A continuum mechanics based framework for optimizing boundary and finite element meshes associated with underground excavations—accuracy,efficiency and applications

A framework was developed to address the automatic optimization of the level of geometric detail required for stress analysis of underground excavations in mining, which was presented in the companion paper. The motivation for optimizing the mesh geometry stems from the over‐discretization of computational domain as the digital mine model is built while our knowledge of some of the input parameters is quite limited. Thus, the accuracy of the solution is not expected to be increased with a finely discretized mesh, only the computation time does. Therefore, it is acceptable if the results obtained from an optimized model have accuracy comparable to the uncertainty in input data (e.g. rock mass properties, geology, etc.). Although the mesh optimization framework automates the geometry optimization and reduces computation time, the accuracy of the solution from the resulting geometry must be evaluated to ensure the quality of the solution at the ‘region of interest’. Both a priori (mesh quality) and a posteriori (solution quality) measures are employed along with recording the mesh optimization time. Finally, the applicability of the mesh optimization framework is demonstrated by analysing a number of mining and civil engineering underground models. Copyright © 2005 John Wiley & Sons, Ltd.     

A continuum mechanics‐based framework for optimizing boundary and finite element meshes associated with underground excavations‐framework

Many field problems, from stress analysis, heat transfer to contaminant transport, deal with disturbances in a continuum caused by a ‘source’ (defined by its discrete geometry) and a ‘region of interest’ (where a solution is sought). Depending on the location of ‘regions of interest’ in relation to the ‘sources’, the level of geometric detail necessary to represent the ‘sources’ in a model can vary considerably. A practical application of stress analysis in mining is the evaluation of the effects of continuous excavation on the states of stress around mine openings. Labour intensive model preparation and lengthy computation coupled with the interpretation of analysis results can have considerable impact on the successful operation of an underground mine, where stope failures can cost tens of millions of dollars and possibly lead to closure of the mine. A framework is proposed based on continuum mechanics principles to automatically optimize the level of geometric detail required for an analysis by simplifying the model geometry using expanded and modified algorithms that originated in computer graphics. This reduction in model size directly translates to savings in computational time. The results obtained from an optimized model have accuracy comparable to the uncertainty in input data (e.g. rock mass properties, geology, etc.). This first paper defines the optimization framework, while a companion paper investigates its efficiency and application to practical mining and excavation‐related problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Zero thickness interface elements—numerical stability and application

Many methods have been proposed to model joints in rocks or the interface between soil and a structure. Many analysts have reported numerical problems when using zero thickness interface elements while others have presented satisfactory results without comment of such difficulties. The numerical behaviour of zero thickness interface elements is further investigated in this paper. Some simple examples illustrate the application of interface elements to practical situations and highlight the numerical difficulties that may be encountered. Both ill-conditioning of the stiffness matrix and high stress gradients were found to cause numerical instability. Ill-conditioning can be reduced by careful selection of the size of the 2D elements adjacent to the interface. The problem of steep stress gradients is entirely one of inadequate mesh design. Contrary to other reports, this paper shows that the Newton–Cotes integration scheme has no benefit over Gaussian integration. Analyses of a retaining wall using interface elements confirm the analytical values of active and passive earth pressure coefficients which are commonly used in analysis and design of retaining walls.     

A continuum mechanics‐based framework for boundary and finite element mesh optimization in two dimensions for application in excavation analysis

The determination of the optimum excavation sequences in mining and civil engineering using numerical stress analysis procedures requires repeated solution of large models. Often such models contain much more complexity and geometric detail than required to arrive at an accurate stress analysis solution, especially considering our limited knowledge of rock mass properties. This paper develops an automated framework for estimating the effects of excavations at a region of interest, and optimizing the geometry used for stress analysis. It eliminates or simplifies the excavations in a model while maintaining the accuracy of analysis results. The framework can equally be applied to two‐dimensional boundary and finite element models. The framework will have the largest impact for non‐linear finite element analysis. It can significantly reduce computational times for such analysis by simplifying models. Error estimators are used in the framework to assess accuracy. The advantages of applying the framework are demonstrated on an excavation‐sequencing scenario. Copyright © 2005 John Wiley & Sons, Ltd.     

An efficient finite–discrete element method for quasi‐static nonlinear soil–structure interaction problems

An efficient finite–discrete element method applicable for the analysis of quasi‐static nonlinear soil–structure interaction problems involving large deformations in three‐dimensional space was presented in this paper. The present method differs from previous approaches in that the use of very fine mesh and small time steps was not needed to stabilize the calculation. The domain involving the large displacement was modeled using discrete elements, whereas the rest of the domain was modeled using finite elements. Forces acting on the discrete and finite elements were related by introducing interface elements at the boundary of the two domains. To improve the stability of the developed method, we used explicit time integration with different damping schemes applied to each domain to relax the system and to reach stability condition. With appropriate damping schemes, a relatively coarse finite element mesh can be used, resulting in significant savings in the computation time. The proposed algorithm was validated using three different benchmark problems, and the numerical results were compared with existing analytical and numerical solutions. The algorithm performance in solving practical soil–structure interaction problems was also investigated by simulating a large‐scale soft ground tunneling problem involving soil loss near an existing lining. Copyright © 2011 John Wiley & Sons, Ltd.     

A computational framework to predict the water‐leakage pressure of segmental joints in underwater shield tunnels using an advanced finite element method

The construction of shield tunnels under riverbeds and seabeds has considerably increased over the past decades. Due to the ultra‐high water head, water leakage through tunnel joints is a major concern during a tunnel's service life. One practical solution to prevent groundwater penetration is to implement ethylene‐propylene‐diene‐monomer gaskets at the segmental joints. However, numerical simulation of fluid pressure penetration into rubber materials remains a challenging problem in computational mechanics. Severe mesh distortions can occur due to large deformation. Consequently, a convergent solution is difficult to achieve. This paper presents an Abaqus‐based numerical framework to solve the previously mentioned problem using the implicit finite element solver. The key aspects of this framework are twofold: (1) a remesh and re‐map algorithm to overcome the excessive mesh distortion, and (2) simulation of fluid penetration into the contact interface of the gaskets to reproduce the water‐leakage process at the tunnel joints. The proposed framework is first tested to simulate the gasket‐in‐groove mechanical behavior and is then validated using experimental data and the solution produced by an explicit finite element solver. The developed framework is then adopted to predict the water‐leakage pressure at gasketed tunnel joints to illustrate the practical applications. Finally, the numerical results are compared with experimental data to demonstrate the accuracy and robustness of the proposed method and confirm its superiority and effectiveness over existing methods. This novel method can be used by tunnel designers to analyze and estimate the waterproof behavior of gasketed joints in shield tunnels without performing extensive experimental testing works.     

三维稳定渗流的 p 型自适应有限元分析

建立了三维稳定渗流的p型自适应有限元分析的误差估计器。充分利用p型有限元前处理少、计算精度高和收敛速度快的优点，分析了p型有限元法求解渗流问题的技术特点，给出了求解三维稳定渗流的自适应升阶策略。通过具体算例，研究了单元阶次及网格尺度对计算结果的影响，验证了三维p型有限元法在求解渗流问题时的可行性。     

A mesh deformation algorithm for free surface problems

Employing the simple iterative technique of adjusting the element positions using computed potentials to locate the free surface can lead to finite elements with large aspect ratios as the free surface drops towards the base of the mesh. In particular, free surface modelling of earth dams with base drains suffer from this problem. The paper suggests a number of steps which can be taken to alleviate mesh distortion problems and improve the numerical stability of the iterative finite element analysis. This leads to a mesh deformation algorithm which adjusts element widths in a simple fashion depending on the free surface height as the iterations proceed. The algorithm is specialized to the sloped earth dam problem, but may find application to other geometries. © 1997 by John Wiley & Sons, Ltd.     

基于Ansys的渗流自由面计算方法

对渗流自由面计算的单元传导矩阵调整法进行了改进。在局部坐标系下,将单元离散为一系列面元（三维为体元）,根据面元或体元形心处的压力水头值来确定面元或体元的权系数,然后按加权平均确定单元的等效渗透系数,该方法既可以避免复合单元中自由面上下面积或体积的计算,面元或体元的离散密度也可以任意选择,简化了计算程序。通过预设的收敛参数来控制迭代过程,可避免相邻两次计算的渗透系数变化太大而难于收敛。以Ansys为开发平台,用APDL语言编写了相应的计算程序,扩展了Ansys的计算功能,为渗流自由面计算提供了新途径。算例表明：该方法的收敛性较好、计算精度较高,能满足工程需要。     

Kinematic modelling of shear band localization using discrete finite elements

Modelling shear band is an important problem in analysing failure of earth structures in soil mechanics. Shear banding is the result of localization of deformation in soil masses. Most finite element schemes are unable to model discrete shear band formation and propagation due to the difficulties in modelling strain and displacement discontinuities. In this paper, a framework to generate shear band elements automatically and continuously is developed. The propagating shear band is modelled using discrete shear band elements by splitting the original finite element mesh. The location or orientation of the shear band is not predetermined in the original finite element mesh. Based on the elasto‐perfect plasticity with an associated flow rule, empirical bifurcation and location criteria are proposed which make band propagation as realistic as possible. Using the Mohr–Coulomb material model, various results from numerical simulations of biaxial tests and passive earth pressure problems have shown that the proposed framework is able to display actual patterns of shear banding in geomaterials. In the numerical examples, the occurrence of multiple shear bands in biaxial test and in the passive earth pressure problem is confirmed by field and laboratory observations. The effects of mesh density and mesh alignment on the shear band patterns and limit loads are also investigated. Copyright © 2003 John Wiley & Sons, Ltd.     

基于局部单元劈裂的FEM/DEM自适应分析方法

为了模拟岩体中裂纹的萌生、扩展,Munjiza提出了有限元法/离散元法(FEM/DEM)耦合分析方法。因为裂纹是沿单元边界进行扩展的,亦即裂纹扩展具有网格依赖性,为获得较好的裂纹扩展形态,需要划分密集的初始网格。为解决上述难题,基于FEM/DEM耦合分析方法,提出了基于局部单元动态劈裂的FEM/DEM自适应分析方法,以克服裂纹扩展形态对网格的依赖性。该方法在最初建模时无需划分很密的初始网格,随着荷载的施加,对裂纹尖端附近的局部单元进行动态劈裂,为裂纹的后续扩展提供了更多可能的扩展方向,使得裂纹扩展不必沿着初始网格的单元边界扩展,即可以沿着单元内部进行扩展,裂纹扩展形态更为平滑,与实际情况更为接近。同时相对原FEM/DEM耦合分析方法一开始就划分很密的网格而言,新方法可以划分较为稀疏的初始网格,计算成本降低。最后,通过巴西劈裂算例与原FEM/DEM耦合分析方法对比,分析表明,新方法在一定程度上克服了裂纹扩展形态对初始网格的依赖性。     

基于FLAC3D和DEM数据的缓倾斜煤层开采沉陷分析

宁夏石嘴山矿区位于西部黄河流域,其煤矿采空区沉陷导致地表生态和环境问题频发,对其采煤沉陷分析将对西部黄河流域煤矿区的环境修复有一定的积极作用。为研究缓倾斜煤层采空区围岩应力与位移场演化特征,以宁夏石嘴山矿区为对象,基于FLAC3D数值模拟软件,建立缓斜煤层开采三维数值模型,计算分析采空区围岩应力、塑性区及位移变化规律,并基于两时相DEM叠加统计分析地表位移变化,与数值模拟结果进行相互验证。结果表明:地下开采引起应力重分布,采空区顶板及煤柱出现明显的应力集中现象,最大主应力呈现从煤层顶板向地表递减的变化趋势;越靠近采空区顶部的岩层垂直位移越大,随着远离采空区逐渐减少,开采完成后地表垂直位移最大值约12 m;随着采空区面积的不断增大,采空区四周及角隅处塑性区逐步延伸扩大,且以剪切破坏为主;地面沉陷盆地不对称,2个沉降中心均发生在沉陷盆地中部且偏下山方向,下山方向比上山方向影响范围更大;数值模拟计算的沉降量与两时相DEM叠加统计分析的变化量结果及趋势基本一致,研究成果可为煤炭安全开采提供参考依据,为地表沉降监测提供新方法。      

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 data exploration framework for validation and setup of hydrological models

Over the course of hydrological research projects often a large number of heterogeneous data sets are acquired from sources as diverse as boreholes, gauging stations or satellite imagery. This data then need to be integrated into models for the simulation of hydrological processes. We propose a framework for exploration of geoscientific data and visually guided preparation of such models. Data sets from a large number of sources can be imported, combined and validated to avoid potential problems due to artefacts or inconsistencies between data sets in a subsequent simulation. Boundary conditions and domain discretisations for surface and subsurface models can be created and tested regarding criteria indicating possible numerical instabilities. All data sets including simulation results can be integrated into a user-controlled 3D scene and aspects of the data can be enhanced using a number of established visualisation techniques including thresholding and user-defined transfer functions. We present the application of this framework for the preparation of a model for simulation of groundwater flow in a river catchment in southwest Germany investigated in the scope of the WESS project.     

基于CT扫描的土石混合体三维数值网格的建立

根据CT扫描得到土石混合体切面图像信息,通过二值化和砾石边界识别技术提取砾石表面点云数据,应用逆向工程软件重构砾石的三维模型。编写APDL代码,在ANSYS中快速生成土石混合体的几何模型,借助ANSYS ICEM CFD强大、灵活的网格划分功能,对砾-土界面处网格进行加密,得到土石混合体的三维数值网格,采用接口程序,将数值网格导入FLAC3D中进行模拟计算。通过与无砾石模型计算结果的对比,说明了建立表征土石混合体结构特征的精细三维数值网格可以更加准确地描述土体,特别是砾-土界面的受力情况和变形特征,从而提高土石混合体数值模拟的精确度及可靠度。     

模拟三维裂纹问题的扩展有限元法

扩展有限元法是一种在常规有限元框架内求解强和弱不连续问题的新型数值方法,其计算网格与不连续面相互独立,因此模拟移动不连续面时无需对网格进行重新剖分。给出了模拟三维裂纹问题的扩展有限元法。在常规有限元位移模式中,基于单位分解的思想加进一个阶跃函数和二维渐近裂尖位移场,反映裂纹处位移的不连续性。用两个水平集函数表示裂纹。采用线性互补法求解裂纹面非线性接触条件,不需要迭代,提高了计算效率。采用两点位移外推法计算裂纹前缘应力强度因子。给出了3个三维弹性静力问题算例,其结果显示了所提方法能获得高精度的应力强度因子,并能有效地处理裂纹面间的接触问题,同时表明扩展有限元结合线性互补法求解不连续问题具有较好的前景。     

A single numerically efficient equation for approximating the Mohr–Coulomb and the Matsuoka–Nakai failure criteria with rounded edges and apex

This paper presents a novel formulation for defining soil failure. It plots in the principal stress space as a surface with the shape ranging between an approximation of the Matsuoka–Nakai and of the Mohr–Coulomb criteria depending on the value of a single parameter. The new function can be used as a replacement of the original equations of these well‐established criteria for implementing in a program for numerical analyses, and it is particularly effective for approximating the Matsuoka–Nakai criterion. Both the Mohr–Coulomb and the Matsuoka–Nakai failure criteria present numerical difficulties during implementation and also at run‐time. In the case of the Matsuoka–Nakai, the new formulation plots in the first octant only, whereas the original criterion plots in all octants, which causes severe convergence problems particularly for those Gauss points with low stress state, such as those on the side of a shallow footing. When the shape parameter is set to reproduce the Mohr–Coulomb failure criterion, on the other hand, the new formulation plots as a pyramid with rounded edges. Moreover, as the new function is at least of class C², the second derivatives are continuous, thus ensuring quadratic convergence of the Newton's method used within the integration scheme of the constitutive law. The proposed formulation can also provide both sharp and rounded apex of the surface at the origin of the stress space by setting accordingly one additional parameter. Copyright © 2013 John Wiley & Sons, Ltd.     

A superposition scheme to obtain fundamental boundary element solutions in multi‐layered elastic media

A superposition scheme is proposed to obtain a fundamental solution for boundary elements in multi‐layered elastic media. A three‐layered elastic region is obtained by superposing two sets of bonded half‐planes and subtracting one infinite plane. Therefore, the solution for an element in a layered media can be expressed in terms of bonded half‐plane solutions and an infinite‐plane solution. The major advantages of this superposition scheme are: (1) it is unnecessary to introduce elements at the interface, (2) it can be extended to higher‐order element, and (3) it may be applicable to three dimensions easily. The accuracy and performance of the developed model is illustrated by two examples. For the problem of a pressurized two‐dimensional crack within a three‐layered system, the comparison with other numerical results shows the model is quite accurate and efficient. The model is also used for a study of a practical two‐dimensional mining problem in South Africa, i.e. stoping through a dyke with material properties different from the host rock. Copyright © 2000 John Wiley & Sons, Ltd.     

Effects of boundary conditions and partial drainage on cyclic simple shear test results—a numerical study

Owing to imperfect boundary conditions in laboratory soil tests and the possibility of water diffusion inside the soil specimen in undrained tests, the assumption of uniform stress/strain over the sample is not valid. This study presents a qualitative assessment of the effects of non‐uniformities in stresses and strains, as well as effects of water diffusion within the soil sample on the global results of undrained cyclic simple shear tests. The possible implications of those phenomena on the results of liquefaction strength assessment are also discussed. A state‐of‐the‐art finite element code for transient analysis of multi‐phase systems is used to compare results of the so‐called ‘element tests’ (numerical constitutive experiments assuming uniform stress/strain/pore pressure distribution throughout the sample) with results of actual simulations of undrained cyclic simple shear tests using a finite element mesh and realistic boundary conditions. The finite element simulations are performed under various conditions, covering the entire range of practical situations: (1) perfectly drained soil specimen with constant volume, (2) perfectly undrained specimen, and (3) undrained test with possibility of water diffusion within the sample. The results presented here are restricted to strain‐driven tests performed for a loose uniform fine sand with relative density D_r=40%. Effects of system compliance in undrained laboratory simple shear tests are not investigated here. Copyright © 2004 John Wiley & Sons, Ltd.