The elastic distortion problem with large rotation in discontinuous deformation analysis

Conventional discontinuous deformation analysis (DDA) results in a change of block volume, which is known as free expansion, during rotation calculations because of the use of a linear displacement function to simulate the behavior of a block with a rigid body and elastic behaviors. This study demonstrates that the linear displacement function also generates unsolved elastic distortion, especially when the block undergoes large rotation in each calculation step. The distortion disturbs the contact judgment in the open–close iteration and update calculations of vertex coordinates, stresses, velocities, etc. at the end of each calculation step. A new procedure follows the flow chart of the original DDA, but it adopts additional codes for the coordinate-transformation calculations in vertex coordinate, stress, and velocity updates. When the vertex coordinates are updated, vertex displacements caused by strains are calculated before involving the block-rotation term in the displacement function to mitigate the elastic distortion. In addition, new codes compile formulas to transform stresses and velocities with block rotation. The new DDA ensures the correctness of rotating elastic calculations to solve practical falling rock problems with a large rotational angle in each calculation step.     

非连续变形分析中块体大转动问题研究

原有非连续变形分析（DDA）采用一阶近似后的位移增量表达式更新块体构形,推导相关子矩阵,且对不同时步计算出的应变增量直接叠加,当模拟的块体发生大转动时往往会产生较大误差。为考虑块体转动与变形的耦合作用,引入先变形、后转动的块体位移增量表达式。重新推导了惯性力子矩阵,将块体转动时的离心力与科氏力加到荷载矩阵中。计算时对应变分量及其相关变量进行坐标变换与修正,并采用新引入的位移增量表达式计算块体顶点位移,进行后接触修正与更新块体构形。数值算例表明,改进后的程序能够消除转动带来的误差,自动考虑了块体转动时离心力和科氏力引起的变形,应变计算精度更高。改进方法克服了块体体积自由膨胀、应变场畸变等问题,给出了合理的块体应变。     

Nodal-based three-dimensional discontinuous deformation analysis (3-D DDA)

This paper presents a new numerical model that can add a finite element mesh into each block of the three-dimensional discontinuous deformation analysis (3-D DDA), originally developed by Gen-hua Shi. The main objectives of this research are to enhance DDA block’s deformability. Formulations of stiffness and force matrices in 3-D DDA with conventional Trilinear (8-node) and Serendipity (20-node) hexahedral isoparametric finite elements meshed block system due to elastic stress, initial stress, point load, body force, displacement constraints, inertia force, normal and shear contact forces are derived in detail for program coding. The program code for the Trilinear and Serendipity hexahedron elements have been developed, and it has been applied to some examples to show the advantages achieved when finite element is associated with 3-D DDA to handle problems under large displacements and deformations. Results calculated for the same models by use of the original 3-D DDA are far from the theoretical solutions while the results of new numerical model are quite good in agreement with theoretical solutions; however, for the Trilinear elements, more number of elements are needed.     

Discontinuous deformation analysis based on the multiplicative decomposition of the displacement

Updating the block configuration on the basis of additive decomposition and its linearized expression of the displacement increment leads to the low calculation accuracy of the original discontinuous deformation analysis (DDA) and false volume expansion. In this study, the displacement expressions of a small deformation, a large rotation, and the corresponding velocity and acceleration terms on the basis of the initial configuration are presented using multiplicative decomposition. With the use of the principle of virtual work, the stiffness matrix, mass matrix, and force vector of blocks are obtained. Compared with the original DDA, each of the block deformation parameters has obvious physical meaning as a parameter of mechanics, which can be obtained by adding the incremental deformation components of each time step directly without co-ordinate transformation. Moreover, the proposed modification automatically considers the block deformation produced by centrifugal and Coriolis forces. The analysis of some typical numerical examples have verified the accuracy of the strain and stress calculated by the proposed method, and the current configuration is updated by the total displacements, which completely overcomes the false volume expansion and provides reasonable linear strains.     

Boundary Element based Discontinuous Deformation Analysis

A Boundary Element based Discontinuous Deformation Analysis (BE‐DDA) method is developed by implementing the improved dual reciprocity boundary element method into the open close iterations based DDA. This newly developed BE‐DDA is capable of simulating both the deformation and movement of blocks in a blocky system. Based on geometry updating, it adopts an incremental dynamic formulation taking into consideration initial stresses and dealing with external concentrated and contact forces conveniently. The boundaries of each block in the discrete blocky system are discretized with boundary elements while the domain of each block is divided into internal cells only for the integration of the domain integral of the initial stress term. The contact forces among blocks are treated as concentrated forces and the open–close iterations are applied to ensure the computational accuracy of block interactions. In the current method, an implicit time integration scheme is adopted for numerical stability. Three examples are used to show the effectiveness of the algorithm in simulating block movement, sliding, deformation and interaction of blocks. At last, block toppling and tunnel stability examples are conducted to demonstrate that the BE‐DDA is applicable for simulation of blocky systems. Copyright © 2016 John Wiley & Sons, Ltd.     

Investigation of highly efficient algorithms for solving linear equations in the discontinuous deformation analysis method

Large‐scale engineering computing using the discontinuous deformation analysis (DDA) method is time‐consuming, which hinders the application of the DDA method. The simulation result of a typical numerical example indicates that the linear equation solver is a key factor that affects the efficiency of the DDA method. In this paper, highly efficient algorithms for solving linear equations are investigated, and two modifications of the DDA programme are presented. The first modification is a linear equation solver with high efficiency. The block Jacobi (BJ) iterative method and the block conjugate gradient with Jacobi pre‐processing (Jacobi‐PCG) iterative method are introduced, and the key operations are detailed, including the matrix‐vector product and the diagonal matrix inversion. Another modification consists of a parallel linear equation solver, which is separately constructed based on the multi‐thread and CPU‐GPU heterogeneous platforms with OpenMP and CUDA, respectively. The simulation results from several numerical examples using the modified DDA programme demonstrate that the Jacobi‐PCG is a better iterative method for large‐scale engineering computing and that adoptive parallel strategies can greatly enhance computational efficiency. Copyright © 2015 John Wiley & Sons, Ltd.     

公路隧道局部塌方洞段的围岩稳定性评价

结合某公路隧道工程,以现场地质调查和监测资料为基础,在结构面发育且可能发生大规模塌方的围岩洞段,根据现场实际统计的优势结构面产状,采用关键块体理论、非连续变形方法（DDA）首先确定围岩主要的失稳破坏模式,然后运用连续介质分析程序FLAC,在DDA方法确定的围岩实际变形破坏塌方形态的基础上进行计算分析,评价围岩的整体稳定性及加固措施的有效性。分析表明,围岩破坏的主要模式为受结构面控制的局部块体的失稳,工程开挖后围岩不会发生大变形,这与实际情况基本一致,为安全施工提供了保证。采用非连续变形方法和连续介质计算相结合的分析方法对隧道工程塌方稳定性进行研究,取得了较好的效果,研究方法和经验可为类似工程参考并借鉴。     

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。     

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。     

Block fracturing analysis using nodal‐based discontinuous deformation analysis with the double minimization procedure

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.     

New point-to-face contact algorithm for 3-D contact problems using the augmented Lagrangian method in 3-D DDA

This paper presents a new point-to-face contact algorithm for contacts between two polyhedrons with planar boundaries. A new discrete numerical method called three-dimensional discontinuous deformation analysis (3-D DDA) is used and formulations of normal contact submatrices based on the proposed algorithm are derived. The presented algorithm is a simple and efficient method and it can be easily coded into a computer program. This approach does not need to use an iterative algorithm in each time step to obtain the contact plane, unlike the ‘Common-Plane’ method applied in the existing 3-D DDA. In the present 3-D DDA method, block contact constraints are enforced using the penalty method. This approach is quite simple, but may lead to inaccuracies that may be large for small values of the penalty number. The penalty method also creates block contact overlap, which violates the physical constraints of the problem. These limitations are overcome by using the augmented Lagrangian method that is used for normal contacts in this research. This point-to-face contact model has been programmed and some illustrative examples are provided to demonstrate the new contact rule between two blocks. A comparison between results obtained by using the augmented Lagrangian method and the penalty method is presented as well.     

基于显式时间积分的球颗粒DDA计算方法

非连续变形分析方法（DDA）是一种平行于有限元法的新型数值计算方法,该方法基于最小势能原理,把每个离散块体的变形、运动和块体之间的接触统一到平衡方程中进行隐式求解。然而,传统DDA方法在计算过程中需组装整体刚度矩阵并联立求解方程组,在用于大型岩土工程问题的三维数值模拟时占用内存较大、耗时较长、计算效率极低。因此,提出一种基于显式时间积分的三维球颗粒DDA方法。该方法在求解过程中不需要组装整体刚度矩阵,在求解加速度时,由于质量矩阵为对角矩阵,可存储为一维向量占用内存较少,且可分块逐自由度求解,效率较高,在接触判断上采用最大位移准则简化了接触算法,采用较小的时步,保证了计算的精确性;通过几个典型算例验证了该方法的准确性及计算效率。     

Effect of block rotation on the pitch of slickenlines

Rotation of faults or pre-existing weakness planes produce two effects on the slickenlines of fault planes. First, the rotation leads to changes in the pitch of slickenlines. As a result, the aspect of the pre-existing fault may change. For example, after rotation, a normal fault may show features of an oblique fault, a strike-slip fault, or a thrust fault. Second, due to rotation, stress states on the fault planes are different from those before the rotation. As a consequence some previous planes may be reactivated. For an isolated plane, the reactivation due to rotation can produce new sets of slickenlines. With block rotation, superimposed slickenlines can be generated in the same tectonic phase. Thus, it is not appropriate to use fault-slip data from slickenlines to analyze the stress tensor in a region where there is evidence of block rotation. As an example, we present the data of slickenlines from core samples in the Tunich area of the Gulf of Mexico. The results wrongly indicate that the calculated stress tensor deviates from the far-field stress tensor.     

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。     

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.     

鄂尔多斯南缘构造带现今地块运动模式

依据研究区内断裂体系及GPS地体运动矢量数据,将鄂尔多斯南缘构造带分割成AK共11个断块群,将差异性GPS数据转化为断块之间的细部相对运动,给研究地块细部运动差异提供一种全新的研究思路与研究模式。鄂尔多斯南缘中东部现今构造应力场整体为挤压应力环境。在渭河盆地西部,从西南方向中秦岭动力楔传递而来的挤压应力峰点在武功咸阳口镇一带形成应力主轴,应力主轴两侧楔形地块普遍存在侧向挤出逃逸,逃逸使GPS运动矢量偏离区域主应力方向。渭河盆地东部则存在左行走滑,显示出豫西断隆的阻隔碰撞作用。鄂尔多斯南缘现今地块相对运动规律符合"三头碰撞"模式,四川地块、鄂尔多斯地块、豫西断隆三个地块的"三头碰撞"形成高应力区,"头间逃逸带"形成走滑带和相对低应力区。本次提出地体之间的"多头碰撞"和"头间逃逸"构造模式有一定的普遍性,研究清楚地块之间哪里是"头"接触点、哪里是"头间逃逸带",可以成为研究大中地块周围震源机制分布规律的有效途径。     

A detailed investigation of block dynamic sliding by the discontinuous deformation analysis

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.     

Modified predictor-corrector solution approach for efficient discontinuous deformation analysis of jointed rock masses

The high computational costs associated with the implicit formulation of discontinuous deformation analysis (DDA) have been one of the major obstacles for its implementation to engineering problems involving jointed rock masses with large numbers of blocks. In this paper, the Newmark-based predictor-corrector solution (NPC) approach was modified to improve the performance of the original DDA solution module in modeling discontinuous problems. The equation of motion for a discrete block system is first established with emphasis on the consideration of contact constraints. A family of modified Newmark-based predictor-corrector integration (MNPC) scheme is then proposed and implemented into a unified analysis framework. Comparisons are made between the proposed approach and the widely used constant acceleration (CA) integration approach and central difference (CD) approach, regarding the stability and numerical damping features for a single-degree-of-freedom model, where the implications of the proposed approach on open-close iteration are also discussed. The validity of the proposed approach is verified by several benchmarking examples, and it is then applied to two typical problems with different numbers of blocks. The results show that the original CA approach in DDA is efficient for the simulation of quasi-static deformation of jointed rock masses, while the proposed MNPC approach leads to improved computational efficiency for dynamic analysis of large-scale jointed rock masses. The MNPC approach therefore provides an additional option for efficient DDA of jointed rock masses.     

Seismic landslide simulations in discontinuous deformation analysis

Discontinuous deformation analysis (DDA) is a numerical approach used to simulate the post-failure behavior of a blocky assembly. Three available algorithms incorporate seismic impacts into DDA simulations for earthquake-induced slope failure. The following methods are used: directly applying time-dependent accelerations to falling/sliding blocks (Method 1); adding time-dependent accelerations to base block (Method 2); and time-dependently constraining seismic displacements of the base block (Method 3). However, incorrect absolute movements of falling/sliding and base blocks were obtained using Method 1. Additionally, relative movements between falling/sliding blocks and the base block are opposite to those simulated by the other two algorithms—Methods 2 and 3. Since locating an earthquake-induced landslide before an earthquake is extremely difficult, the seismic movements of base rock are recorded. Method 1 applies recorded seismic data to sliding blocks in conflict with d’Alembert’s principle of mechanics. Additionally, in Method 2, when the computation time step must be longer than the time in seismic data, computational results reveal abnormal base block displacements due to the non-zero velocity recorded at the end time of seismic data in seismic DDA. In this study, a novel algorithm to diminish the velocity of the base rock in the seismic analysis is utilized to modify Method 2. Furthermore, this work confirms that DDA with the modified Methods 2 and 3 is a practical approach for earthquake-induced landslide simulations.     

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.