Computation of the safety factor for slope stability using discontinuous deformation analysis and the vector sum method

This study introduces the vector sum method into discontinuum-based methods by considering the sliding vector and the stress state of the discrete block system. The sliding direction computation and force projection in the new approach are detailed, and the safety factor is solved by explicit equations. The vector sum method is implemented in the discontinuous deformation analysis (DDA) program and is used to compute the safety factors for two numerical examples. A comparison of the solutions obtained with the theoretical analysis and limit equilibrium analysis demonstrates that the new method is suitable for calculating the safety factor of a slope.     

Comparative study of Sarma's method and the discontinuous deformation analysis for rock slope stability analysis

This paper presents a comparative study of two methods, Sarma's method and the discontinuous deformation analysis (DDA), for rock slope stability analysis. The comparison concerns the stability analysis of two classic rock slopes. The study shows that the DDA, which accounts for the block kinematics, provides a very different factor of safety as compared with Sarma's method. More realistic reaction forces around each rock block can be obtained by the DDA, including the thrust forces between rock blocks and the forces between the base and the blocks. The DDA's result shows two possible directions for the relative movement between two contiguous blocks at the initiation of slope failure. It also indicates that the limit equilibrium condition may not occur along the interfaces of rock blocks at the initiation of slope failure. The determination of realistic interaction forces around each block will be very important in rock slope stability analysis if nonlinear failure criteria are considered.     

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.     

基于线性互补的非连续变形分析

非连续变形分析(DDA)方法是一种新的用来分析块体系统运动和变形的非连续介质数值计算方法。研究的核心工作是致力于对现有DDA接触问题处理方法的改进。DDA主要采用罚函数法和Lagrange乘子法处理接触问题,合理设定罚参数很困难,此外,因开闭迭代而引起的刚度矩阵的不连续变化也会导致收敛方面的困难。为避免引入罚参数及传统意义上的开闭迭代,用混合线性互补模型(LCDDA)对DDA方法进行了重新描述。在此基础上,综合基于非光滑分析的Newton法的局部平方收敛和最速下降法的全局线性收敛的优势,提出求解LCDDA模型的有效算法。根据上述思想及理论研究成果编制了完整的计算程序,算例计算结果证明了方法的精度及可行性。     

Boundary setting method for the seismic dynamic response analysis of engineering rock mass structures using the discontinuous deformation analysis method

Large deformations and discontinuous problems can be calculated using the discontinuous deformation analysis (DDA) method by solving time steps, and this method is suitable for simulating the seismic dynamic response of engineering rock mass structures. However, the boundary setting must be carefully analyzed. In this paper, four boundary settings for the DDA method are investigated. First, the contributions to the DDA equations for nonreflecting boundaries (including the viscous boundary and the viscoelastic boundary) are deduced based on the Newmark method. Second, a free‐field boundary is introduced in the DDA method with boundary grid generation and coupling calculation algorithms to accurately simulate external source wave motion, such as earthquakes. Third, seismic input boundary treatments are intensively examined, and the force input method is introduced based on nonreflecting boundaries. Finally, the static‐dynamic unified boundary is implemented to ensure consistent boundary transformation. The boundary setting method in the DDA method is discussed, and the suggested treatments are used to analyze the seismic dynamic response of underground caverns. Copyright © 2015 John Wiley & Sons, Ltd.     

一种改进步长自调的非连续变形分析法

非连续变形分析（DDA）是一种针对块体系统变形和位移求解的数值计算方法。引入Newmark方法于结构动力学微分方程中,考虑惯性力和阻尼力作用,改进时间步长自动调节,并实现DDA求解程序;比较研究Newmark方法中的线性加速法、常加速法和平均加速法在DDA程序中计算的收敛速度,讨论块体系统动力学计算过程中DDA方法对惯性力和阻尼力的添加和删除,并提出根据计算精度要求的误差控制实现方案。将改进的DDA方法模拟一个典型的煤与瓦斯突出过程,取得了满意的计算结果,该改进算法为DDA方法处理动力学问题提供新的途径     

Three-dimensional discontinuous deformation analysis (3-D DDA) using a new contact resolution algorithm

This paper presents a new contact calculating algorithm for contacts between two polyhedra with planar boundaries in the three-dimensional discontinuous deformation analysis (3-D DDA). In this algorithm, all six type contacts in 3-D (vertex-to-face, vertex-to-edge, vertex-to-vertex, face-to-face, edge-to-edge, and edge-to-face) are simply transformed into the form of point-to-face contacts. The presented algorithm is a simple and efficient method and it can be easily coded into a computer program. In this paper, formulations of normal contact, shear contact and frictional force submatrices based on the new method are derived and the algorithm has been programmed in VC⁺⁺. Examples are provided to demonstrate the new contact rule between two blocks.     

A new model of edge-to-edge contact for three dimensional discontinuous deformation analysis

A robust contact theory can be regarded as key to three dimensional discontinuous deformation analyses (3D-DDA). Not only must this theory provide an efficient algorithm to judge the type and location of contacts but also it must be able to present comprehensive formulations for every kind of contact (open, sliding and locked contact). There are six types of contact in three dimensional discontinuous deformation analyses (vertex-to-vertex, vertex-to-edge, vertex-to-face, edge-to-edge, edge-to-face and face-to-face) that can be converted to vertex-to-face and edge-to-edge contacts. This paper presents a new model of edge-to-edge contact to three dimensional discontinuous deformation analyses (3D-DDA). This new model considers both kinds of edge-to-edge contact (cross-over and parallel edge-to-edge contact) and presents a criterion for inter-penetration. Sub matrices of normal and shear spring and friction force are derived by geometrical analysis and penalty method. This new model is implemented in a 3D-DDA computer programme, and the example results demonstrate the validity of the model.     

Kinetic Energy Dissipation and Convergence Criterion of Discontinuous Deformations Analysis (DDA) for Geotechnical Engineering

The discontinuous deformation analysis (DDA) is a numerical method for modeling discontinuous deformation behaviour of jointed rocks. In this paper, two basic problems are discussed related to kinetic energy dissipation and the convergence criterion for the DDA method when it is applied to geotechnical engineering. In view of the fact that the deformation and progressive failure can be treated as a quasi-static process with low kinetic energy dissipation rates, this paper introduces a viscous damping component to absorb discrete blocks’ kinetic energy, establishes the global equations of motion of the discrete block system that take damping effects into account, investigates the energy dissipation mechanism when solving a static or quasi-static problem, and defines the convergence criteria of displacement, kinetic energy and unbalanced force for DDA solutions when the system arrives at a stable state.     

Discontinuous deformation analysis with second‐order finite element meshed block

The discontinuous deformation analysis (DDA) with second‐order displacement functions was derived based on six‐node triangular mesh in order to satisfy the requirement for the accurate calculations in practical applications. The matrices of equilibrium equations for the second‐order DDA were given in detail for program coding. By close comparison with widely used finite element method and closed form solutions, the advantages of the modified DDA were illustrated. The program coding was carried out in C++ environment and the new code applied to three examples with known analytical solutions. A very good agreement was achieved between the analytical and numerical results produced by the modified DDA code. Copyright © 2006 John Wiley & Sons, Ltd.     

数值流形方法框架下散体系统与连续介质共同作用模拟

数值流形方法是包含流形元、有限元及DDA在内的数值方法体系,建立流形元与DDA块体的接触方程,则可实现流形方法框架下的连续介质和散体系统共同作用模拟。针对填石路堤工程,编制了大型数值计算程序,采用块体随机生成、块体粒径控制及块体自然堆积的方法建立散体系统的DDA模型,对路堤的分层铺设、碾压及工后沉降变形等进行模拟分析。通过算例表明,在数值流形方法框架下,采用流形元与DDA共同作用的方法,可以很好地对同时存在连续变形和散体大变形的体系进行计算分析,其对该类问题的模拟更接近分析对象的实际情况,有助于从根本上揭示分析对象变形的细观机制和规律,并能考察更多因素对工程问题的影响。     

三维极限平衡理论及其在块体稳定分析中的应用

岩体与土体失稳的滑稳面形状大多具有三维特征，用二维极限平衡理论难以得到与实际相应的分析结果。通过将滑动块体离散为一系列柱体，对每一条柱体，分析作用于条柱上的作用力和力矩平衡条件，提出块体稳定分析的三维极限平衡分析方法。该方法不需要对滑裂面作形状的假定，运用迭代解法，可以方便地求出可能滑动块体稳定系数。应用三维极限平衡分析方法对某电站坝基深层滑动块体的稳定性进行了分析，得到了一些有益的结论。     

非连续变形分析方法中的阻尼问题研究

定量研究了非连续变形分析(DDA)方法中的黏性阻尼与数值阻尼。首先,基于Newmark直接积分法,推导了块体系统的运动方程。其次,通过动力学中的黏性阻尼理论建立了DDA中动力系数、时间步长与黏性阻尼比的关系式,探讨了DDA中的常加速度积分方案的数值阻尼分区及阻尼比计算方法,进而得到两种阻尼共同作用时的阻尼比表达式,并分析了频域内阻尼比的分布情况。最后,以谐振激励下的块体振动为例,通过对比不同阻尼作用下块体位移的DDA计算值与理论解,验证了本文提出的阻尼比计算公式的正确性。研究表明：黏性阻尼对低频的衰减作用明显,数值阻尼则可以很快地消除高频干扰,而二者共同作用下可降低阻尼的频率相关性。该研究成果为DDA的振动、波动等动力计算的阻尼取值提供了理论依据。     

基于OpenMP的非连续变形分析并行计算方法

非连续变形分析（DDA）方法严格满足平衡要求和能量守恒,具有完全的运动学及数值可靠性,但对大规模岩土工程问题的数值模拟耗时太长,尤其是线性方程组求解,并行计算可以很好地解决该问题。首先基于DDA方法的基本理论,阐述了适用于DDA方法中的基于块的行压缩法和基于“试验-误差”迭代格式的非零位置记录;其次,引入块雅可比迭代法并行求解DDA方法的线性方程组,并改进了相应的非零存储方法;最后,基于OpenMP实现了DDA线性方程组求解并行计算,并将其应用于地下洞室群的破坏过程分析,以加速比为并行效率的指标评价,结果表明,该并行计算策略可以极大提高DDA的计算效率,而且适合各种规模的问题。     

混合多位移模式的非连续变形分析法研究

传统的非连续变形分析法（DDA）法采用简单的线性位移模式计算效率高,描述大块体的高阶多项式位移模式在一定程度保留了该特点,并提高了计算精度。近年来流行的耦合有限元、自然单元的DDA法实质上是引入相应的插值形函数构成块体位移函数,计算相对低效,但具有计算更精细、更容易施加边界条件等优点。为结合传统DDA法与DDA耦合法各自的优点,建立了一种同时利用传统DDA法线性位移模式与耦合型DDA法非线性位移模式的混合法。该方法非线性模式主要针对大块体,采用了自然单元插值,缘于其具有一定无网格特征,且效率比有限元高。建立了混合模式下的整体矩阵并推导出接触等因素刚度子矩阵和荷载子向量的具体表达式。该方法建模更加方便合理,计算精度、效率介于线性模式的传统DDA法和非线性位移模式的耦合法之间。通过基本算例验证了混合法的有效性,并给出了节理围岩-隧道衬砌整体分析模型的计算结果,体现了新方法的优越性。     

Non-rigid disk-based DDA with a new contact model

In this paper, a new disk-based DDA formulation is presented. In the original disk-based DDA, disks are considered to be rigid and the penalty method is used to enforce disk contact constraints. In order to improve the accuracy of the disk-based DDA, new formulations of stiffness and force matrices for non-rigid disks using a new efficient contact model are presented in this paper. Blocks are considered deformable without need to do more computations for contact detection. In the proposed contact model, disk–disk and disk–boundary contacts are transformed into the form of point-to-line contacts and normal spring, shear spring and frictional force sub-matrices are derived by vector analysis. The penalty method is quite simple to implement, but has some major disadvantages. In the presented contact model, not only the simplicity of the penalty method is retained but also the limitations are overcome by using the augmented Lagrangian method. Moreover, unlike the contact model used in the original disk-based DDA, reference line can be obtained directly by using only coordinates of disk centers and their radii, and no more computations are needed. The validity and capability of the new disk-based DDA formulation are demonstrated by several illustrative examples.     

基于DDA方法一种流-固耦合模型的建立及裂隙体渗流场分析和应用

以非连续变形分析方法（DDA）为基础并采用稳态流体计算方法将二者结合进行裂隙岩体流-固耦合分析。利用DDA方法生成裂隙岩体模型,在此基础上采用矩阵搜索等方法形成新的裂隙水通网络模型。采用稳态迭代算法和立方定律求得裂隙水压力,并把裂隙水压力作为线载荷施加到块体边界,在DDA算法中每个迭代步完成后更新裂隙开度和水压值,与DDA算法结合研究裂隙水与块体之间相互作用关系。利用以上裂隙岩体流-固耦合计算方法研究了某水封油库开挖和运行过程洞室围岩流量和密封性,为该工程预测水封效果提供了有益的主要依据,也是国内首次采用DDA方法做大型工程的流-固耦合模型分析。     

Initial stress formulae of high-order NMM and DDA for large deformation analysis

With high-order numerical manifold method (NMM) or high-order discontinuous deformation analysis method (DDA), computational accuracy of structure deformation can be improved greatly. However, poor accuracy is obtained and even computation is not convergent while treating large deformation problems, due to inaccurate or incorrect high-order initial stress formulae. Based on 2D triangular mathematical meshes and polynomial cover functions in high-order NMM, exact formulae for high-order initial stresses are deduced to depict configuration change of structures under large deformations. The formulae are expressed in polynomial forms so as to be used in simplex integrations. The approach is also extended to high-order DDA. Comparing with analytical solutions, accurate results for large deformation of a cantilever beam prove the validity of the proposed formulae.     

强度折减DDA方法及其在稳定分析中的应用

在自行研制的非连续变形分析（discontinuous deformation analysis,DDA）程序中实现了自动强度折减法以模拟边坡的稳定性和安全系数。通过锦屏高边坡和某重力坝深层滑动的计算分析以及与刚体极限平衡法和有限元方法的比较研究可以看出,强度折减DDA法可以较高精度求出岩石高边坡的安全系数,但求得的重力坝深层抗滑稳定安全系数要低于刚体极限平衡法的结果。基于强度折减技术的DDA与FEM的耦合方法,可以较准确地求出重力坝深层抗滑稳定安全系数。     

关于强度折减有限元方法中边坡失稳判据的讨论

在边坡稳定性分析中采用强度折减弹塑性有限元方法时,所得到的总体安全系数在一定程度上依赖于所采用的失稳评判标准,通常以数值计算的收敛性作为边坡失稳判据。而有限元计算的数值收敛性受多种因素的影响,因而由此所得到的安全系数的合理性及其唯一性受到了质疑。为了考查目前各种失稳判据的合理性及其适用性,分别依据计算的收敛性、特征部位位移的突变性和塑性区的贯通性等3个失稳判据,针对某一典型边坡算例,采用强度折减弹塑性有限元方法进行稳定性分析,并与Spencer极限平衡法所得到的总体安全系数进行了对比。对比分析表明,以有限元数值计算的收敛性作为失稳判据在某些情况下所得到的安全系数可能误差较大,而采用特征部位位移的突变性或塑性区的贯通性作为失稳判据所得到的边坡安全系数与Spencer极限平衡法的计算结果比较接近,考虑到实用性与简便性,建议在边坡稳定性分析的强度折减有限元方法中联合采用特征部位位移的突变性和塑性区的贯通性作为边坡的失稳判据。