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Discontinuous deformation analysis based on complementary theory
Abstract
The contact between blocks is treated by the open-close iteration in the conventional discontinuous deformation analysis (DDA),
which needs to introduce spurious springs between two blocks in contact and to assume the normal stiffness and the tangential
stiffness (the penalty factors). Unreasonable values of stiffness would result in numerical problems. To avoid the penalty
factors and the open-close iteration, we reformulate the DDA as a mixed complementary problem (MiCP) and then choose the path
Newton method (PNM) to solve the problem. Some examples including those originally designed by Shi are reanalyzed, which proves
feasibility of the proposed procedure.
... Furthermore, a series of models have been proposed based on the complementarity theory. 16,20,21 However, efficient solvers dedicated to these problems need to be further developed, as described in the work of Li and Zheng. 16 Recently, Zheng et al 22 proposed a dual form of DDA based on variational inequalities. ...
... In the same way, the contact forces are recovered here as part of the solution, namely, as the Lagrange multipliers associated with the inequality constraints. In conclusion, governing equation (17) can be cast into two equivalent optimization problems, ie, the force-based problem (19) and the displacement-based problem (20). ...
... It is clear that the minimization part of (A2) is solved as the first set of KKT equations, ie, Kd − f − C T p. Including it into the remaining minimization part of the problem as a constraint results in the force-based problem (19). Alternatively, combining optimization conditions and minimization part of the problem results in the displacement-based problem (20). ...
... 44 Further, a so-called augmented Lagrangian method (ALM) 45 was proposed to deal with the contact problems in DDA, which combined the advantages of both Lagrangian multiplier and penalty method. Zheng et al. proposed the DDA method based on the complementarity theory 46 and dual form of DDA, 47 and they all have removed the artificial springs. ...
... Similarly, the coordinates of the centroid of Contactor block are assumed as ðx c0 ; y c0 ; z c0 Þ, and substitute it into Eq. (1), so it can be obtained that (46) where W c , T c and D c are the corresponding local displacement functions, basis functions and degrees of freedom, respectively. For the Contactor block, potential energy Π c caused by the normal contact force can be expressed as ...
The three-dimensional discontinuous deformation analysis (3D-DDA) method was developed for the deformation simulation of rock block system cut by the natural discontinuities in rock mass engineering. In the conventional DDA, open-close iteration is used to deal with the contact constraints, which needs to apply or remove the normal or tangential springs repeatedly to meet the equilibrium equations at each time step. DDA provides a time step adjustment strategy to meet the fast convergence of open-close iterations, but when solving large-scale problems, the adjusted time step often reaches a very small order of magnitude, which makes the calculation time-consuming increase sharply. In the framework of the original 3D-DDA, a new contact potential based three-dimensional discontinuous deformation analysis method (3D-CPDDA) is developed. The proposed method not only retains the advantage of the original DDA method in defining local displacement functions on a single patch, but also integrates the simplicity and rapidity of potential based contact processing. The improved method is easier to be implemented in the parallel way, which can further improve the computational efficiency. Numerical examples have confirmed the correctness and feasibility of the proposed procedure.
... On the other hand, the conventional open-close iteration for solving contact force in the original DDA heavily relies on values of contact springs that are usually specified by experience. To bypass this limitation, some sophisticated strategies, such as the complementary theory [26], the variational inequality formation [27,28], and the secondorder cone programming [29], have been proposed along with the development of DDA to calculate the contact force as accurately as possible. However, convergence rate of these solution algorithms, i.e. the path Newton method in [26] or the compatibility iteration in [27,28], is still an obstacle to their practical application. ...
... To bypass this limitation, some sophisticated strategies, such as the complementary theory [26], the variational inequality formation [27,28], and the secondorder cone programming [29], have been proposed along with the development of DDA to calculate the contact force as accurately as possible. However, convergence rate of these solution algorithms, i.e. the path Newton method in [26] or the compatibility iteration in [27,28], is still an obstacle to their practical application. Meanwhile, the second-order cone programming in [29] is difficult to implement. ...
... To enhance the block deformability in the DDA method, the incorporation of artificial joints, a higher order displacement function or finite element meshing was attempted [6][7][8] . With regard to calculation of contact forces of the DDA method, Lin et al. [9] developed the augmented Lagrange multiplier method-based DDA, Cai et al. [10] presented the Lagrange multiplier method-based DDA, and Zheng and Jiang [11] proposed the linear complementarity formulation-based DDA. For addressing the indeterminacy of vertex-vertex contact, enhancements to the shortest path method or strengthening the movement trend of blocks were suggested [12,13] . ...
... Furthermore, the contributions from other energy sources are also calculated from the initial calculation information as in original DDA. The velocities at the end of this time step is calculated by Eq. (11) . To ensure the equilibrium condition, the accelerations at the end of the time step are computed by ...
An explicit discontinuous deformation analysis (DDA) that uses an explicit time integration procedure and an explicit calculation of interaction forces between blocks is proposed to overcome the limitations of conventional implicit DDA in simulating large-scale problems. The advantages of the explicit DDA are that (1) the global equilibrium equations are unnecessary to be assembled and the solving for unknowns of every block can be performed independently and conveniently, thereby reducing the computational effort and memory requirement; (2) the open-close iteration process is avoided because the interaction forces between blocks are calculated explicitly according to the initial information at the start of the current time step. The efficient parallel computing is very appropriate for the explicit DDA. To further improve its computational efficiency, the explicit DDA is paralleled based on OpenMP. The accuracy of the explicit DDA is verified through several numerical examples with analytical solutions, experimental data or field observation. Further, the computational efficiency is demonstrated by a series of models and the parallel speedup factor on 6 OpenMP threads is approximately 4.2. Conclusively, the explicit DDA is promising for analyzing blocky systems in large scale.
... Few studies have focused on improving the open-close iteration; instead, the majority of research work is still focused on the study of contact judgment [16][17][18]. Zheng [19,20] proposed the discontinuous deformation analysis method based on linear complementary theory, which originates from the variational equations of momentum conservation. The linear complementary relationship is used to describe the normal and tangential contact constraint and avoids the implicit open-close iteration; however, iteration is still needed when determining the valid access line of edge-toedge contact. ...
... When the friction angle is 54 ∘ , the block remains at the initial position, and the normal and tangential embedding distances of the two vertexes at the bottom can be calculated to validate the calculation accuracy of the contact force. The theoretical embedding distance [32] could be obtained using (20), which is written as follows: ...
This paper proposes an improved DDA method based on explicitly solving contact constraints. The potential energy function generated by contacting, which contains only displacement variables as an unknown, is deduced based on the approximated step function and Lagrange interpolation, and the displacement variables and contact constraints are obtained via the variable metric method by analyzing the potential energy extremum. There is no need to conduct the open-close iteration during the process of calculation. The improved DDA method based on explicitly solving contact constraints has high precision and a more stable and more robust computational convergence. The accuracy and iterative stability of the improved DDA method are verified using two numerical examples.
... The original DDA repeatedly applied/ removed the spring between the contact pairs to describe the mechanical behavior of the joint. The literature on contact theory can be divided into two types: research on the traditional methods, with new algorithms for more efficient and accurate detection and identification of complex contact patterns (Keneti et al. 2008;Zheng et al. 2017;Lin et al. 2019;Ni et al. 2020;Shi 2015Shi , 2021, and new contact methods, such as the Lagrange and augmented Lagrange multiplier method (Cai et al. 1996;Lin et al. 1996), complementarity method (Zheng and Jiang 2009;Zheng and Li 2015;Li and Zheng 2015), variational inequality method (Jiang and Zheng 2011;Zheng et al. 2016), and generalized contact potential method Zheng et al. 2018). In engineering applications, DDA and its generalized form, the NMM, have been extensively adopted to reveal the cracking process in geomaterials or concretes (Pearce et al. 2000;Wu et al. 2018), instability in soil and rock slopes (Sitar et al. 2005;Zhang et al. 2013Zhang et al. , 2014Do and Wu 2020;Gong et al. 2020a;Yang and Wu 2022;Yang et al. 2023a, b;Ma et al. 2023a), rockfall and rock burst mechanisms (Ning et al. 2011;Chen et al. 2013Chen et al. , 2018Hatzor et al. 2017a), and mechanical responses in soils (Guo et al. 2019(Guo et al. , 2023Yu et al. 2022). ...
Tensile cracks significantly influence stability of mining, tunneling, and other geotechnical engineering scenarios. Discontinuous deformation analysis (DDA) is widely employed to analyze, predict, and prevent the deformation and progressive failure of such geomaterials. However, an accurate and efficient simulation of the cracking process remains a challenge, requiring appropriate fracture models. This study presents an improved DDA algorithm featuring a nonlinear cohesive zone model (CZM) designed for the intricate deformation and fragmentation analysis of geotechnical structures composed of quasi-brittle materials. Building upon fracture mechanics theory, an exponential-type CZM that accounts for both post-peak softening behaviors and nonlinear relations inherent in quasi-brittle materials is introduced. This exponential-type CZM is integrated into the original DDA method, addressing the co-edge blocks connected with cohesive springs. The ensuing unbalanced force in normal contact arising from the nonlinear cohesive spring is mitigated using a nonlinear iterative algorithm. The formulation, the implementation, and the iteration of the exponential-type CZM within the DDA framework are elucidated. Finally, the feasibility and the accuracy of the improved DDA for analyzing quasi-brittle crack propagation are demonstrated and validated by comparing several examples with experimental and numerical results.
... Currently, machine learning (ML) methods are widely employed for predicting structural fragility curves owing to their perfect data-fitting capabilities [16][17][18][19][20][21][22][23][24][25][26][27][28][29]. Given a set of damage data from a specific structure (e.g., input data could be IM and output data could be labels representing the specific damage limit states), ML can adaptively fit the input data to the output data forming a function that captures the data pattern between them. ...
... In recent years, great efforts have been contributed to different aspects of DDA. For example, a variety of DDA forms have been proposed to overcome the challenges associated with employing penalty springs, including the complementaritybased DDA (Fan et al. 2020;Zheng and Jiang 2009), variational inequality-based DDA (Jiang and Zheng 2011), dual form DDA (Zheng et al. 2016) and contact potential based DDA Zheng et al. 2020). Numerous contact constitutive models have been devised to replicate the rock failure process (Hu et al. 2023;Jiao et al. 2012;Zheng et al. 2021). ...
Dynamic failure widely exists in rock engineering, such as excavation, blasting, and rockburst. However, the quantitative measurement of the dynamic damage process using experimental methods remains a challenge. In this study, a SHPB modeling technique is established based on Voronoi-based DDA to study the damage evolution of Fangshan granite under dynamic loading. The assessment of cracking along the artificial joints among Voronoi sub-blocks is conducted using the modified contact constitutive law. A calibration procedure has been implemented to investigate the rock dynamic properties quantitatively. The dispersion and damping effect can be effectively eliminated by regular discretization in SHPB bars, based on which the dynamic stress equilibrium can be satisfied. To reproduce the loading rate effect of the dynamic compressive strength, which has been observed in the experiment, a modification strategy considering the influence of the rate effect on the strength meso-parameters is proposed. Using this strategy, the peak stresses of the transmitted waves predicted by DDA match well with those obtained from experiments conducted at different loading rates. The simulation results show that more microcracks are generated and the proportion of tensile cracks decreases as the loading rate increases. Furthermore, the dynamic mechanical behavior and fracturing process have also been discussed and compared with the experiments. The results show that the established SHPB system is a powerful tool for quantitative analysis of rock dynamics problems and can handle more complex problems in the future.
... Numerous enhancements have been put forward to improve the performance of traditional DDA. To alleviate the sensibility of penalty parameters, contacts between the blocks were remodeled using the Lagrange multiplier method [14], Augmented Lagrangian method [15], Complementary theory [16], and Variational Inequality theory [17]. Apart from modifying kinetic velocity using the dynamic factor [18], the damping effect was used to reflect the energy dissipation by imposing viscous boundary conditions [19] and by adding viscous forces into the equilibrium equation [20]. ...
To avoid disadvantages caused by rotational degrees of freedom in the original Discontinuous Deformation Analysis (DDA), a new block displacement mode is defined within a time step, where displacements of all the block vertices are taken as the degrees of freedom. An individual virtual element space
V
1(Ω) is defined for a block to illustrate displacement of the block using the Virtual Element Method (VEM). Based on VEM theory, the total potential energy of the block system in DDA is formulated and minimized to obtain the global equilibrium equations. At the end of a time step, the vertex coordinates are updated by adding their incremental displacement to their previous coordinates. In the new method, no explicit expression for the displacement
u
is required, and all numerical integrations can be easily computed. Four numerical examples originally designed by Shi are analyzed, verifying the effectiveness and precision of the proposed method.
... The main difference between them is that, DEM is based on Newton's second law and it uses an explicit approach to obtain the displacement of each block; while, DDA is an implicit-schemed method, which uses the minimum potential energy principle to establish simultaneous equations and solves these equations, thus obtaining the displacements of blocks. Compared with DEM, DDA has more rigorous theoretical fundamental, higher computational precision and unconditional stability [Bobet et al. (2009);Zheng and Jiang, (2009)]. However, the computational efficiency of DDA is much lower than that of DEM in most cases. ...
The efficiency of contact search is one of the key factors related to the computational efficiency of three-dimensional sphere discontinuous deformation analysis (3D SDDA). This paper proposes an efficient contact search algorithm, called box search algorithm (BSA), for 3D SDDA. The implementation steps and data structure for BSA are designed, with a case study being conducted to verify its efficiency. The data structure also has been improved for parallelizing the computation in contact search. For the demonstration of the proposed algorithm (BSA), six cases with various sphere numbers are simulated. Simulation results show that the time consumed in contact search using BSA (CTofBSA) is much less than that by the direct search algorithm (DSA) (CTofDSA). For the case with 12,000 spheres, CTofBSA is 1.1[Formula: see text]h, which is only 1.3% of CTofDSA (84.62[Formula: see text]h). In addition, the proportion of the computation quantity of contact search in the entire computation (Pcs) is 91.3% by using DSA, while this value by BSA is only 12.4%, which demonstrates the contribution of BSA. The efficiency brought about by BSA (time consumed and computation quantity) may enable 3D SDDA to simulate large-scale problems.
... DEM is based on Newton's second law and it uses an explicit scheme to simulate the movement of blocks and DDA often uses the minimum potential energy principle to establish a set of simultaneous equations and then solves these equations to obtain the displacements of the blocks. Compared with DEM, DDA has a more rigorous mathematical assumption and higher computation accuracy (Bobet et al. 2009;Zheng and Jiang 2009). In addition, similar to FEM, an implicit scheme is often used in DDA, resulting in the unconditional stability of DDA even when a large time-step size is used without applying artificial damping. ...
A new C++ programming strategy with high modularization and good portability, and a novel data storage format for simultaneous equations with little computer memory consumption, no sorting operation, and simple addressing algorithm are proposed for the three-dimensional sphere discontinuous deformation analysis (3D SDDA) to overcome the shortcomings of existing computation programs. An object-oriented data structure for the 3D SDDA computing code that is highly modular and easily transplanted is designed. Then, to demonstrate the portability of the 3D SDDA computing code, two computation architectures are respectively constructed to form two independent computation programs for 3D SDDA. Finally, several benchmark tests are conducted to verify the correctness of the 3D SDDA model in the new computation program, and a 170,725-sphere landslide example is simulated on a desktop computer to demonstrate the capability of the new computation program in large-scale engineering applications. Comparison between the new and existing computation programs regarding computer memory and time consumed demonstrates the great advantages brought about by the new computation program.
... Among these numerical methods the most widely used methods includes; the finite element method (Roy 1999, Sukumar 2003, Lin 2014, the direct and indirect boundary element method (Altiero 1982, Aliabadi 1993, Tan 1998, Fatehi Marji et al. 2006, Fatehi Marji. 2014, Haeri et al. 2013, Haeri et al. 2014a, 2014b, 2015a, 2015b, 2015c, 2015d, the discontinuous deformation analysis (Shi 1988, Zheng 2009, Amadei 1996, the explicit discrete element modelling, (Cundall 1979, Ghazvinian 2012, Lin 2013, Sarfarazi 2014, 3D numerical manifold method (He 2010). ...
Brazilian disc test is one of the most widely used experiments in the literature of geo-mechanics. In this work, the pre-holed concrete Brazilian disc specimens are numerically modelled by a two-dimensional discrete element approach. The cracks initiations, propagations and coalescences in the numerically simulated Brazilian discs (each containing a single cylindrical hole and or multiple holes) are studied. The pre-holed Brazilian discs are numerically tested under Brazilian test conditions. The single-holed Brazilian discs with different ratios of the diameter of the holes to that of the disc radius are modelled first. The breakage load in the ring type disc specimens containing an internal hole with varying diameters is measured and the crack propagation mechanism around the wall of the ring is investigated. The crack propagation and coalescence mechanisms are also studied for the case of multi-holes' concrete Brazilian discs. The numerical and experimental results show that the breaking mechanism of the pre-holed disc specimens is mainly due to the initiation of the radially induced tensile cracks which are growth from the surface of the central hole. Radially cracks propagated toward the direction of diametrical loading. It has been observed that for the case of disc specimens with multiple holes under diametrical compressive loading, the breaking process of the modelled specimens may occur due to the simultaneous cracks propagation and cracks coalescence phenomena. These results also show that as the hole diameter and the number of the holes increases both the failure stress and the crack initiation stress decreases. The experimental results already exist in the literature are quit agree with the proposed numerical simulation results which validates this simulation procedure.
... However, this approach inevitably leads to solving a nonsymmetric system. Zheng and Jiang (2009) and Zheng and Li (2015) removed the penalty parameter and the open-close iteration by considering discontinuous system as a nonlinear mixed complementarity problem in which the contact conditions are expressed by the complementarity equations. Due to the adoption of contact forces as the independent variables and rank deficiency of the Jacobi matrix, the solution would become inefficient. ...
Modelling discontinuous systems involving large frictional sliding is one of the key requirements for numerical methods in geotechnical engineering. The contact algorithms for most numerical methods in geotechnical engineering is based on the judgement of contact types and the satisfaction of contact conditions by the open–close iteration, in which penalty springs between contacting bodies are added or removed repeatedly. However, the simulations involving large frictional sliding contact are not always convergent, particularly in the cases that contain a large number of contacts. To avoid the judgement of contact types and the open–close iteration, a new contact algorithm, in which the contact force is calculated directly based on the overlapped area of bodies in contact and the contact states, is proposed and implemented in the explicit numerical manifold method (NMM). Stemming from the discretization of Kuhn–Tucker conditions for contact, the equations for calculating contact force are derived and the contributions of contact force to the global iteration equation of explicit NMM are obtained. The new contact algorithm can also be implemented in other numerical methods (FEM, DEM, DDA, etc.) as well. Finally, five numerical examples are investigated to verify the proposed method and illustrate its capability.
... Because of its promise, the validation of the DDA method has drawn significant research attention that has proven its effectiveness in various ways under both 2D and 3D conditions (Beyabanaki et al. 2009;Liu et al. 2012;MacLaughlin and Doolin 2006;Yeung et al. 2003;Zhang et al. 2015). Extensions of the DDA method have also been developed to include additional functions: modifications of the calculation procedure (Zheng and Jiang 2009), constraints (Doolin 2005), contact treatment Wang et al. 2019), the joint constitutive model (Ma et al. 2017a, b;Peng et al. 2019b), earthquake input and wave propagation Doolin 2005;Fu et al. 2015Fu et al. , 2017bJiao et al. 2007) and deformation adjustment (Cheng and Zhang 2000;Ke 1995;MacLaughlin and Sitar 1996;Wu 2015;Wu et al. 2017;Zhao and Gu 2009), as well as coupling of DDA with other methods (Fu et al. 2017a;Grayeli and Hatami 2008;Grayeli and Mortazavi 2006;Sun et al. 2016;Tian et al. 2014;Peng et al. 2019c) have all been studied. The calculation efficiency is essential to the implementation of DDA, so parallelization technology is introduced (Peng et al. 2019a). ...
Edge-to-edge contact is a fundamental contact type in blocky systems. In two-dimensional discontinuous deformation analysis (2D DDA, and hereinafter DDA for short), an edge-to-edge contact is transformed into two separated vertex-to-edge contacts by applying two pairs of concentrated springs. Although this simplification facilitates the DDA algorithm, it is not always sufficiently accurate and can even yield irregular results. To solve this problem, a distributed-spring contact model (DSCM) that exerts distributed instead of concentrated forces on contact edges is proposed in this paper for the edge-to-edge contact in DDA. Submatrices for the force matrix and stiffness matrix are obtained by minimizing the potential energy of the distributed contact forces and are incorporated into an improved DDA (I-DDA) code. Four examples are evaluated to illustrate the validations and advantages of the I-DDA. The first example is a single square impacting on a base block. Deformation of the contact area is evaluated by comparison with the theoretical deformation solution, and the results calculated by the I-DDA show better agreement with the analytical solution than the original DDA (O-DDA). The second example is an impact validation, proving that the I-DDA is more adaptable to discrete systems containing blocks of different sizes. Then an example and an experiment about block rebounding are provided, demonstrating that the errors in rotation and rebounding exhibited in the O-DDA results are avoided when using the I-DDA, indicating that the I-DDA provides more realistic solutions. The results of this study suggest that the proposed I-DDA incorporating the DSCM is quite accurate and capable of improving calculation accuracy compared to the O-DDA.
... Too stiff springs can turn governing equations into ill-conditioned, while soft springs can undermine its accuracy. To address the stiffness dependent problem, the Lagrangian multiplier method, 31 the augmented Lagrangian method, 32,33 and the complementarity theory [34][35][36] have been proposed. Most recently, DDA is reformulated based on the theory of finite-dimensional quasi-variational inequality. ...
For Discrete Element Methods (DEM), integrating the equation of motion based on the Newton's second law is an integral part of the computation. Accelerations and velocities are involved even for modelling static mechanics problems. As a consequence, accuracy of numerical results can be ruined and numerous calculation steps are required to converge. In this study, we propose a static discrete element method based on Discontinuous Deformation Analysis (DDA). The force of inertia is removed to develop a set of static equilibrium equations for distinct blocks. It inherits the advantages of DDA in dealing with distinct block system such as jointed rock structures. Furthermore, the critical numerical artifact in DDA, i.e., artificial springs between contact blocks are avoided. Accurate numerical solution can be achieved in mere one calculation step. Last but not the least, since the method is formulated in the framework of mathematical programming, the implementation can be easily conducted with standard and readily available solvers. Its accuracy and efficiency are verified against a series of benchmarks found in the literature.
... For example the finite element method have been carried out by Roy (1999), Sukumar (2003), Lin (2014), the boundary element method by Altiero (1982), Aliabadi (1993), Haeri et al. (2015a, b, c), and the finite difference method (Cundall 1979, Ghazvinian 2012, Sarfarazi 2014. Some other specially developed sophisticated numerical methods for cracks analyses in brittle materials include the discontinuous deformation analysis (Shi 1988, Zheng 2009, Amadei 1996, discrete numerical modelling, (Cundall 1979, Ghazvinian 2012, Lin 2013, Sarfarazi 2014, 3D numerical manifold method (He 2010) and the displacement discontinuity method (Crouch 1976, Fatehi Marji 1989, Tan 1998, Fatehi Marji et al. 2014. However, in the present work, the three dimensional particle flow code (PFC3D) which is based on the sophisticated discrete element method (DEM) is used to simulate the crack propagation mechanism in the modelled disc type specimens containing single hole or multiple holes under compressive loading condition. ...
In the previous studies on the porous rock strength the effect of pore number and its diameter is not explicitly defined. In this paper crack initiation, propagation and coalescence in Brazilian model disc containing a single cylindrical hole and or multiple holes have been studied numerically using PFC3D. In model with internal hole, the ratio of hole diameter to model diameter was varied between 0.03, 0.17, 0.25, 0.33, and 0.42. In model with multiple hole number of holes was different in various model, i.e., one hole, two holes, three holes, four holes, five holes, six holes, seven holes, eight holes and nine holes. Diameter of these holes was 5 mm, 10 mm and 12 mm. The pre-holed Brazilian discs are numerically tested under Brazilian test. The breakage load in the ring type disc specimens containing an internal hole with varying diameters is measured. The mechanism of cracks propagation in the wall of the ring type specimens is also studied. In the case of multi-hole Brazilian disc, the cracks propagation and b cracks coalescence are also investigated. The results shows that breaking of the pre-holed disc specimens is due to the propagation of radially induced tensile cracks initiated from the surface of the central hole and propagating toward the direction of diametrical loading. In the case of disc specimens with multiple holes, the cracks propagation and cracks coalescence may occur simultaneously in the breaking process of model under diametrical compressive loading. Finally the results shows that the failure stress and crack initiation stress decreases by increasing the hole diameter. Also, the failure stress decreases by increasing the number of hole which mobilized in failure. The results of these simulations were comprised with other experimental and numerical test results. It has been shown that the numerical and experimental results are in good agreement with each other.
... On the contrary, softening the contact stiffness may allow too much block penetration, resulting in the failure of contact detection algorithm and even complete interpenetration of a small block into a much larger block. This shortcoming of virtual contact spring is overcome by complementary form DDA 56 and dual form DDA, 57 which apply contact forces as basic variables and use an iterative algorithm called "compatibility iteration" to fulfill the inequality contact constraints. The proposed NPC approach and MNPC approach can be regarded as explicit solution approaches. ...
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 predictorcorrector 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 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.
... The first assumption also implies that the displacement and deformation of each block are so small that the higher order terms regarding the block rotation and block deformation can be neglected. Contact interaction can be solved in the DDA framework by various techniques, such as the penalty function method [2,[53][54][55][56][57][58], Lagrange multiplier method [60], augmented Lagrange multiplier method [61] or complementary theory [62]. In contrast to the traditional penalty function scheme, the complementary form DDA [63], dual form DDA [64], and explicit contact constraints scheme [65] can solve contact interaction without using the implicit OCI process or penalty springs. ...
Algorithmic robustness is important in the contact interaction analysis of polyhedral blocks using discontinuous computation methods. Several robustness issues of contact analysis associated with the identification of four contact types are discussed here, including rounding errors of floating-point operations, criteria and tolerances in contact type identification, and restrictions of input parameters. This paper also proposes revised criteria to identify contact types and general rules to specify tolerances regarding contact searches, quasi-parallel edges, overlapping angle and maximum displacement in a time step. These rules facilitate robust contact analysis of polyhedral blocks in the discontinuous deformation analysis framework.
... The open-close iteration is not always convergent, especially when many point-to-point contact candidates are involved in the problem. Many improvements have been applied to avoid the contact spring (Cai et al. 2000;Lin et al. 1996;Zheng and Jiang 2009), but the open-close iteration is still necessary to determine the contact state and it may fail due to rank deficiency of the stiffness matrix Zheng and Li 2015;Zheng et al. 2016). Furthermore, it has no efficient way to solve the point-to-point contact condition (Bao and Zhao 2012). ...
The combined finite–discrete-element method (FDEM) has made a groundbreaking progress in the computation of contact interaction. However, FDEM has a strict requirement on the element type, and the simulation result may be inconsistent due to a deficiency of physical meaning of the potential function. To address this problem, a new 3D discrete-element method based on a distance potential is proposed for a system consisting of a large number of arbitrary convex polyhedral elements. In this approach, a well-defined distance potential is proposed as a function of the penetration between the contact pairs. It exhibits a clear physical meaning and a precise measurement of the embedding between the elements in contact. The newly presented method provides a holonomic and accurate contact interaction without being influenced by the element shape. Therefore, the restraint of the element type in FDEM is removed and the proposed method can be used for arbitrary convex polyhedrons. In addition, an improved contact detection algorithm for non-uniform block discrete elements is provided to overcome the constraint of elements with the same size in the Munjiza-No Binary Search contact detection method. The new approach retains the merits of the FDEM and avoids its deficiencies. It is validated with well-known benchmark examples including an impact simulation, a friction experiment, a joint structure of a sliding rock mass, pillar impact, block accumulation, and analysis for the failure process of wedge slope. The results of this proposed method are in excellent agreement with the existing experimental measurements and analytical solutions.
... For example, instead of a penalty spring, Jiao et al. [37] presented a new twodimensional contact constitutive model, which consists of a two-phase force-displacement and the Mohr-Coulomb criterion in the normal and shear direction, respectively, to simulate the fragmentation of jointed rock. To avoid artificial parameters and the open-close iteration, Zheng et al. [38,39] a new NMM contact search algorithm with double-ended spatial sorting and improve contact detection efficiency. ...
In the classical numerical manifold method (NMM), contact conditions are enforced by applying or releasing contact springs repeatedly. Penalty spring stiffness is constant during the calculation process. In this study, a modified contact model describing joint mechanical behavior is proposed using the NMM. Two rock joint constitutive models, namely the BB model and hyperbolic model, are used to modify normal and shear stiffness, respectively. In the modified contact model, stiffness varies with normal and shear inter-penetration distance in each iteration step. Therefore, stiffness varies for different contact points. With increasing inter-penetration distance, the normal stiffness increases and shear stiffness decreases. As a result, the force needed for trigger sliding along the joint increased. Finally, three example simulations are conducted to validate the effectiveness of the modified contact model.
... Hsiung (2001) replaced the first-order displacement function in the traditional DDA method with a higherorder displacement function, while Grayeli and Hatami (2008) adopted finite-element mesh to partition DDA blocks to obtain more accurate stress distribution. Apart from the penalty method, contacts between blocks are modeled by Cai et al. (2000) using a Lagrange multiplier method, by Lin et al. (1996) and Bao et al. (2014) using an augmented Lagrangian method (ALM), and by Zheng and Jiang (2009) and Zheng and Li (2015) using a complementary theory. Moreover, many efforts have been made to develop a 3D DDA method (Zhu et al. 2016;Yeung et al. 2003Yeung et al. , 2007Jiang and Yeung 2004;Beyabanaki et al. 2010). ...
In the traditional discontinuous deformation analysis (DDA) method, the implicit time integration scheme is used to integrate equations of motion for modeling the mechanical behavior of a highly discrete rock block system. This requires that global equations be constantly solved. Hence, the computational efficiency of the traditional DDA method will decrease, especially when large-scale discontinuous problems are involved. Based on the explicit time integration scheme, an explicit version of the DDA (EDDA) method is proposed to improve computational efficiency of the traditional DDA method. Since a lumped mass matrix is used, there is no need to assemble global mass and stiffness matrices. More importantly, solving large-scale simultaneous algebraic equations can be avoided. The open-close iteration, which can assure the correct arrangement of constraints, is kept in the EDDA method. In addition, the simplex integration method, which is capable of conducting exact integration over an arbitrarily shaped block, is employed. Two numerical examples, including a sliding problem with an analytical solution and an underground cavern, are solved. The numerical results indicate the accuracy and robustness of the proposed EDDA method.
... The one temporary spring method [28] and the angle-based method [18] were used to tackle the indeterminacy of vertex-vertex contact. In order to improve the accu- racy of contact force, the Lagrange multiplier method [29], the augmented Lagrange multiplier method [30], the variational inequality method [31,32] and the complementarity method [33][34][35] were also considered. Recently, the strain-rotation decomposition theorem was introduced into the DDA [36,37]. ...
The distributed contact force that is determined from the contact potential can be used to deal with the contact between discrete bodies. The current study defines the generalized contact potential (GCP). The GCP is independent of the shape and size of body and possesses the thorough geometric locality. If and only if the contact regions are exactly the same, the contact potentials are identical. By combining discontinuous deformation analysis (DDA) and GCP a new DDA, named GCP-DDA, is produced, in which the generalized-α method is adopted to discretize the time domain. The GCP-DDA can make the global controlling equation and the open-close iteration (OCI), which may cause the reduction of time step size and the rebuilding and solving of global controlling equation, become unnecessary. The intractable issues related to convex-convex contact in the original DDA can be accordingly bypassed.
... Solutions have been proposed to restrain false volume expansion when simulating large rotation, such as adopting the second-order approximation of sin and cos in the shape function [11], executing a postadjustment once the open-close iteration converges [12], introducing a new displacement variable based on the trigonometric function transformations [13] and accumulating the total strain components in fixed local frames [14]. To avoid the introduction of virtual springs, which are commonly used in classical DDA, the DDA has been successfully reconfigured using the Lagrange multiplier method [15], the complementarity theory [16], and the variational inequality theory [17]. Recently, a dual-form DDA has been proposed by Zheng [18], in which the contact forces instead of the block displacements are utilized as the basic variables and the compatibility iteration for the quasi-variational inequality guarantees the convergence and solution efficiency. ...
In the original discontinuous deformation analysis (DDA) method, the complete first-order displacement function is used to describe block movement and deformation, which induce constant stress and strain throughout the block. To achieve a more detailed stress distribution, Wachspress interpolation displacement function is employed to express the displacement of blocks in DDA, and the interactions between blocks are still governed under the original DDA. Displacements of the vertexes of all blocks constitute new freedom vectors, and the stiffness and force matrix formulations are derived again. In the new formulation, Wachspress interpolation ensures that the edges of the blocks are straight; therefore, contact detection can be processed based on the original DDA. Several classical examples are analyzed. The results show that the new formulation obtains similar configurations as the original DDA but provides more detailed and continuous stress distributions within block element.
... [40][41][42] In particular, the SR-DDA formulation helps to void the small deformation assumption. The issue of contact nonlinearity has been tackled more recently by such methods including the open-close iteration (OCI), 31 the augmented Lagrange multiplier method, 43 the Lagrange multiplier method, 44 the complementarity method, [45][46][47] and the variational inequality method. 48 Other latest developments in DDA include the angle-based method 49 in addressing the indeterminacy of vertex-vertex contact, the new contact theory, 50 and the generalized contact potential-based DDA 51 to address the potential contact. ...
For modeling discrete particle‐block systems, a new framework of Discontinuous Deformation Analysis (DDA) is established on the basis of finite‐dimensional variational inequality. The presented method takes into account the contacts, the rolling resistance, and the tensile resistance of cemented interface among particles and blocks using the corresponding variational or quasi‐variational inequalities. The new formulation avoids using the artificial springs that are usually indispensable in many conventional methods dealing with similar discrete problems, and conveniently integrates the rigid circle particles, the non‐rigid ring particles and the arbitrary shape blocks into a uniform framework. The proposed DDA approach is further coupled with the finite element method (FEM) using a node‐based composite contact matrix and several simple transformation matrices to solve practical problems. A particle/block‐based composite contact matrix is constructed to further broaden the application of the proposed method. The accuracy, robustness and capability of the presented method are demonstrated with examples.
... For instance, instead of the penalty contacting spring, Jiao et al. [3][4][5] used a contact constitutive model, which consists of a two-phase force-displacement relation in the normal direction and the Mohr-Coulomb criterion in the shear direction, respectively, to simulate the whole process of rock fragmentation, and they also used the bonded DDA algorithm and hydro-mechanical coupling approach to simulate rock hydraulic fracturing processes. To avoid the penalty factors and the open-close iteration, Zheng and Jiang [6] reformulated the DDA as a mixed complementary problem (MiCP) and chose the path Newton method (PNM) to solve this problem. Contacts between any two neighboring blocks are the fundamental problem for discontinuous deformation analysis. ...
Discontinuous deformation analysis (DDA) method is a newly developed discrete element method which employs the implicit time-integration scheme to solve the governing equations and the open-close iteration (OCI) method to deal with contact problem, its computational efficiency is relatively low. However, spherical element based discontinuous deformation analysis (SDDA), which uses very simple contact type like point-to-point contact, has higher calculation speed. In the framework of SDDA, this paper presents a very simple contact calculation approach by removing the OCI scheme and by adopting the maximal displacement increment (MDI). Through some verification examples, it is proved that the proposed method is correct and effective, and a higher computational efficiency is obtained.
... Because DDA uses penalty method to substitute actual force and use the open-close iteration method, some control parameters [13] (i.e. spring stiffness, time step, etc.) have great effect on the kinematic characteristics. ...
... The scheme cannot guarantee that the iteration is always convergent. In this paper, after the explicit expression of block displacement containing dynamic contact force is deduced, the contact constraint between the dynamic contact force and the block displacement [33,34] is introduced to form the contact constraint equations, which can play the role of open-close iteration. ...
This paper proposes an explicit dynamic DDA method considering dynamic contact force, which aims at solving the problems of low efficiency of dynamic contact detection and the simulation of dynamic contact force in the conventional DDA method. The mutual contact between blocks can be regarded as the application of point loading on a single block, and the corresponding contact submatrix can be calculated and the simultaneous equations of the block system can be integrated. The central difference method is adopted to deduce the explicit expression of block displacement containing dynamic contact force. With the relationship between displacement and dynamic contact force, contact constraint equations of a block system are obtained to calculate the dynamic contact force and the corresponding block displacement. The accuracy of the explicit dynamic DDA method is verified using two numerical cases. The calculation results show that the new DDA method can be applied in large-scale geotechnical engineering.
... Some convergence criterions 30 , the trick of contact state recovery 31 , and the strategy of strengthening the movement trend 32 speeded up the open-close iteration. The augmented Lagrange multiplier method 33 , the Lagrange multiplier method 34 , the complementarity method [35][36][37] the variational inequality method 38 improved the accuracy of contact force. The one temporary spring method 39 and the angle-based method 32 handled the inde-terminacy of vertex-vertex contact. ...
With the development of 3D discontinuous deformation analysis (DDA) in precise stress fields and crack propagation problems, it has also demonstrated outstanding capabilities in solving continuous–discontinuous problems. However, currently, 3D DDA modeling primarily focuses on generating rock joint networks and developing 3D cutting algorithms. Correspondingly, 3D geological modeling methods are not yet mature, and establishing 3D models often demands substantial time. The lack of supporting preprocessing modeling methods and corresponding visual operation interfaces significantly hampers the development of 3D DDA. This method builds upon advanced research achievements in unmanned aerial vehicle oblique photography, 3D reconstruction, 3D cutting, computer graphics, and visualization program design. This research establishes a 3D geological entity modeling method for 3D DDA and constructs a comprehensive program using relevant C++ libraries and C language interfaces. In this method, a 3D geological model that incorporates geological elements such as strata and faults is initially established using non‐uniform rational B‐splines (NURBSs) surfaces as the boundary of the solid model. Subsequently, finite element meshing is applied, followed by corresponding topology transformation, resulting in a 3D block system model suitable for 3D DDA calculation. To cater for diverse application scenarios, continuous–discontinuous models integrated with subblocks and models of arbitrary polyhedra can be established. The proposed method has been validated through several typical modeling examples, showing its ability to rapidly and generate 3D high‐precision geological reality models suitable for 3D DDA calculations. Additionally, some techniques used in this method can be extended for modeling other numerical simulation methods, warranting further research.
Dynamic failure widely exists in rock engineering, such as excavation, blasting, and rockburst. However, the quantitative measurement of the dynamic damage process using experimental methods remains a challenge. In this study, a SHPB modeling technique is established based on Voronoi-based DDA to study the damage evolution of Fangshan granite under dynamic loading. The assessment of cracking along the artificial joints among Voronoi sub-blocks is conducted by employing the modified contact constitutive law. A calibration procedure has been implemented to investigate the rock dynamic properties quantitatively. The dispersion and damping effect can be effectively eliminated by regular discretization in SHPB bars, based on which the dynamic stress equilibrium can be satisfied. To reproduce the loading rate effect of the dynamic compressive strength, which has been observed in the experiment, a modification strategy considering the influence of the rate effect on the strength meso-parameters is proposed. Using this strategy, the peak stresses of the transmitted waves predicted by DDA match well with those obtained from experiments conducted at different loading rates. The simulation results show that more microcracks are generated and the proportion of tensile cracks decreases as the loading rate increases. Furthermore, the dynamic mechanical behavior and fracturing process have also been discussed and compared with the experiments. The results show that the established SHPB system is a powerful tool for quantitative analysis of rock dynamics problems and is capable of handling more complex problems in the future.
Since its introduction, discontinuous deformation analysis (DDA) has been widely used in different areas of rock mechanics. By dividing large blocks into subblocks and introducing artificial joints, DDA can be applied to rock fracture simulation. However, parameter calibration, a fundamental issue in discontinuum methods, has not received enough attention in DDA. In this study, the parameter calibration of DDA for intact rock is carefully studied. To this end, a subblock DDA with Voronoi tessellation is presented first. Then, a modified contact constitutive law is introduced, in which the tensile and shear meso-strengths are modified to be independent of the bond lengths. This improvement can prevent the unjustified preferential failure of short edges. A method for imposing confining pressure is also introduced. Thereafter, sensitivity analysis is performed to investigate the influence of the calculated parameters and meso-parameters on the mechanical properties of modeled rock. Based on the sensitivity analysis, a unified calibration procedure is suggested for both cases with and without confining pressure. Finally, the calibration procedure is applied to two examples, including a biaxial compression test. The results show that the proposed Voronoi-based DDA can simulate rock fracture with and without confining pressure very well after careful parameter calibration.
The major obstacle for the application of discontinuous deformation analysis (DDA) in engineering problems is the high computational cost and poor efficiency. In this paper, the main algorithms of disk‐based DDA are redesigned and implemented on graphics processing unit (GPU) to improve its performance. First, a contact pair‐wise scheme is proposed to assemble the stiffness matrix on GPU. Second, a buffer strategy and a GPU version of grid‐based contact detection algorithm are developed to improve the efficiency of contact detection. Third, for solving the simultaneous equations, two iterative methods are considered along with the direct solver method. The parallel performances of proposed strategies are tested and compared with the CPU counterparts. The results show that the maximum speedup ratio is 14 for the assembly of the stiffness matrix and 215 for contact detection. The speedup ratio for solving simultaneous equations depends on several factors, and the preconditioned conjugate gradients method (pcg) is suggested. Finally, the effectiveness and performance of the proposed GPU accelerated disk‐based DDA is further demonstrated by several examples, one of which consisted of over 500,000 particles. The results show that the proposed method can achieve a satisfactory speedup ratio, and is ready for large‐scale problems.
In nature, some flaws (such as fissures, joints, weak surfaces, and faults, etc.) are pre‐existed in the rock masses, which have been proved an important influence on the mechanical properties of rock masses. In this paper, a three‐dimensional (3D) joint model of a fractured specimen is built by means of the computed tomography (CT) scanning technic and digital image processing. The model shows three joints exist in the specimen: an essential joint which can be found from the specimen surface and two extra‐essential joints can only be detected from the 3D joint model. The 3D joint model is integrated into 3D particle flow code (PFC3D) to build a jointed numerical specimen and triaxial compression tests under different confining pressures are simulated. The results show that the linear phase of the stress‐strain curve is more significant under larger confining pressure. The peak strength of the jointed specimen increases linearly with the confining pressure. In addition, the peak strength of the jointed specimen shows highly consistency with the residual strength of the corresponding intact specimen. For both the intact and jointed specimen, Young's modulus increases nonlinearly with the confining pressure, and remains constant after confining pressure reaching 20 MPa.
Since the basic theory of the discontinue deformation analysis (DDA) method was proposed, the DDA open source has gone through a long development process. At present, different kinds of programs have been widely applied in rock mass engineering such as slope, dam, and tunnel. This paper introduces the solution principle of DDA motion equations in detail, as well as the development status of the 2D open-source program. Numerical simulation of shaking table test of rock mass engineering using 2D DDA program is highlighted, and investigations of seismic wave pre-processing and seismic input method are carried out. First, based on the Newmark integration scheme, the integration algorithms of synthetic or measured seismic wave time history, correction function of seismic wave, and DDA simulation are unified. Then, three seismic input methods are implanted in the DDA program, and the applicability of various seismic input methods is discussed. On this basis, using the improved seismic 2D DDA program, a shaking table test of typical rock mass engineering is simulated. Through the comparison between the theoretical/test data and simulation results, the reliability of the improved DDA program in seismic response analysis is verified; the large mass method and the large stiffness method are more suitable for rigid foundation, such as shaking table test; the propagation of the seismic wave presents a significant amplification effect due to the reflection, refraction, and diffraction in the tunnel. The research results provide DDA theory and an open-source program for analyzing the seismic response of rock mass engineering.
Liu et al. (2018) preliminarily proposed a bonded-particle model (BPM) for disk-based discontinuous deformation analysis (DDA). This BPM can resist relative rotation of contact pairs by introducing a rolling spring. In this technical note, the ability of disk-based DDA with this BPM to simulate the fracturing process of rock is explored. The detailed formulations of the BPM fitted to DDA are derived. A simple elastic-brittle constitutive model is employed as the bond failure criterion, with the help of which the fracture of rock can be explicitly represented by the progressive failure of the bonds. Sensitivity analysis is performed to investigate the influence of micro parameters on the macro parameters, such as the elastic parameters (Young's modulus E and Poisson's ratio v) and the strength parameters (uniaxial compression strength and indirect tensile strength). Based on the sensitivity analysis, a calibration process is suggested. Finally, several numerical examples are presented to simulate the rock failure process under both static and dynamic loading. The results show that disk-based DDA is a good candidate for rock failure simulation.
Fracture and fragmentation of rock under various loadings or in multi-field conditions are important topics in many rock engineering scenarios. Numerical tools are commonly applied to evaluate the rock deformation and progressive failure process, while the accuracy and efficiency of the analysis are closely related to the applied fracture models. In this paper, an improved discontinuous deformation analysis (DDA) program with the distributed bond (DB) is proposed for rock deformation and rock fragmentation analysis. The jointed rock model is regarded as an assembly of triangular elements connected with DBs and contacts. The DB uses distributed constraints to bond the co-edge elements for continuous deformation modeling, and the fracturing and fragmentation of rock material is fulfilled by progressive DB failure with linear and nonlinear bond constitutive models. The theory, formulation, and implementation of DB in DDA framework are given in detail. The feasibility and accuracy of the proposed DB in simulating rock deformation and failure process are verified by several examples.
This study improved two parts of the virtual element method (VEM) to ensure that the stability of a stony soil slope, especially under excavation conditions, could be better studied. One of the improvements was the specific realization of the excavation algorithm, including the determination of how to identify polygonal elements that needed to be excavated, and the “deactivate and reactivate polygonal elements” used to calculate the excavation load during the slope excavation process. The second improvement was the calculation of the factor of safety (FOS) of a slope by combining shear strength reduction and the φ−ν inequality so that the stability of the stony soil slope could be better evaluated. The principal goal was to extend the latest developed virtual element method to the analysis and calculation of the excavation of a stony soil slope. Because the VEM was shape independent, it also allowed the number of nodes in an element to be chosen flexibly and fluidly, so the number of nodes could be changed easily during a simulated procedure. Two numerical examples of the one-step excavation of a homogeneous slope and the two-step excavation of a stony soil slope were solved with the improved VEM. The numerical results showed that the VEM could accurately simulate the excavation process of the slope.
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.
The rotation degree of freedom in discontinuous deformation analysis may cause false volume expansion when a block undergoes a large rotation. We propose a block displacement function to prevent this defect. Specifically, the degrees of freedom in a block are redefined by incremental displacements at its vertices, and displacement is formulated based on the mean value coordinates. In addition, the finite element method with updated Lagrangian formulation is employed to derive the equilibrium equations, while the contact analysis and implicit time integration for dynamics is maintained from the original discontinuous deformation analysis. After each time step, the block configuration is updated by adding the new degrees of freedom to the previous coordinates of the block vertices. Results from numerical examples confirm the effectiveness of the proposed approach to prevent false volume expansion, ensure correctness of contact analysis, and provide realistic stress results when simulating large rotations.
The first-order displacement function of the original discontinuous deformation analysis (DDA) results in uniform stress within each block, and existing remedies to obtain a stress distribution for each block cause the number of contacts to increase. To remedy this issue, a mean value coordinate-based DDA (Mvc-DDA) approach is proposed for 2D in this study. New degrees of freedom are defined by using the displacements at the block vertices. Mvc-DDA obtains detailed stress distributions within the blocks and ensures that the block boundary edges are straight so that contact detection can be performed as it is performed in the original DDA. The results of numerical examples verify that Mvc-DDA is effective in generating more detailed stress distributions within blocks than is possible with the original approach, regardless of the block shapes involved.
The efficiency of solving equations plays an important role in implicit‐scheme discontinuous deformation analysis (DDA). A systematic investigation of six iterative methods, namely, symmetric successive over relaxation (SSOR), Jacobi (J), conjugate gradient (CG), and three preconditioned CG methods (ie, J‐PCG, block J‐PCG [BJ‐PCG], and SSOR‐PCG), for solving equations in three‐dimensional sphere DDA (SDDA) is conducted in this paper. Firstly, simultaneous equations of the SDDA and iterative formats of the six solvers are presented. Secondly, serial and OpenMP‐based parallel computing numerical tests are done on a 16‐core PC, the result of which shows that (a) for serial computing, the efficiency of the solvers is in this order: SSOR‐PCG > BJ‐PCG > J‐PCG > SSOR>J > CG, while for parallel computing, BJ‐PCG is the best solver; and (b) CG is not only the most sensitive to the ill‐condition of the equations but also the most time consuming under both serial and parallel computing. Thirdly, to estimate the effects of equation solvers acting on SDDA computations, an application example with 10 000 spheres and 200 000 calculation steps is simulated on this 16‐core PC using serial and parallel computing. The result shows that SSOR‐PCG is about six times faster than CG for serial computing, while BJ‐PCG is about four times faster than CG for parallel computing. On the other hand, the whole computation time using BJ‐PCG for parallel computing is 3.37 hours (ie, 0.061 s per step), which is about 36 times faster than CG for serial computing. Finally, some suggestions are given based on this investigation result.
Discontinuous deformation analysis (DDA) computes the mechanical behaviours of discrete deformable-block systems. It also has been used to simulate continuous rock fracturing, e.g., through the sub-block approach. In the present study, the sub-block DDA method is first verified in simulations of continua. The influences of the contact penalty spring stiffness and mesh size on the stress and strain calculations are investigated. Thereafter, the fracturing modelling algorithm of the sub-block DDA is improved, which determines the tensile or shear fracturing failure along artificial joints based on the stress state of adjacent sub-blocks according to the maximum tensile stress criterion and the Mohr–Coulomb criterion, respectively. The tensile, shear, mixed-mode, as well as dynamic fracturing failures of rock samples are simulated, and the results are calibrated theoretically or experimentally. Compared with former sub-block DDA fracturing modelling algorithms which determine the fracturing failure along artificial joints based on the contact stresses between sub-blocks, the improved algorithm has better accuracy in terms of the failure strength, and it significantly reduces the influence of the distribution of the pre-set artificial joints on the failure strength and fracturing route simulation results. This work makes DDA a better candidate for use in rock fracturing problem simulations.
The complexity of polyhedral block systems (e.g., small blocks, flat blocks with small angles, edges, or faces) poses challenges in the kinetic analysis of rock block systems. This paper proposed an improved potential-based penalty function approach within an explicit three-dimensional (3D) discontinuous deformation analysis (DDA) framework for efficient and robust kinetic analysis of rock block systems. An explicit formulation of 3D DDA based on velocity verlet algorithm is first derived. A novel definition of potential function is then proposed with details of the key algorithms for overlap judgment of convex polyhedron, construction of intersection polyhedron and numerical integral for computation of contact force. The improved potential-based penalty function method is robust and efficient for complex convex polyhedral shapes. Several benchmark and application examples verify the feasibility, accuracy and robustness of the proposed methods in solving contact of polyhedral block systems.
In conventional discontinuous deformation analysis (DDA), the procedure called open–close iteration is adopted to enforce the contact condition, which needs to repeatedly fix and remove the artificial springs between blocks in contact to determine real contact states. The open–close iteration belongs to the category of trial-and-error methods, in which convergence cannot be always guaranteed. Meanwhile, the contact force is treated as concentrated force, leading to the difficulties in determining the shear strength from cohesion and stresses in the contact area. The so-called potential contact force concept adopted in the combined finite-element method and discrete-element method (FEM-DEM) has been proved efficient and robust. In the FEM-DEM the contact force is treated as a distributed contact force, which is more realistic and was utilized in this study to tackle contacts. A major advantage over the conventional DDA lies in the elimination of the need to handle singular contact types that would incur huge difficulties in three-dimensional simulations. Therefore, a contact potential–based DDA (CPDDA) was developed by introducing potential contact forces. Some typical examples, including those originally designed by the DDA inventor, are reanalyzed, proving the feasibility of CPDDA.
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.
A partition-of-unity (PU) based “FE-Meshfree” three-node triangular element (Trig3-RPIM) was recently developed for linear elastic problems. This Trig3-RPIM element employs hybrid shape functions that combine the shape functions of three-node triangular element (Trig3) and radial-polynomial basis functions for the purpose of synergizing the merits of both finite element method and meshfree method. Although Trig3-RPIM element is capable of obtaining higher accuracy and convergence rate than the Trig3 element and four-node iso-parametric quadrilateral element without adding extra nodes or degrees of freedom (DOFs), the nodal stress field through Trig3-RPIM element is not continuous and extra stress smooth operations are still needed in the post processing stage. To further improve the property of Trig3-RPIM element, a new PU-based triangular element with continuous nodal stress, called Trig3-RPIMcns, is developed. Numerical examples including several linear, free vibration and forced vibration test problems, have confirmed the correctness and feasibility of the proposed Trig3-RPIMcns element.
The numerical manifold method (NMM) is suitable for the solution of both continuous and discontinuous problems in geotechnical engineering. In the conventional NMM, the contact between blocks is treated with the open-close iteration, which needs to fix or remove spurious springs between two blocks in contact and to assume properly the normal stiffness and the tangential stiffness (the penalty parameters). Unreasonable values of stiffness would result in numerical problems. To avoid the penalty parameters, contacts are treated in a direct way in which contact forces are primal variables. Numerical examples have confirmed the correctness and feasibility of the proposed procedure.
The key to three-dimensional discontinuous deformation analysis (3D DDA) is a rigorous contact theory that governs the interaction of many three-dimensional blocks. This theory must provide algorithms to judge contact types and locations and the appropriate state of each contact, which can be open, sliding or locked. This paper presents a point-to-face contact model, which forms a part of the contact theory, to be used in 3D DDA. Normal spring, shear spring and frictional force submatrices are derived by vector analysis and the penalty method. Also given are the open-close iteration criteria and operations performed for different changes in contact state. Sliding at a contact can occur in any direction parallel to the contact face, as opposed to one of two directions in two-dimensional DDA. This point-to-face contact model has been implemented into a 3D DDA computer program, and numerical results from several test cases demonstrate the validity of the model and the capability of the program.
Detection of contacts between 3D blocks is a key problem in 3D DDA (discontinuous deformation analysis) and DEM (discrete element method) analysis. The limitations of the approaches commonly used to detect 3D block contacts are discussed. A new approach called penetration edges method is put forward for the detection of contacts in 3D blocks system, and the contacts between two 3D blocks are classified into seven types. The principle of this new approach is robust and can overcome the limitations of the commonly used methods. This new method can greatly reduce the amount of calculation and is easy to be coded for analysis.
In discontinuity based numerical methods, determining the contact points between interacting blocks and the associated contact forces at each time step is an important and time consuming calculation. This paper presents a new algorithm for detection of all contact patterns between any two convex blocks. In the new developed method one of the two approaching faces is considered as “main plane” and vertices of another face that are within the tolerance of the main plane are stored. Then, according to the number of gathered points, different algorithms for searching real contact points and the type in global coordinate system are applied. Finally, the dominant contact of neighboring blocks is identified using the contacts found between approaching faces.
This article has been retracted at the request of the Editors-in-Chief. Please see Elsevier Policy on Article Withdrawal (http://www.elsevier.com/locate/withdrawalpolicy).
Reason: The authors have plagiarized part of a paper that had already appeared in Comput. Geotech., 31 (2004) 575–587, doi:10.1016/j.compgeo.2004.08.002. One of the conditions of submission of a paper for publication is that authors declare explicitly that their work is original and has not appeared in a publication elsewhere. Re-use of any data should be appropriately cited. As such this article represents a severe abuse of the scientific publishing system. The scientific community takes a very strong view on this matter and we apologize to readers of the journal that this was not detected during the submission process.
Discontinuous deformation analysis (DDA) by Shi is an alternative to the commonly used discontinuity based discrete element method. The author discovered that there are some problems in both the theory and the actual implementation of this method. In this paper, the author will discuss some modifications and improvements to the original method so that the problems of DDA can be overcome and it can become a practical tool for general types of problems. DDA is also applied to a classical problem in continuum and compare with UDEC and classical solution to demonstrate the applicability of this method towards general problems.
A numerical model of deformable block systems that gives a unique solution for large displacement, large deformation and failure computations is presented. The forces acting on each block, from external loading or contact with other blocks, satisfy the equilibrium equations. Equilibrium is also achieved between external forces and the block stresses. Furthermore, the analysis fulfills constraints of no tension between blocks and no penetration of one block into another. Also, Coloumb's law is fulfilled at all contact positions for both static and dynamic computations. The program ready algorithms with brief derivations are stated in this paper.
This article presents a new vertex-to-face contact searching algorithm for the three-dimensional (3-D) discontinuous deformation analysis (DDA). In this algorithm, topology is applied to the contact rule when any two polyhedrons are close to each other. Attempt is made to expand the original contact searching algorithm from two-dimensional (2-D) to 3-D DDA. Examples are provided to demonstrate the new contact rule for vertex-to-face contacts between two polyhedrons with planar boundaries. Copyright © 2005 John Wiley & Sons, Ltd.
The key to three-dimensional discontinuous deformation analysis (3D DDA) is a rigorous contact theory that governs the interaction of many three-dimensional blocks. Such a theory must provide ways to judge contact types and contact locations and also formulas for contact submatrices. A part of this contact theory is an edge-to-edge contact model. This paper presents the details of an edge-to-edge contact model to be used in 3D DDA, including the detection of edge-to-edge contact type, judging of the first entrance faces, and the criteria for inter-penetration. Contact submatrices are derived by vector analysis and the penalty function method. This edge-to-edge contact model has been implemented into a 3D DDA computer program. Three test cases demonstrate the reasonableness of the model and the capability of 3D DDA to deal with large movements and multi-block interaction.
A dynamic, two-dimensional, stability analysis of a highly discontinuous rock slope is demonstrated in this paper. The studied rock slope is the upper terrace of King Herod's Palace in Masada, situated on the western margins of the seismically active Dead Sea Rift. The slope consists of sub-horizontally bedded and sub-vertically jointed, stiff, dolomite blocks. The dynamic deformation of the slope is calculated using a fully dynamic version of DDA in which time-dependent acceleration is used as input. The analytically determined failure modes of critical keyblocks in the jointed rock slope are clearly predicted by DDA at the end of the dynamic calculation. It is found however that for realistic displacement estimates some amount of energy dissipation must be introduced into the otherwise fully elastic, un-damped, DDA formulation. Comparison of predicted damage with actual slope performance over a historic time span of 2000 years allows us to conclude that introduction of 2% kinetic damping should suffice for realistic damage predictions. This conclusion is in agreement with recent results of Tsesarsky et al. (In: Y.H. Hatzor (Ed.), Stability of Rock Structures: Proceedings of the Fifth International Conference of Analysis of Discontinuous Deformation, Balkema Publishers, Lisse, 2002, pp. 195–203) who compared displacements of a single block on an inclined plane subjected to dynamic loading obtained by DDA and by shaking table experiments. Using dynamic DDA it is shown that introduction of a simple rock bolting pattern completely stabilizes the slope.
The three-part wedge limit equilibrium method for seismic stability analysis of the landfill along liners is presented. The
approximate solutions of the factor of safety and the yield acceleration coefficient are obtained. Parametric studies show
that the interface strength of liners, the shear strength of waste and the height of retaining wall can influence the seismic
stability of landfill along liners. The density and the shear wave velocity of the field waste are obtained by the borehole
investigation and the spectral analysis of surface wave (SASW), respectively. The strain-dependent shear modulus and damping
ratio of the artifical waste are obtained by the moderate-scale dynamic triaxial tests. The one-dimensional (1D) equivalent
linear dynamic response analysis is used to calculate the horizontal equivalent seismic coefficient-time history of the sliding
landfill during earthquake. The seismic permanent displacement of the landfill along liners with different site conditons
and heights is evaluated by the Newmark method. The catculated results show that ratio of k
y/k
max, site conditions, the amplitude and frequency content of the bedrock motion can affect the seismic permanent displacement
of the landfill along liners in some degree. Finally, the seismic stability and permanent displacements of three expanded
configurations of a certain landfill case are analyzed.
Discontinuous deformation analysis (DDA) provides a powerful numerical tool for the analysis of discontinuous media. This
method has been widely applied to the 2D analysis of discontinuous deformation. However, it is hindered from analyzing 3D
rock engineering problems mainly due to the lack of reliable 3D contact detection algorithms for polyhedra. Contact detection
is a key in 3-D DDA analysis. The limitations and advantages of existing contact detection schemes are discussed in this paper,
and a new approach, called the incision body (IB), is proposed, taking into account the advantages of the existing methods.
A computer code 3DIB, which uses the IB scheme as a 3D contact detection algorithm, was programmed with Visual C++. Static and dynamic stability analysis for three realistic engineering problems has been carried out. Furthermore, the focus
is on studying the stability of a gravity dam on jointed rock foundation and dynamic stability of a fractured gravity dam
subject to earthquake shaking. The simulation results show that the program 3DIB and incision body scheme are capable of detecting
3D block contacts correctly and hence simulating the open-close and slide process of jointed block masses. In addition, the
code 3DIB could provide an effective tool for evaluating the safety of 3D dam structures, which is quite important for engineering
problems.
A powerful numerical method that can be used for modeling rock-structure interaction is the discontinuous deformation analysis (DDA) method developed by Shi in 1988. In this method, rock masses are treated as systems of finite and deformable blocks. Large rock mass deformations and block movements are allowed. Although various extensions of the DDA method have been proposed in the literature, the method is not capable of modeling water-block interaction, sequential loading or unloading and rock reinforcement; three features that are needed when modeling surface or underground excavation in fractured rock. This paper presents three new extensions to the DDA method. The extensions consist of hydro-mechanical coupling between rock blocks and steady water flow in fractures, sequential loading or unloading, and rock reinforcement by rockbolts, shotcrete or concrete lining. Examples of application of the DDA method with the new extensions are presented. Simulations of the underground excavation of the ‘Unju Tunnel’ in Korea were carried out to evaluate the influence of fracture flow, excavation sequence and reinforcement on the tunnel stability. The results of the present study indicate that fracture flow and improper selection of excavation sequence could have a destabilizing effect on the tunnel stability. On the other hand, reinforcement by rockbolts and shotcrete can stabilize the tunnel. It is found that, in general, the DDA program with the three new extensions can now be used as a practical tool in the design of underground structures. In particular, phases of construction (excavation, reinforcement) can now be simulated more realistically. However, the method is limited to solving two-dimensional problems.
Studying Impact Problem by LDDA Method, Discontinuous Deformation analysis (DDA) and Simulations of Discontinuous Media
- Y E Cai
- G P Liang
- G H Shi
- Y. E. Cai
Cai Y E, Liang G P, Shi G H, et al. Studying Impact Problem by
LDDA Method, Discontinuous Deformation analysis (DDA) and
Simulations of Discontinuous Media. Albuquerque: The Transformative Studies Institute (TSI) Press, 1996
Modeling fracture of rock masses with the DDA method
- B Amadei
- C Lin
- S Sture
- B. Amadei
Amadei B, Lin C, Sture S, et al. Modeling fracture of rock masses with
the DDA method. In: Nelson L. ed. Rock Mechanics. Rotterdam:
Balkema, 1994. 583-590
A fast common plane identification algorithm for 3D contact problems
- J. Liu
Studying Impact Problem by LDDA Method, Discontinuous Deformation analysis (DDA) and Simulations of Discontinuous Media. Albuquerque: The Transformative Studies Institute
- Y. E. Cai
Defection of 3D rock block contacts by penetration edges
- W S Chen
- H Zheng
- Y M Cheng
- W. S. Chen