A framework for automatic modeling of underground excavations and optimizing three‐dimensional boundary and finite element meshes derived from them—framework

Many problems in mining and civil engineering require using numerical stress analysis methods to repeatedly solve large models. Widespread acceptance of tunneling methods, such as New Austrian Tunneling Method, which depend heavily on numerical stress analysis tools and the fact that the effects of excavation at the face of a tunnel are distinctively three–dimensional (3D), necessitates the use of 3D numerical analysis for these problems. Stress analysis of a practical mining problem can be very lengthy, and the processing time can be measured in days or weeks at times. A framework is developed to facilitate efficient modeling of underground excavations and to create an optimal 3D mesh by reducing the number of surface and volume elements while keeping the result of stress analysis accurate enough at the region of interest, where a solution is sought. Fewer surface and volume elements mean fewer degrees of freedom in the numerical model, which directly translates into savings in computational time and resources. The mesh refinement algorithm is driven by a set of criteria that are functions of distance and visibility of points from the region of interest, and the framework can be easily extended by adding new types of criteria. This paper defines the framework, whereas a second companion paper will investigate its efficiency, accuracy and application to a number of practical mining problems. Copyright © 2012 John Wiley & Sons, Ltd.     

A new stereo‐analytical method for determination of removal blocks in discontinuous rock masses

The paper provides a new stereo‐analytical method, which is a combination of the stereographic method and analytical methods, to separate finite removable blocks from the infinite and tapered blocks in discontinuous rock masses. The methodology has applicability to both convex and concave blocks. Application of the methodology is illustrated through examples. Addition of this method to the existing block theory procedures available in the literature improves the capability of block theory in solving practical problems in rock engineering. Copyright © 2003 John Wiley & Sons, Ltd.     

基于三维裂隙网络模拟的有限块体面积判断法

水工建筑物深开挖边坡中大量出现的随机楔体,他们是由广泛分布于裂隙岩体中的结构面切割而成。首先应用随机不连续面三维网络模拟技术建立不连续的概率模型,在此基础上,搜索出临空面迹线上所有的闭合凸多变形,根据文中提出的“面积判断”法,判断块体是否为有限块体。工程实例模拟的结果说明了该方法的合理性。     

Developing an algorithm for reconstruction blocky systems in discontinuous media: three‐dimensional analysis

Block detection is one of the important steps in all discontinuous methods of analysis such as discontinuous deformation analysis and discrete element method. It is in fact a pre‐processing step for these methods. This paper describes a new approach to the problem of geometrically defining polyhedral rock blocks created by the intersection of planar discontinuities in a rock mass. An approach is developed based on the concept of using matrices with integer elements that mostly represent vertices, edges, or face numbers and their connections. Using square matrices with integer elements and performing edge/face regularization reduce the size of the matrices because of elimination of unnecessary faces, edges, and vertices; speed and accuracy of block tracing operation will be increased. This algorithm is able to trace and identify all kinds of blocks including convex and concave blocks formed by limited or unlimited fractures. The simplicity of the procedure makes it very attractive. The algorithm was programmed in C#.Net by over 8100 code lines; several examples are presented to show application of the algorithm in different situations. Copyright © 2011 John Wiley & Sons, Ltd.     

Cauchy problem of three‐dimensional critical slip surfaces of slopes

Using the continuum‐based finite element approach in conjunction with the strength reduction technique, one can easily find out the factor of safety of a slope with rather high precision, but will encounter some obstacles in the accurate location of the corresponding three‐dimensional critical slip surface (CSS). On the basis of the Mohr–Coulomb failure criterion and the stress field in the limit equilibrium state of the slope, it is deduced that the three‐dimensional CSS is the solution of the Cauchy problem of a quasi‐linear partial differential equation (PDE) of first order. By means of the method of characteristics for the problem, the three‐dimensional CSS that may take any shape can be determined without the specification of its geometry. Copyright © 2010 John Wiley & Sons, Ltd.     

Semi‐analytical far field model for three‐dimensional finite‐element analysis

A challenging computational problem arises when a discrete structure (e.g. foundation) interacts with an unbounded medium (e.g. deep soil deposit), particularly if general loading conditions and non‐linear material behaviour is assumed. In this paper, a novel method for dealing with such a problem is formulated by combining conventional three‐dimensional finite‐elements with the recently developed scaled boundary finite‐element method. The scaled boundary finite‐element method is a semi‐analytical technique based on finite‐elements that obtains a symmetric stiffness matrix with respect to degrees of freedom on a discretized boundary. The method is particularly well suited to modelling unbounded domains as analytical solutions are found in a radial co‐ordinate direction, but, unlike the boundary‐element method, no complex fundamental solution is required. A technique for coupling the stiffness matrix of bounded three‐dimensional finite‐element domain with the stiffness matrix of the unbounded scaled boundary finite‐element domain, which uses a Fourier series to model the variation of displacement in the circumferential direction of the cylindrical co‐ordinate system, is described. The accuracy and computational efficiency of the new formulation is demonstrated through the linear elastic analysis of rigid circular and square footings. Copyright © 2004 John Wiley & Sons, Ltd.     

Investigation on strength and stability of jointed rock mass using three‐dimensional numerical manifold method

This paper proposes a numerical model for jointed rock masses within the 3‐D numerical manifold method (NMM) framework equipped with a customized contact algorithm. The strength of rock sample containing a few sets of discontinuities is first investigated. The results of models with simple geometries are compared with the available analytical solutions to verify the developed computer code, whereas models with complex geometries are simulated to better understand the fundamental behavior and failure mechanism of jointed rock mass. Furthermore, the stability of jointed rock mass in an underground excavation is studied, where rock failure process is determined by the 3‐D NMM simulation. The simulation results provide valuable guidance on excavation process design and stabilization design in rock engineering practice. Copyright © 2012 John Wiley & Sons, Ltd.     

A three‐dimensional staggered finite element approach for random parametric modeling of thermo‐hygral coupled phenomena in porous media

The aim of this paper is to present a three‐dimensional (3D) finite element modeling of heat and mass transfer phenomena in partially saturated open porous media with random fields of material properties. Randomness leads to transfer processes within the porous medium that naturally need a full 3D modeling for any quantitative assessment of these processes. Nevertheless, the counterpart of 3D modeling is a significant increase in computations cost. Therefore, a staggered solution strategy is adopted which permits to solve the equations sequentially. This appropriate partitioning reduces the size of the discretized problem to be solved at each time step. It is based on a specific iterative algorithm to account for the interaction between all the transfer processes. Accordingly, a suitable linearization of mass convective boundary conditions, consistent with the staggered algorithm, is also derived. After some validation tests, the 3D numerical model is used for studying the drying process of a cementitious material with regard to its intrinsic permeability randomness. Copyright © 2011 John Wiley & Sons, Ltd.     

Implementation of the finite element method in the three‐dimensional discontinuous deformation analysis (3D‐DDA)

A modified three‐dimensional discontinuous deformation analysis (3D‐DDA) method is derived using four‐noded tetrahedral elements to improve the accuracy of current 3D‐DDA algorithm in practical applications. The analysis program for the modified 3D‐DDA method is developed in a C++ environment and its accuracy is illustrated through comparisons with several analytical solutions that are available for selected problems. The predicted solutions for these problems using the modified 3D‐DDA approach all show satisfactory agreement with the corresponding analytical results. Results presented in this paper demonstrate that the modified 3D‐DDA method with discontinuous modeling capabilities offers a useful computational tool to determine stresses and deformations in practical problems involving fissured elastic media with reasonable accuracy. Copyright © 2008 John Wiley & Sons, Ltd.     

A three‐phase thermo‐hydro‐mechanical finite element model for freezing soils

Artificial ground freezing (AGF) is a commonly used technique in geotechnical engineering for ground improvement such as ground water control and temporary excavation support during tunnel construction in soft soils. The main potential problem connected with this technique is that it may produce heave and settlement at the ground surface, which may cause damage to the surface infrastructure. Additionally, the freezing process and the energy needed to obtain a stable frozen ground may be significantly influenced by seepage flow. Evidently, safe design and execution of AGF require a reliable prediction of the coupled thermo‐hydro‐mechanical behavior of freezing soils. With the theory of poromechanics, a three‐phase finite element soil model is proposed, considering solid particles, liquid water, and crystal ice as separate phases and mixture temperature, liquid pressure, and solid displacement as the primary field variables. In addition to the volume expansion of water transforming into ice, the contribution of the micro‐cryo‐suction mechanism to the frost heave phenomenon is described in the model using the theory of premelting dynamics. Through fundamental physical laws and corresponding state relations, the model captures various couplings among the phase transition, the liquid transport within the pore space, and the accompanying mechanical deformation. The verification and validation of the model are accomplished by means of selected analyses. An application example is related to AGF during tunnel excavation, investigating the influence of seepage flow on the freezing process and the time required to establish a closed supporting frozen arch. Copyright © 2013 John Wiley & Sons, Ltd.     

Analysis of soil‐structural interface behavior using three‐dimensional DEM simulations

The shear behavior at the interface between the soil and a structure is investigated at the macroscale and particle‐scale levels using a 3‐dimensional discrete element method (DEM). The macroscopic mechanical properties and microscopic quantities affected by the normalized interface roughness and the loading parameters are analyzed. The macro‐response shows that the shear strength of the interface increases as the normalized roughness of the interface increases, and stress softening and dilatancy of the soil material are observed in the tests that feature rough interfaces. The particle‐scale analysis illustrates that a localized band characterized by intense shear deformation emerges from the contact plane and gradually expands as shearing progresses before stabilizing at the residual stress state. The thickness of the localized band is affected by the normalized roughness of the interface and the normal stress, which ranges between 4 and 5 times that of the median grain diameter. A thicker localized band is formed when the soil has a rough shearing interface. After the localized band appears, the granular material structuralizes into 2 regions: the interface zone and the upper zone. The mechanical behavior in the interface zone is representative of the interface according to the local average stress analysis. Certain microscopic quantities in the interface zone are analyzed, including the coordination number and the material fabric. Shear at the interface creates an anisotropic material fabric and leads to the rotation of the major principal stress.     

Generalized contact model for polyhedra in three‐dimensional discontinuous deformation analysis

We present a generalized contact computation model for arbitrarily shaped polyhedra to simplify the contact analysis in discontinuous deformation analysis. A list of generalized contact constraints can be established for contacting polyhedra during contact detection. Each contact constraint contains information for 2 contact points, unique contact plane, and related contact modes (open, locked, or sliding). Computational aspects of the generalized contact model include identification of contact positions and contact modes, uniform penalty formulation of generalized contact constraint, and uniform updating of contact modes and contact planes in the open‐close iteration. Compared with previous strategies, the generalized contact computation model has a simpler data structure and fewer memory requirements. Meanwhile, it simplifies the penalty formulation and facilitates the open‐close iteration check while producing enough accuracy. Illustrative examples show the ability of the method to handle the full range of polyhedral shapes.     

A two‐dimensional coupled hydromechanical discontinuum model for simulating rock hydraulic fracturing

Within the framework of our discontinuous deformation analysis for rock failure algorithm, this paper presents a two‐dimensional coupled hydromechanical discontinuum model for simulating the rock hydraulic fracturing process. In the proposed approach, based on the generated joint network, the calculation of fluid mechanics is performed first to obtain the seepage pressure near the tips of existing cracks, and then the fluid pressure is treated as linearly distributed loads on corresponding block boundaries. The contribution of the hydraulic pressure to the initiation/propagation of the cracks is considered by adding the components of these blocks into the force matrix of the global equilibrium equation. Finally, failure criteria are applied at the crack tips to determine the occurrence of cracking events. Several verification examples are simulated, and the results show that this newly proposed numerical model can simulate the hydraulic fracturing process correctly and effectively. Although the numerical and experimental verifications focus on one unique preexisting crack, because of the capability of discontinuous deformation analysis in simulating block‐like structures, the proposed approach is capable of modeling rock hydraulic fracturing processes. Copyright © 2014 John Wiley & Sons, Ltd.     

An integral equation solution for three‐dimensional heat extraction from planar fracture in hot dry rock

In the numerical simulation of heat extraction by circulating water in a fracture embedded in geothermal reservoir, the heat conduction in the reservoir is typically assumed to be one‐dimensional and perpendicular to the fracture in order to avoid the discretization of the three‐dimensional reservoir geometry. In this paper we demonstrate that by utilizing the integral equation formulation with a Green's function, the three‐dimensional heat flow in the reservoir can be modelled without the need of discretizing the reservoir. Numerical results show that the three‐dimensional heat conduction effect can significantly alter the prediction of heat extraction temperature and the reservoir life as compared to its one‐dimensional simplification. Copyright © 2003 John Wiley & Sons, Ltd.     

Sarma‐based key‐group method for rock slope reliability analyses

The methods used in conducting static stability analyses have remained pertinent to this day for reasons of both simplicity and speed of execution. The most well‐known of these methods for purposes of stability analysis of fractured rock masses is the key‐block method (KBM). This paper proposes an extension to the KBM, called the ‘key‐group method’ (KGM), which combines not only individual key‐blocks but also groups of collapsable blocks into an iterative and progressive analysis of the stability of discontinuous rock slopes. To take intra‐group forces into account, the Sarma method has been implemented within the KGM in order to generate a Sarma‐based KGM, abbreviated ‘SKGM’. We will discuss herein the hypothesis behind this new method, details regarding its implementation, and validation through comparison with results obtained from the distinct element method. Furthermore, as an alternative to deterministic methods, reliability analyses or probabilistic analyses have been proposed to take account of the uncertainty in analytical parameters and models. The FOSM and ASM probabilistic methods could be implemented within the KGM and SKGM framework in order to take account of the uncertainty due to physical and mechanical data (density, cohesion and angle of friction). We will then show how such reliability analyses can be introduced into SKGM to give rise to the probabilistic SKGM (PSKGM) and how it can be used for rock slope reliability analyses. Copyright © 2005 John Wiley & Sons, Ltd.     

Understanding heavy mineral enrichment using a three‐dimensional numerical model

Layered deposits of relatively light and heavy minerals can be found in many aquatic environments. Quantification of the physical processes which lead to the fine‐scale layering of these deposits is often limited with flumes or in situ field experiments. Therefore, the following research questions were addressed: (i) how can selective grain entrainment be numerically simulated and quantified; (ii) how does a mixed bed turn into a fully layered bed; and (iii) is there any relation between heavy mineral content and bed stability? Herein, a three‐dimensional numerical model was used as an alternative measure to study the fine‐scale process of density segregation during transport. The three‐dimensional model simulates particle transport in water by combining a turbulence‐resolving large eddy simulation with a discrete element model prescribing the motion of individual grains. The granular bed of 0·004 m in height consisted of 200 000 spherical particles (D50 = 500 μ m). Five suites of experiments were designed in which the concentration ratio of heavy (5000 kg m⁻³) to light particles (i.e. 2560 kg m⁻³) was increased from 6%, 15%, 35%, 60% to 80%. All beds were tested for 10 sec at a predefined flow speed of 0·3 m sec⁻¹. Analysis of the particle behaviour in the interior of the beds showed that the lighter particles segregated from the heavy particles with increasing time. The latter accumulated at the bottom of the domain, forming a layer, whereas the lighter particles were transported over the layer forming sweeps. Particles below the heavy particle layer indicated that the layer was able to armour the particles below. Consequentially, enrichment of heavy minerals in a layer is controlled by the segregation of a heavy mineral fraction from the light counterpart, which enhances current understanding of heavy mineral placer formation.     

Effects of non‐uniform traction and specimen height in the direct shear test on stress and deformation in a rock fracture

In the direct shear test (DST), an internal moment is distributed within the rock specimen by non‐coaxial shear loads applied to the specimen, which cause non‐uniform distributions of both the traction on the loading planes and the stress and deformation in the specimen. To examine the validity of the DST for a rock fracture and to clarify the effect of specimen height, both the stress and deformation in a fracture in the DST were analyzed for specimens with three different heights using a three‐dimensional finite element method with quadratic joint elements for a fracture model. The constitutive law of the fracture considers the dependence of the non‐linear behavior of closure on shear displacement and that of shear stiffness on normal stress and was implemented in simulation code to give a conceptional fracture with uniform mechanical properties to extract only the effect of non‐uniform traction on the stress and deformation in the fracture. The results showed that both normal and shear stresses are concentrated near the end edges of the fracture, and these stress concentrations decrease with a decrease in the specimen height according to the magnitude of the moment produced by the non‐coaxial shear loads. Furthermore, although closure is greater near the end edges of the fracture, where normal stress is concentrated, this concentration of closure is not so significant within the range of this study because of the non‐linear behavior of closure, that is, closure does not significantly increase with an increase in normal stress at large normal stresses. Copyright © 2012 John Wiley & Sons, Ltd.     

Semi‐analytical elastostatic analysis of unbounded two‐dimensional domains

Unbounded plane stress and plane strain domains subjected to static loading undergo infinite displacements, even when the zero displacement boundary condition at infinity is enforced. However, the stress and strain fields are well behaved, and are of practical interest. This causes significant difficulty when analysis is attempted using displacement‐based numerical methods, such as the finite‐element method. To circumvent this difficulty problems of this nature are often changed subtly before analysis to limit the displacements to finite values. Such a process is unsatisfactory, as it distorts the solution in some way, and may lead to a stiffness matrix that is nearly singular. In this paper, the semi‐analytical scaled boundary finite‐element method is extended to permit the analysis of such problems without requiring any modification of the problem itself. This is possible because the governing differential equations are solved analytically in the radial direction. The displacement solutions so obtained include an infinite component, but relative motion between any two points in the unbounded domain can be computed accurately. No small arbitrary constants are introduced, no arbitrary truncation of the domain is performed, and no ill‐conditioned matrices are inverted. Copyright © 2002 John Wiley & Sons, Ltd.     

Coupled formulation and algorithms for the simulation of non‐planar three‐dimensional hydraulic fractures using the generalized finite element method

This paper presents a coupled hydro‐mechanical formulation for the simulation of non‐planar three‐dimensional hydraulic fractures. Deformation in the rock is modeled using linear elasticity, and the lubrication theory is adopted for the fluid flow in the fracture. The governing equations of the fluid flow and elasticity and the subsequent discretization are fully coupled. A Generalized/eXtended Finite Element Method (G/XFEM) is adopted for the discretization of the coupled system of equations. A Newton–Raphson method is used to solve the resulting system of nonlinear equations. A discretization strategy for the fluid flow problem on non‐planar three‐dimensional surfaces and a computationally efficient strategy for handling time integration combined with mesh adaptivity are also presented. Several three‐dimensional numerical verification examples are solved. The examples illustrate the generality and accuracy of the proposed coupled formulation and discretization strategies. Copyright © 2015 John Wiley & Sons, Ltd.     

An approach to the modeling of viscoelastic damage. Application to the long‐term creep of gypsum rock materials

A three‐dimensional phenomenological model is developed to describe the long‐term creep of gypsum rock materials. The approach is based on the framework of continuum damage mechanics where coupling with viscoelasticity is adopted. Specifically, a local damage model based on the concept of yield surface is proposed and deeply investigated. Among the many possibilities, we choose in this work its coupling with a generalized Kelvin–Voigt rheological model to formulate the whole behavior. Long‐term as well as short‐term relaxation processes can be integrated in the model by means of as many as necessary viscoelastic processes. The numerical discretization is described for an easy integration within a finite element procedure. Finally, a set of numerical simulations is given to show the possibilities of the presented model. It shows good agreement with some experimental results found in the literature. Copyright © 2012 John Wiley & Sons, Ltd.