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.     

Review of validation of the discontinuous deformation analysis (DDA) method

Over the last decade, researchers in the discontinuous deformation analysis (DDA) community have dedicated a great deal of effort to document the accuracy of the method by performing validation studies. This paper contains a summary of more than 100 published and unpublished validation studies which comprise the body of DDA validation information to which the authors have access. The studies are grouped into three general categories: (a) validation with respect to analytical solutions, (b) validation with respect to results of other numerical techniques, and (c) validation with respect to laboratory and field data. Three general techniques for validation are described: qualitative assessment visually examining runtime behaviour of simulations, semi‐quantitative assessment comparing numerical results of simulations, and quantitative where numerical simulation results are evaluated in detail with respect to similar analytical, laboratory or field results. We find that for many of the problems addressed by the papers in this review, DDA performs more than adequately for engineering analysis. Copyright © 2006 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.     

Unified displacement boundary constraint formulation for discontinuous deformation analysis (DDA)

Displacement boundary constraints in discontinuous deformation analysis (DDA) are applied using stiff penalty springs. A co‐ordinate‐free formulation for displacement boundary constraints is presented here for DDA, which unifies previous derivations for points of fixity, and for points constrained to induce or prohibit block motion in specified directions as a function of location or time. Examples for each type of constraint are used to illustrate the behaviour of the algorithm and provide a link with previous formulations for each case. The new, unified formulation has five benefits: (1) simple to express algorithmically; (2) easy to program and verify; (3) penalty values in different directions may be chosen to allow fixed points, lines, curves or planes; (4) formulation works for 2D and 3D; (5) displacement constraint may be a function of time or location or both. Feedback in the algorithm may induce internal resonance in homogeneously deformable discrete elements used in DDA, and resonance in block‐to‐block contact interactions. Consequently, high mass problems with insufficient damping may suffer from excessive ‘vibrational hammering’, inducing physically implausible behaviour such as elastic rebound. Copyright © 2005 John Wiley & Sons, Ltd.     

节理岩质边坡变形的DDA模拟

对原非连续变形分析方法（DDA方法）中边界约束方法进行了推广，将其应用于节理岩质边坡变形规律的数值模拟，并利用模型实验结果验证了推广方法的正确性。     

Hydraulic fracturing modeling using the discontinuous deformation analysis (DDA) method

Recent attempts to couple discontinuous deformation analysis (DDA) blocks with finite element meshes have been regarded as fruitful. The hybrid model has been proven to be reliable in modeling continuum–discontinuum problems. In this paper, a hydraulic crack initiation–propagation extension complete with a coupled hydro-mechanical analysis algorithm is introduced to the hybrid model for the simulation of fracturing problems initiated by hydraulic pressure. Several simulation cases, of which the results appear to be consistent with existing theories of hydraulic fracturing mechanisms, are presented to demonstrate the capability of the proposed algorithm in modeling hydraulic fracturing processes.     

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.     

Modeling the interaction between fluid flow and particle movement with discontinuous deformation analysis (DDA) method

Coupled hydro‐mechanical processes in granular media represent the interaction between solid particles movement and fluid flow during external and/or internal loading. The processes attract attention in geotechnical engineering since they cause many disasters such as landslide, slope collapse, boiling/quicksand and soil liquefaction, etc. This study presents a numerical method combining discontinuous deformation analysis (DDA) for mechanical calculation and finite element method for fluid flow simulation to model the interaction between solid particles' movement and fluid flow from microscopic point of view. The term of hydraulic pressure was newly formulated and introduced to the original DDA. The simplified examples are given to verify the new method, and the computational results correlate well with the theoretical calculations, further development is also considered. Copyright © 2010 John Wiley & Sons, Ltd.     

膨胀土边坡稳定性非连续变形分析新方法

许多膨胀土边坡在长期的地质和干湿循环的自然气候作用下裂隙逐渐发展,以致被裂隙分割成块体系统。将膨胀土边坡看成是一个具有非饱和土本构关系、块体间存在摩擦力和膨胀力和具有牵引式滑动及时空效应特点的块体系统,建立了一种新的边坡稳定性非连续变形分析方法,并应用于某高速公路段膨胀土边坡稳定性的分析中,获到了很好的效果。     

Applying disk-based discontinuous deformation analysis (DDA) to simulate Donghekou landslide triggered by the Wenchuan earthquake

In this paper, for the first time, disk-based discontinuous deformation analysis (DDA) is applied to simulate a real landslide triggered by an earthquake. For this purpose, the kinematic behaviour of the Donghekou landslide triggered by the Wenchuan earthquake is simulated and the results obtained using disk-based DDA are compared with actual data. The comparisons show that there is a good agreement between the results obtained using disk-based DDA and observed data. The simulation results provided an understanding of the failure behaviour and temporal evolution of the landslide. This study shows that disk-based DDA is a practical numerical tool that can be used to simulate the post-failure behaviour of landslides triggered by an earthquake.     

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。     

Variational inequality-based particle discontinuous deformation analysis

Orienting the circular and rigid particle medium, the variational inequality-based discontinuous deformation analysis (DDA) is established. In the proposed DDA, the global stiffness matrix, the penalty parameters, and the open-close iteration are successfully avoided. The contact constraint is transferred into the problem of variational or quasi-variational inequalities. And explicit variational expression on the contact force is firstly established. To speed up the rate of solving contact force, on the basis of the two-stage prediction-correction method, we design a compatibility iteration algorithm (PPC-CI). The C++ code is developed in multicore environment through the open multi-processing (OpenMP) in order to take advantage of the parallelizable features of the new DDA. Numerical tests suggest that the presented DDA is effective and promising.     

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.     

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.     

Formulation of discontinuous deformation analysis (DDA) — an implicit discrete element model for block systems

Continuum-based numerical methods have played a leading role in the numerical solution of problems in rock mechanics and engineering geology. However, for fractured rocks, a continuum assumption often leads to difficult parameters to define and over-simplified geometry to be realistic. In such case, discrete representations of fractures and individual blocks must be adopted. In this paper, a newly emerged member in the family of discrete element methods (DEM), the discontinuous deformation analysis (DDA), is presented, including its variational principle, governing equations, solution techniques and contact representation and detection algorithms. Its relative advantages and shortcomings are compared with the explicit distinct element method and the finite element method. An example of the analysis of tunnel stability is provided to demonstrate the capability of this new method.     

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.     

Sphere-boundary edge and sphere-boundary corner contacts model in DDA for simulating particulate media in 3-D

In modelling particulate media, such as soils, using 3-D DDA, there are four types of contacts: sphere to sphere, sphere to boundary face, sphere to boundary edge, and sphere to boundary corner contacts. The first two were studied by the authors in a previous work (Beyabanaki and Bagtzoglou 2012). In this paper, we present a new contact model for sphere-boundary edge and sphere-boundary corner contacts in sphere-based 3-D DDA. The model includes a new algorithm to search for contacts, detect the contact types and calculate contact points. Moreover, formulas for contact sub-matrices are derived. The proposed contact model has been implemented into a sphere-based 3-D DDA program and three test cases are studied in order to verify the workability of the new contact model. The numerical results obtained demonstrate the capability of the model to deal with sphere-boundary interaction in particulate media.     

Formulation of a Three-dimensional Numerical Manifold Method with Tetrahedron and Hexahedron Elements

Summary  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. Author’s address: Y. M. Cheng, Department of Civil and Structural Engineering, Hong Kong Polytechnic University, Room TV707, Hung Hom Kowloon, Hong Kong, P.R. China     

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。