by means of approximation theory and very few new mathematical concepts are used in this paper. So, it may be understood by most engineering researchers. By close comparison with widely used Finite Element Method, the advantages of MM can be seen very clearly in the following aspects: (1) the capability of processing large deformation and handing discontinuities like block oriented Discontinuous Deformation Analysis method; (2) making element meshes easily and (3) using high-order displacement functions easily. The C program codes for the second-order MM has been developed, and it has been applied to the example of a beam bending under a central point loading. The calculated results are quite good in agreement with theoretical solutions. By contrast, the results calculated for the same model by use of the original first-order MM are far from the theoretical solutions. Therefore, it is important and necessary to develop high-order Manifold Method for the complicated deformation problems. © 1998 John Wiley & Sons, Ltd.
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paper_title: Implementation of the numerical manifold method for thermo-mechanical fracture of planar solids

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Abstract The numerical manifold method (NMM) is developed to solve thermo-mechanical fracture problems in two-dimensional setting. The temperature fields are firstly determined through steady heat conduction analysis and then imported into the mechanical simulation. The temperature and displacement discontinuities across crack faces are essentially captured due to the dual cover systems in the NMM. To account for the singularity of thermal flux and stresses at the crack tip, the associated asymptotic basis is adopted in the cover functions for thermal and mechanical computation, respectively. The stress intensity factors are calculated with the domain-independent interaction integral. The accuracy of the proposed method is tested through several numerical examples including single, double and multiple-branched cracks and the results agree well the available solutions from the literature. The superiority of the NMM in the multi-field discontinuity modeling is preliminarily verified through the present work.
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paper_title: On generation of lumped mass matrices in partition of unity based methods

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Summary ::: The partition of unity based methods, such as the extended finite element method and the numerical manifold method, are able to construct global functions that accurately reflect local behaviors through introducing locally defined basis functions beyond polynomials. In the dynamic analysis of cracked bodies using an explicit time integration algorithm, as a result, huge difficulties arise in deriving lumped mass matrices because of the presence of those physically meaningless degrees of freedom associated with those locally defined functions. Observing no spatial derivatives of trial or test functions exist in the virtual work of inertia force, we approximate the virtual work of inertia force in a coarser manner than the virtual work of stresses, where we inversely utilize the ‘from local to global’ skill. The proposed lumped mass matrix is strictly diagonal and can yield the results in agreement with the consistent mass matrix, but has more excellent dynamic property than the latter. Meanwhile, the critical time step of the numerical manifold method equipped with an explicit time integration scheme and the proposed mass lumping scheme does not decrease even if the crack in study approaches the mesh nodes — a very excellent dynamic property. Copyright © 2017 John Wiley & Sons, Ltd.
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paper_title: The numerical manifold method for elastic wave propagation in rock with time-dependent absorbing boundary conditions

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Abstract In this study, a modified first-order Higdon׳s absorbing boundary scheme is proposed and incorporated into the numerical manifold method (NMM) to reduce reflections from artificial boundaries induced by truncating infinite media. The modified time-dependent absorbing boundary scheme can not only consider the absorbing boundary and input boundary at the same artificial boundary, but also take the effects of the incident angles into consideration by adjusting the velocities and strains of points at the boundary automatically. For illustrating the efficiency of the proposed time-dependent absorbing boundary scheme, comparisons between the results of the proposed method and the widely used viscous boundary conditions for different incident angles are presented. The developed NMM is then used to investigate wave attenuation and transmission across a joint in an infinitely long rock bar.
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paper_title: New strategies for some issues of numerical manifold method in simulation of crack propagation

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SUMMARY ::: Aiming to solve, in a unified way, continuous and discontinuous problems in geotechnical engineering, the numerical manifold method introduces two covers, namely, the mathematical cover and the physical cover. In order to reach the goal, some issues in the simulation of crack propagation have to be solved, among which are the four issues to be treated in this study: (1) to reduce the rank deficiency induced by high degree polynomials as local approximation, a new variational principle is formulated, which suppresses the gradient-dependent DOFs; (2) to evaluate the integrals with singularity of 1/r, a new numerical quadrature scheme is developed, which is simpler but more efficient than the existing Duffy transformation; (3) to analyze kinked cracks, a sign convention for argument in the polar system at the crack tip is specified, which leads to more accurate results in a simpler way than the existing mapping technique; and (4) to demonstrate the mesh independency of numerical manifold method in handling strong singularity, a mesh deployment scheme is advised, which can reproduce all singular locations of the crack with regard to the mesh. Corresponding to the four issues, typical examples are given to demonstrate the effectiveness of the proposed schemes. Copyright © 2013 John Wiley & Sons, Ltd.
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paper_title: Prediction of rank deficiency in partition of unity-based methods with plane triangular or quadrilateral meshes

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In the partition of unity (PU)-based methods, the global approximation is built by multiplying a partition of unity by local approximations. Within this framework, high-order approximations are achieved by directly adopting high-order polynomials as local approximations, and therefore nodes along sides or inside elements, which are usually adopted in the conventional finite element methods, are no more required. However, the PU-based approximation constructed in this way may suffer from rank deficiency due to the linear dependence of the global degrees of freedom. In this paper, the origin of the rank deficiency in the PU-based approximation space is first dissected at an element level, and then an approach to predict the rank deficiency for a mesh is proposed together with the principle of the increase of rank deficiency. Finally, examples are investigated to validate the present approach. The current work indicates such a fact that the rank deficiency is an unrelated issue to the nullity of the global matrix. It can be resolved in its own manner.
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paper_title: Dynamic high order numerical manifold method based on weighted residual method

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SUMMARY ::: High Order Numerical Manifold Method (HONMM) is a powerful method to solve static problems. A development of HONMM to achieve a dynamic solution with high accuracy and less computational cost is addressed in the current paper. In the developed method, the global approximation is obtained through increasing the order of local approximation functions without any Linear Dependence (LD) of the unknowns. The weighted residual formulations are modified to be used in dynamic high order simulation. Moreover, a modified Newmark method formulation is adjusted for time integration of high order equations. The superiority of the proposed method over the conventional NMM is demonstrated through a special beam example. The dynamic free fall block example is used to exhibit the removal of mass matrix singularity. As cases of dynamic analysis, beam free and forced vibrations are illustrated which include a moving load. Finally, a non-uniform cross-section beam under dynamic variable loads with accelerated motion is solved while demonstrating the capability of the new method such as simplicity, accuracy and time efficiency for simulation of complex dynamic problems. Copyright © 2014 John Wiley & Sons, Ltd.
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paper_title: A COMPLEX VARIABLE MESHLESS MANIFOLD METHOD FOR FRACTURE PROBLEMS

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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.
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paper_title: Modeling progressive failures in rock slopes with non-persistent joints using the numerical manifold method

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SUMMARY ::: Rock slope failure is a complex process that usually involves both opening/sliding along pre-existing discontinuities as well as fracturing of intact rock bridges. Discontinuity persistence is an important factor governing rock slope instabilities. However, traditional slope failure analysis assumes persistent discontinuities, and rock slope fails along a predefined persistent continuous potential failure surface because of the limitations of the analysis tools. This paper proposes the numerical manifold method (NMM) incorporated with a Mohr–Coulomb criterion-based fracturing algorithm to simulate the progressive failure of rock slopes with non-persistent joints. Detailed fracturing algorithm is first presented. Then, the NMM enabling fracturing is calibrated through simulating an edge-cracked plate and the Brazilian test. Lastly, the developed code is applied to investigate the failure process of rock slopes involving non-persistent joints. Numerical results indicate that the proposed method can capture the opening/sliding along existing discontinuities, the fracturing in intact rock bridges and the final kinematic release. Progressive slope failure is well exhibited. Copyright © 2013 John Wiley & Sons, Ltd.
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paper_title: Application of the numerical manifold method for stress wave propagation across rock masses

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In this paper, the numerical manifold method (NMM) is extended to study wave propagation across rock masses. First, improvements to the system equations, contact treatment, and boundary conditions of the NMM are performed, where new system equations are derived based on the Newmark assumption of the space–time relationship, the edge-to-edge contact treatment is further developed for the NMM to handle stress wave propagation across discontinuities, and the viscous non-reflection boundary condition is derived based on the energy minimisation principle. After the modification, numerical comparisons between the original and improved NMM are presented. The results show that the original system equations result in artificial numerical damping, which can be overcome by the Newmark system equations. Meanwhile, the original contact scheme suffers some calculation problems when modelling stress wave propagation across a discontinuity, which can be solved by the proposed edge-to-edge contact scheme. Subsequently, the influence of the mesh size and time step on the improved NMM for stress wave propagation is studied. Finally, 2D wave propagation is modelled, and the model's results are in good agreement with the analytical solution.
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paper_title: Footwall slope stability analysis with the numerical manifold method

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Abstract This paper extends the numerical manifold method (NMM) for footwall slope stability analysis. A fracturing algorithm based on the Mohr–Coulomb criterion with a tensile cutoff is implemented into the NMM code. The developed program is first calibrated by simulating four typical crack problems. It is then applied to analyze the potential footwall slope instability. Parametric studies with respect to dip of a bedding plane, orientation of a predominant joint set, the bedding profile, etc., are carried out. Numerical results indicate that the developed program can simulate the opening and sliding along pre-existing discontinuities, fracturing through intact rock, as well as kinematics of the failed slope, and can also reproduce the major failure mechanisms observed in footwall slope collapses. The NMM is promising for such problems and deserves to be further developed to be practically used in natural/excavated rock slope stability analysis and open pit slope design.
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paper_title: Dynamic Study on Fracture Problems in Viscoelastic Sedimentary Rocks Using the Numerical Manifold Method

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The viscoelastic deformation behavior of a sedimentary rock under different loading rates is numerically modeled and investigated by the numerical manifold method (NMM). By incorporating a modified 3-element viscoelastic constitutive mode in the NMM, crack initiation and propagation criteria, and crack identification and evolution techniques, the effects of the loading rates on the cracking behavior of a sedimentary rock, such as crack open displacement, crack sliding displacement, crack initiation, crack propagation and final failure mode, are successfully modeled. The numerical results reveal that under a high loading rate (>1,000 MPa/s), due to the viscoelastic property of the sedimentary rock, not only the structural behavior deviates from that of elastic model, but also different cracking processes and final failure modes are obtained.
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paper_title: Thermal shock analysis of 2D cracked solids using the numerical manifold method and precise time integration

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Abstract The numerical manifold method (NMM), combined with the precise time integration method (PTIM), is proposed for thermal shock fracture analysis. The temperature and displacement discontinuity across crack faces is naturally portrayed attributing to the cover systems in the NMM. The crack tip singularities are characterized through the use of asymptotic bases in the approximations. The discrete equations for transient thermal analysis are firstly solved with the PTIM and then the thermoelastic study is performed. With the interaction integral, the stress intensity factors are computed. Several examples are tested and the nice consistency between the present and existing results is found.
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paper_title: Wilson non‐conforming element in numerical manifold method

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Numerical manifold method (NMM) by Shi tries to unify the solution of problems in both continuous and discontinuous domains. Presently, NMM is commonly adopted on finite triangular and quadrilateral covers and the shape functions used in classical finite element analysis are also commonly adopted in manifold analysis. In this paper, the Wilson non-conforming element is used in quadrilateral covers. The corresponding matrices of equilibrium equations are derived and a cantilever slab is considered. The results from the use of non-conforming element in NMM are in good agreement with theoretical solution as compared with the classical Q4 isoparametric element. Copyright © 2002 John Wiley & Sons, Ltd.
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paper_title: DEVELOPMENT OF 3D NUMERICAL MANIFOLD METHOD

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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.
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paper_title: A high order local approximation free from linear dependency with quadrilateral mesh as mathematical cover and applications to linear elastic fractures

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Abstract The numerical manifold method falls into the category of the partition of unity methods. In order to enhance accuracy, high order polynomials can be specified as the local approximations. This, however, would incur rank deficiency of the stiffness matrix. In this study, a local displacement approximation is constructed over a physical patch generated from a four quadrilateral mathematical mesh. All the degrees of freedom are physically meaningful. The stresses are continuous at all nodes, suggesting that no stress polish is required. The proposed approximations have the same accuracy as the first-order polynomials, but no linear dependency inherent in the latter.
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paper_title: Finite cover method for linear and non-linear analyses of heterogeneous solids

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We introduce the finite cover method (FCM) as a generalization of the finite element method (FEM) and extend it to analyse the linear and non-linear mechanical behaviour of heterogeneous solids and structures. The name ‘FCM’ is actually an alias for the manifold method (MM) and the basic idea of the method has already been established for linear analyses of structures with homogeneous materials. After reviewing the concept of physical and mathematical covers for approximating functions in the FCM, we present the formulation for the static equilibrium state of a structure with arbitrary physical boundaries including material interfaces. The problem essentially involves the discontinuities in strains, and possibly has the discontinuities in displacement caused by interfacial debonding or rupture of material interfaces. We simulate such non-linear mechanical behaviour after presenting simple numerical examples that demonstrate the equivalence between the approximation capabilities of the FCM and those of the FEM. Copyright © 2003 John Wiley & Sons, Ltd.
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paper_title: A GENERALIZED AND EFFICIENT METHOD FOR FINITE COVER GENERATION IN THE NUMERICAL MANIFOLD METHOD

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The numerical manifold method (NMM) based on the concept of finite covers and the partition of unity (PU) provides a unified framework to analyze continuum and discontinuum without changing predefined mesh in a discretized way. The NMM has been applied in the modeling of fluid structure interaction as well as in rock mechanics including the analysis of block system, jointed rock and fractured body, showing particular advantages over other PU based methods. Unlike other PU methods, the degrees of freedoms in the NMM are associated with the physical covers, rather than the nodes, which allow it to be naturally adapted to the changing geometries in analyzing complex discontinuum such as multiple intersecting cracks and branched cracks. Despite these recent advances, there is no publication available to date describing the physical cover generation of the NMM in a systematic way or giving a general principle of cover numbering, which has practically limited a wider application of the NMM. To address this issu...
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paper_title: A three-node triangular element fitted to numerical manifold method with continuous nodal stress for crack analysis

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Abstract A three-node triangular element fitted to the numerical manifold method with continuous nodal stress called Trig3-CNS (NMM) element for accurately modeling two-dimensional linear elastic fracture problems is presented. By adopting two cover systems, namely, the mathematical cover and physical cover, the numerical manifold method (NMM) could easily solve continuous and discontinuous problems in a unified way. In contrast to the three-node triangular element (Trig3), the Trig3-CNS element has higher order of approximations, much better accuracy and continuous nodal stress. Moreover, it is free from the “linear dependence” which otherwise cripples many of the partition of unity based methods with high order approximations. The purpose of the present work is to synergize the advantages of both the recently developed Trig3-CNS element and the NMM to precisely model two-dimensional linear elastic fracture problems. A number of numerical examples indicate the accuracy and robustness of the present Trig3-CNS (NMM) element.
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paper_title: Mixed mode fracture propagation by manifold method

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The numerical manifold method combined with the virtual crack extension method is proposed to study the mixed mode fracture propagation. The manifold method is a new numerical method, and it provides a unified framework for solving problems dealing with both continuums and jointed materials. This new method can be considered as a generalized finite element method and discontinuous deformation analysis. One of the most innovative features of the method is that it employs both physical mesh and mathematical mesh to formulate the physical problem. These two meshes are separated and independent. They are inter-related through the application of weighting functions. A local mesh refinement and auto-remeshing schemes previously proposed by the authors are adopted in this study. The proposed model is first verified by comparing the numerical stress intensity factors with the benchmark solutions, and the results show satisfactory accuracy. The maximum tangential stress criterion is adopted and the mixed mode fracture propagation problems are then fully investigated. The numerical solutions by the present method agree well with the experimental results.
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paper_title: A rigorous and unified mass lumping scheme for higher-order elements

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Abstract In dynamic analysis with explicit time integration schemes, a lumped mass matrix (LMM) is preferable, because LMM can avoid solving the large scale simultaneous algebraic equations. Mathematically rigorous mass lumping schemes, such as the mass lumping by nodal quadrature and the row-sum technique, are applicable to only linear or bilinear elements. For higher-order elements, such as 8-node serendipity elements, the diagonal scaling procedure is the only lumping method that can be recommended to generate positive definite diagonal element mass matrices. Unfortunately, there is no mathematical theory in support of this approach. This study proposes a general mass lumping scheme applicable to higher order elements, where the virtual work of initial force is integrated over the problem domain that is viewed as the manifold covered by the finite element patches. By a series of numerical experiments, both free and forced vibration problems, it is suggested that even in the implicit time integration scheme the consistent mass matrix (CMM) can be superseded by the proposed LMM. Furthermore, the proposed LMM has much stronger adaptability to distorted meshes.
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paper_title: Particle manifold method (PMM): A new continuum-discontinuum numerical model for geomechanics

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Particle manifold method (PMM) is a new extension of the numerical manifold method (NMM). PMM uses a mathematical cover system to describe the motion and deformation of a particle-based physical domain. By introducing the concept of particle into NMM, PMM takes the advantages of easy topological and contact operations with particles. In this article, the methodology, formulations and implementation of the method are presented, together with modelling examples for validation. It is found that good solutions for both continuous and discontinuous problems are obtained by the new developed PMM. Due to the underlying coupled continuum-discontinuum property of PMM, it has great potential for modelling of geomechanical problems. Copyright (c) 2012 John Wiley & Sons, Ltd.
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paper_title: Particle Manifold Method (PMM) for Multiscale Continuous-Discontinuous Analysis

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Keywords: particle manifold method (PMM) ; continuous and discontinuous analysis ; wave propagation and transmission ; particle contact ; micro failure mechanism ; GPU parallelization ; CUDA ; multiscale analysis ; rock dynamics ; loading rate effect These Ecole polytechnique federale de Lausanne EPFL, n° 5515 (2012)Programme doctoral StructuresFaculte de l'environnement naturel, architectural et construitInstitut d'ingenierie civileLaboratoire de mecanique des rochesJury: J.-F. Molinari (president), H. Konietzky, L. Laloui, A. Munjiza Public defense: 2012-10-5 Reference doi:10.5075/epfl-thesis-5515Print copy in library catalog Record created on 2012-09-27, modified on 2016-08-09
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paper_title: A multiscale manifold method using particle representations of the physical domain

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Multiscale modelling has great advantages by simulating material behaviours on one level using information from another level. The popular multiscale methodology couples the atomistic/meso particle model with the continuum model (e.g., multiscale model by coupling molecular dynamics and finite element analysis). In this paper, a new multiscale method is proposed using the numerical manifold method (NMM) and its micro extension particle manifold method (PMM). Due to the same mathematical and mechanical scheme NMM and PMM adopt, their multiscale coupling has a uniform framework and does not need special treatment at the multiscale boundary. The proposed method has an advantage over PMM in presenting the material’s micro structure and failure description. It also saves the computational resource by using NMM to present the macro part. In the paper, the multiscale methodology is presented with some numerical demonstrations. It is proven that this novel multiscale manifold method is a good tool for analysis of...
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paper_title: Specific Energy as an Index to Identify the Critical Failure Mode Transition Depth in Rock Cutting

paper_content:

Rock cutting typically involves driving a rigid cutter across the rock surface at certain depth of cut and is used to remove rock material in various engineering applications. It has been established that there exist two distinct failure modes in rock cutting, i.e. ductile mode and brittle mode. The ductile mode takes precedence when the cut is shallow and the increase in the depth of cut leads to rock failure gradually shifted to brittle-dominant mode. The threshold depth or the critical transition depth, at which rock failure under cutting changes from the ductile to the brittle mode, is associated with not only the rock properties but also the cutting operational parameters and the understanding of this threshold is important to optimise the tool design and operational parameters. In this study, a new method termed the specific cutting energy transition model is proposed from an energy perspective which is demonstrated to be much more effective in identifying the critical transition depth compared with existing approaches. In the ductile failure cutting mode, the specific cutting energy is found to be independent of the depth of cut; but in the brittle failure cutting mode, the specific cutting energy is found to be dependent on the depth of cut following a power-law relationship. The critical transition depth is identified as the intersection point between these two relationships. Experimental tests on two types of rocks with different combinations of cutting velocity, depth of cut and back rake angle are conducted and the application of the proposed model on these cutting datasets has demonstrated that the model can provide a very effective tool to analyse the cutting mechanism and to identify the critical transition depth.
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paper_title: Finite element modeling of rock cutting and its fragmentation process

paper_content:

Abstract Rock cutting is a challenging problem from a modeling perspective. The challenges come from the complexity of the physics from the tool–rock interaction to the fracture process and propagation of the quasibrittle rocks. This study was aimed at developing a finite element procedure that was capable of providing reasonable estimates of cutting forces and, at the same time, capturing the essential characteristics of the fragmentation process. Published laboratory rock scratch tests were used as modeling targets since these tests encompass all essential characteristics of rock cutting. Both shallow cuts and deep cuts from a rectangular cutter were analyzed first, followed by modeling of shallow cuts from a disc cutter. It was concluded that rock cutting could be reasonably modeled by using a plasticity-damage model, an element erosion scheme that removes an element when its energy release equals fracture energy, together with a proper selection of modeling parameters.
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paper_title: Dynamic tensile behaviours of heterogeneous rocks: The grain scale fracturing characteristics on strength and fragmentation

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Abstract The dynamic tension behaviours of granites are tested with the split Hopkinson pressure bar and the quasi-static responses including the compression and Brazilian splitting are carried out with a material testing system. The experimental results show the tensile strengths behave significant strain rate effect. In order to characterize the realistic fracturing process from the viewpoint of grain scale failure, a multiple scale discrete element model considering the micro heterogeneity is proposed using the digital image processing of mineral scanning for rocks. Comparison of the experimental and numerical tension stress as well as the ultimate fragment state indicates the grain-based model is reasonable in simulation of dynamic tension test on granites. Then the three-wave superposition, crack propagation sequences, end forces and the stress distribution are discussed to confirm the stress equilibrium in the specimen. Using the microheterogeneous model, the micro fracturing process and fragmentation in association with energy dissipation at different strain rates are discussed. It is found that the failure sequence can be divided into five stages as crack initiation, propagation, coalesce, branching and indentation crush on the stress curve in dynamic loading. The intrinsic mechanism of the strain rate effect is believed to be the transitions of the micro fracturing type, orientation and the damage degree in the specimen and in turn exhibiting more energy dissipation as well as fragmentation transition from sparse fracture to pervasive pulverization. Finally, the scaling model of the dynamic increase factor for granite is derived and the characteristic strain rate, increase rate factor values are discussed.
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paper_title: Numerical simulation on rockburst of underground opening triggered by dynamic disturbance

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The dynamic disturbance, which is termed as the time-dependent loading such as explosion, vibration, stress impact from neighboring rockbursts, earthquakes, may trigger the rockbursts around the underground opening at depth. A numerical model capable of studying the dynamic failure process of rock under coupled static geo-stress and dynamic disturbance is proposed, and it is implemented into the Rock Failure Process Analysis (RFPA), a general finite element package to analyze the damage and failure process of engineering materials such as rock and concrete. Based on the consideration of the static geo-stress, the RFPA-Dynamics is used to simulate the rockburst that is deemed to be triggered by dynamic disturbance around the deep underground opening. The effect of lateral pressure coefficient and dynamic disturbance waveform on the development of failure zone around the underground opening is numerically simulated. The numerical results indicate that the dynamic disturbance is one of the most important mechanisms that trigger the rockbursts around underground opening. Therefore, it is of theoretical and practical significance to investigate the effect of dynamic disturbance on the rockbursts of underground opening, especially for the underground excavation at depth where the surrounding rockmass is highly stressed. The numerical results also reveal that the contribution of the dynamic disturbances is closely pertinent to both the static geo-stress condition and the waveforms of the dynamic disturbance. In general, dynamic disturbance brings about the greater influences on the stability of underground opening with its increasing magnitude and prolonged duration. However, with regard to the specific static geo-stress condition and characteristics of dynamic disturbance, the contribution of dynamic disturbance to trigger the rockbursts must be examined based on numerical analysis according to the specific geo-stress conditions and characteristics of the dynamic disturbance.
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paper_title: Reproducing kernel particle methods for structural dynamics

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This paper explores a Reproducing Kernel Particle Method (RKPM) which incorporates several attractive features. The emphasis is away from classical mesh generated elements in favour of a mesh free system which only requires a set of nodes or particles in space. Using a Gaussian function or a cubic spline function, flexible window functions are implemented to provide refinement in the solution process. It also creates the ability to analyse a specific frequency range in dynamic problems reducing the computer time required. This advantage is achieved through an increase in the critical time step when the frequency range is low and a large window is used. The stability of the window function as well as the critical time step formula are investigated to provide insight into RKPMs. The predictions of the theories are confirmed through numerical experiments by performing reconstructions of given functions and solving elastic and elastic–plastic one-dimensional (1-D) bar problems for both small and large deformation as well as three 2-D large deformation non-linear elastic problems. Numerical and theoretical results show the proposed reproducing kernel interpolation functions satisfy the consistency conditions and the critical time step prediction; furthermore, the RKPM provides better stability than Smooth Particle Hydrodynamics (SPH) methods. In contrast with what has been reported in SPH literature, we do not find any tensile instability with RKPMs.
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paper_title: A numerical approach to the testing of the fission hypothesis

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A finite-size particle scheme for the numerical solution of two- and three-dimensional gas dynamical problems of astronomical interest is described and tested. The scheme is then applied to the fission problem for optically thick protostars. Results are given, showing the evolution of one such protostar from an initial state as a single, rotating star to a final state as a triple system whose components contain 60% of the original mass. The decisiveness of this numerical test of the fission hypothesis and its relevance to observed binaries are briefly discussed.
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paper_title: A bonded-particle model for rock

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A numerical model for rock is proposed in which the rock is represented by a dense packing of non-uniform-sized circular or spherical particles that are bonded together at their contact points and whose mechanical behavior is simulated by the distinct-element method using the two- and three-dimensional discontinuum programs PFC2D and PFC3D. The microproperties consist of stiffness and strength parameters for the particles and the bonds. Damage is represented explicitly as broken bonds, which form and coalesce into macroscopic fractures when load is applied. The model reproduces many features of rock behavior, including elasticity, fracturing, acoustic emission, damage accumulation producing material anisotropy, hysteresis, dilation, post-peak softening and strength increase with confinement. These behaviors are emergent properties of the model that arise from a relatively simple set of microproperties. A material-genesis procedure and microproperties to represent Lac du Bonnet granite are presented. The behavior of this model is described for two- and three-dimensional biaxial, triaxial and Brazilian tests and for two-dimensional tunnel simulations in which breakout notches form in the region of maximum compressive stress. The sensitivity of the results to microproperties, including particle size, is investigated. Particle size is not a free parameter that only controls resolution; instead, it affects the fracture toughness and thereby influences damage processes (such as notch formation) in which damage localizes at macrofracture tips experiencing extensile loading.
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paper_title: A review of discrete modeling techniques for fracturing processes in discontinuous rock masses

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Abstract The goal of this review paper is to provide a summary of selected discrete element and hybrid finite–discrete element modeling techniques that have emerged in the field of rock mechanics as simulation tools for fracturing processes in rocks and rock masses. The fundamental principles of each computer code are illustrated with particular emphasis on the approach specifically adopted to simulate fracture nucleation and propagation and to account for the presence of rock mass discontinuities. This description is accompanied by a brief review of application studies focusing on laboratory-scale models of rock failure processes and on the simulation of damage development around underground excavations.
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Title: An Overview of Particle-based Numerical Manifold Method and Its Application to Dynamic Rock Fracturing

Section 1: Introduction
Description 1: Provide an introduction to dynamic rock fracturing and the importance of numerical methods in studying the mechanisms of rock material behavior at the microscopic scale.

Section 2: Fundamental principles
Description 2: Discuss the fundamental principles of the numerical manifold method (NMM), including its dual-cover system and various key features.

Section 3: Recent development and applications
Description 3: Review recent developments and applications of NMM, highlighting improvements in element modeling and boundary conditions and their application to various rock mechanics problems.

Section 4: Particle-based numerical manifold method
Description 4: Introduce the particle-based numerical manifold method (PNMM), detailing how particle concepts are integrated into NMM to simplify geometrical and contact operations and enhance the model.

Section 5: Particle manifold method
Description 5: Elaborate on the particle manifold method (PMM) as a precursor to PNMM, explaining its approach to reduce operational complexity and include particle-to-particle contacts.

Section 6: Applications
Description 6: Discuss various applications of PNMM in dynamic fracturing of rock, including validation studies and parametric analyses of rock behaviors under different conditions.

Section 7: Discussion
Description 7: Compare PNMM with other particle-based models, highlight its unique features, and identify areas for improvement and future research directions.

Section 8: Concluding remarks
Description 8: Summarize the strengths and limitations of PNMM, concluding its potential and application in advanced rock dynamic modeling.

Section 9: Conflicts of interest
Description 9: Declare any potential conflicts of interest and note the absence of significant financial support that could have influenced the study.