Three-dimensional Discontinuous Deformation Analysis Research Articles

Numerical simulation of wedge failure of rock slopes using three-dimensional discontinuous deformation analysis

Numerical simulation of wedge failure of rock slopes using three-dimensional discontinuous deformation analysis

Meta-contact analysis based on contact theory in three-dimensional discontinuous deformation analysis

Accurate contact analysis between nonspherical blocks, such as polyhedral blocks, is crucial for modeling jointed rock masses. The Contact Theory provides a rigorous foundation for contact detection. However, designing a robust and efficient algorithm that conforms to the original Contact Theory remains challenging. This study introduces the meta-contact concept for polyhedral blocks, which corresponds to the contact cover on the entrance block of the Contact Theory. Using the meta-contact concept, contact type identification, determination of contact points and plans, and contact force calculation can be formulated while retaining the advantages of Contact Theory in dealing with irregular blocks with small edges and large penetration.

Directivity effect of the spatial distribution of co-seismic landslides affected by near-fault ground motions

Numerous co-seismic landslides occur in near-fault areas, and their spatial distribution seems to correlate well with the distribution of the energy and pulse of near-fault ground motions (NFGMs) in all directions. To confirm the relationship between the energy and pulse characteristics of NFGMs and the spatial distribution of co-seismic landslides, three-dimensional discontinuous deformation analysis (3D-DDA) was used to investigate the effect of NFGMs on initiating co-seismic landslides through an octagonal platform model and a generalized landslide. This model was subjected to the influence of two sets of representative three-component NFGMs. The outcomes derived from the block sliding processes revealed pronounced variations in the displacements of blocks in different spatial orientations. The greatest difference was up to 4.2 times, and the sliding blocks were more prone to large displacements in the direction with large energy and pulse. This study confirmed that the distribution characteristics of co-seismic landslides are related to the energy and pulse of NFGMs and provided evidence to illuminate the special distribution characteristics phenomenon of co-seismic landslides and theoretical support for the selection of factors for landslide hazard assessment.

Implementation of Viscoelastic Artificial Boundary and Seismic Motion Input Method in Three-Dimensional Discontinuous Deformation Analysis Method

Implementation of Viscoelastic Artificial Boundary and Seismic Motion Input Method in Three-Dimensional Discontinuous Deformation Analysis Method

Development of a Composite Implicit Time Integration Scheme for Three-Dimensional Discontinuous Deformation Analysis

Discontinuous deformation analysis (DDA) is a discontinuum-based and implicit method for investigating the deformational behavior of block systems. The constant acceleration integration (CAI) scheme characterized by unconditional stability is employed in the traditional DDA. In this study, the problems of the CAI scheme regarding the time step in DDA are pointed out. A too large or too small time step size adopted in the CAI scheme will have adverse effects on the DDA computation. To overcome the weaknesses, an alternative composite implicit time integration (CITI) scheme, which is a combination of the trapezoidal rule and the three-point backward Euler method, is implemented in the three-dimensional (3D) DDA method. Verification examples and slope numerical simulations are presented to illustrate the accuracy and effectiveness of the proposed methodology. The results showed that the CITI scheme can overcome the numerical error caused by the large time step size, and the algorithm damping is closely related to the time step size and the selected splitting parameter.

A new coupled depth-integrated model incorporating 3D DDA on debris flow with large boulders

Debris flow frequently occurs in mountainous regions and can cause serious damages to residents and structures in susceptible areas. Conventional debris flow simulation (DFS) with a depth-integrated model has been widely used to study debris flow behavior. In DFS, two-phase fluid of water and solid is considered. However, large boulders are often entrained in debris flows in practice. Obviously, the behaviors of debris flow with large boulders cannot be correctly described by DFS with two-phase fluid model. Therefore, a new coupling method for simulating debris flow with large boulders is developed by combining three-dimensional discontinuous deformation analysis (3D DDA) with conventional DFS. The dynamic behavior of those entrained boulders in debris flow can be solved by 3D DDA while the kinematic and mechanic behaviors of two-phase fluid are predicted by conventional DFS. An interactive module is introduced to address the interaction between solid motion in 3D DDA module and fluid field in DFS module. The imposed interaction forces from fluid on solid block include buoyancy, drag force and static pressure. Drag effect between adjacent blocks in debris flow is evaluated empirically to make the calculated drag force more realistic. For the block resistance to the debris flow, an equivalent solution is proposed by time-varying elevation of terrain node. Subsequently, the developed 3D DDA-DFS method is validated by several simple examples. Further, a case study is carried out to discuss the kinematic and mechanic behaviors of debris flow with large boulders based on a real catchment. The dam safety under different dam material properties is investigated when debris flow with large boulders happens. The simulation results suggest debris flow with large boulders can be more destructive than a pure debris flow. The developed 3D DDA-DFS method is a promising approach to study debris flow with large boulders.

Numerical Study on the Influence of Block Physical Characteristics on Landslide Migration Using Three-Dimensional Discontinuous Deformation Analysis

The physical characteristics of blocks have an important impact on the migration and deposition in landslides, so the damage of landslides to the surrounding environment often has great uncertainty. To explore how the physical properties of blocks affect the landslide movement, we used Compaq Visual Fortran software with a DDA program to simulate the movement of a block landslide in our study. The velocity and location changes of two types of blocks (triangular and square blocks) were simulated by comparison. The contents of the simulation included the following points: (1) the influence of block density and weight on migration, (2) the influence of the elastic-slip characteristics and spring penalty coefficient on velocity, and (3) interaction between block vertices and the sliding surface. The results showed how the shape and physical properties of the blocks affect the migration of those blocks. Although the triangular blocks appeared to be more stable than the square blocks, they only exhibited greater stability on the gentle slope. The weight and elastic modulus of the blocks could improve the velocity and destructive force, whereas the spring penalty coefficient between the blocks and the contact surface had an obscure effect on the velocity. However, the changes of the above three parameters all led to non-linear changes of velocity. The simulation results indicated that the blocks with different numbers of edges and surfaces had complex trajectories because of contact frequency and mobility. The results show that the physical properties of the blocks could be one of the factors that could ultimately change the displacement of landslides.

An improved 3D DDA method considering the unloading effect of tunnel excavation and its application

Tunnel excavation unloading is the primary factor in disasters such as collapse and large deformations of surrounding rock. Therefore, it is a key issue to study the deformation mechanism of the surrounding rock in the process of tunnel excavation unloading. However, the original three-dimensional discontinuous deformation analysis method (3D DDA) can only simulate the displacement of the surrounding rock after the completion of tunnel excavation, which is not suitable for studying the mechanism of the surrounding rock’s deformation caused by tunnel excavation. Therefore, this paper proposes a virtual block removal algorithm that moves the blocks out of the calculation area while ensuring the consistency of the array and develops an improved 3D DDA method (3D DDA_exc) by embedding the algorithm into the original 3D DDA program, which is suitable for the simulation of tunnel excavation. In addition, based on the free fall models of static and sliding blocks, we compare the results of the 3D DDA_exc simulation with theoretical calculation results, which are basically the same. Then, both the 3D DDA_exc method and the discrete element method (DEM) are used to simulate the excavation of the tunnel in a blocky rock mass, and the rock displacements around the tunnel hole are basically the same, which further illustrates the accuracy of the 3D DDA_exc method in simulating tunnel excavation. Finally, taking the Huangjiazhuang Tunnel of the Rizhao-Lankao high-speed railway as a case study, we simulated block collapse during the excavation process of the tunnel using the centre diaphragm method (CD method). Compared with the result of the 3D DDA method without the process of tunnel excavation, block collapse can be avoided if the dangerous blocks are supported and reinforced during the excavation of part ① and part ② of the tunnel, which was of guiding significance to the safe construction of the tunnel.

Simulation of tensile failure problems by improved three-dimensional discontinuous deformation analysis with multi-spring face-to-face contact model

Discontinuities remarkably dominate in jointed or blocky media, yielding an assemblage of blocks, and the mechanical properties of joints can greatly affect the behavior of a blocky system. Discontinuous deformation analysis (DDA) is promising method to analyze the failure process of a blocky system. In three-dimensional (3-D) blocky systems, face-to-face contact between adjacent blocks is common, especially for an ideal rock block system at the very early stage. The accurate modelling of the face-to-face contact is critical. The original 3-D DDA adopts the concentrated spring face-to-face contact model, in which a face-to-face contact is simply converted as the combination of several dominant contacts each with a couple of normal and shear springs. However, such simplification is not always accurate and can yield unreasonable results when simulating the progressive tensile failure between blocks under asymmetrical tension conditions. To overcome this problem, a multi-spring face-to-face contact model incorporated into 3-D DDA is proposed. In the proposed model, multiple couples of normal and shear springs are applied for a face-to-face contact to simulate the distributed force induced between contacting blocks. Several verification examples involving tensile failure are studied to show the advantages of the improved 3-D DDA. The simulated results obtained from the improved 3-D DDA converge to the theoretical solutions or experimental results while the original 3-D DDA gets unreasonable results. Thus, the improved 3-D DDA can provide more reasonable results when analyzing the tensile failure problems. • Improved 3-D DDA by incorporating a multi-spring face-to-face contact model. • Showed limitations of the original concentrated spring face-to-face contact model. • Presented the advantages of the proposed contact model in 3-D DDA. • Improved 3-D DDA can more reasonably simulate tensile failure problems.

Dynamic simulation of the water inrush process in tunnel construction using a three-dimensional coupled discontinuous deformation analysis and smoothed particle hydrodynamics method

Water inrush, which is a major cause of geological disasters during tunnel excavation, can worsen construction conditions and result in casualties. It is thus important to study the water inrush process for mitigating and preventing water inrush disasters in tunnel construction. The tunnel water inrush problem involves solid–fluid interactions. The coupling of the parallel three-dimensional discontinuous deformation analysis (3-D DDA) and 3-D smoothed particle hydrodynamics (3-D SPH) is suitable for analyzing this problem. The critical condition of a block under water pressure was first investigated to check the validity of the 3-D coupled DDA-SPH method. Further, a tunnel water inrush phenomenon was reproduced, and the mechanical behaviors of tunnels were explained with respect to the thickness T of the rock wall, the initial water height H, and the presence and location of a water outlet. The simulation results show that the tunnel deformation increases as T decreases and as H increases. The tunnel deformation was found to be larger when there was a water outlet; however, the location of the outlet had a slight effect on the deformation.

Three-dimensional discontinuous deformation analysis derived from the virtual work principle with a simplex integral on the boundary

Three-dimensional discontinuous deformation analysis (3D-DDA) is a powerful deformable discrete element method that is derived from the principle of minimum potential energy. However, the principle cannot be easily applied for all constitutive models, and certain forces do not have corresponding potential energy, which can impede DDA from being applied to more complex problems. In addition, since simplex integration in 3D can only be applied to tetrahedrons, no accurate integral is presented in 3D-DDA for polygonal boundaries of 3D blocks. Thus, face constraints have rarely been employed in 3D DDA. This study derives the DDA governing equation from the virtual work principle. The submatrices of stress, inertia, point force, face loading, body force, fixed boundary, roller boundary, bolts, implicit contact forces, and explicit contact forces are reformulated. An accurate integral on polygonal boundaries of 3D blocks is specified with the face integral of scalar fields and 2D simplex integration. Several numerical examples corroborate the correctness of the reformulation and merits of the face constraints.

Investigating the Role of Earthquakes on the Stability of Dangerous Rock Masses and Rockfall Dynamics

In recent years, earthquake rockfalls have occurred frequently all over the world, resulting in heavy casualties and property losses. Unfortunately, research on rockfall dynamics upon earthquake events is still rare due to limited field and experimental data, and restricted to numerical simulations of only two dimensions. In order to primarily reveal the role of earthquakes on rockfall, this study focused on the stability of rock blocks on an inclined slope, and following rockfall dynamics by three-dimensional discontinuous deformation analysis (3-D DDA). First, earthquake input methods were discussed and implemented for the triangulated regular network (TRN) in 3-D DDA. The effectiveness was verified by comparison with analytical solution results of a single block on the inclined slope under seismic loads. Further, by discussing the variations of the boundary chart of failure modes, it indicated that the block was more prone to slide even with a large friction angle, became instability under seismic conditions. Moreover, a regular dodecahedron rock block was released on a stochastic roughness slope with two platforms through parallel realizations. The indices of the movement characteristics of the block, such as runout distance, lateral displacement range, and resting position, were investigated. The results showed that the maximum runout distance was not sensitive to seismic load, but the lateral displacement range was significantly sensitive to seismic load and increased appreciably. Through the 3D-DDA numerical simulations, both rock stability and rockfall behaviors under earthquake conditions could be better understood. Furthermore, it will be helpful to by analyzing trajectories and kinetic energies predict earthquake rockfall disasters and design reasonable protective countermeasures under earthquake scenarios.

A Stochastic Rockfall Model Related to Random Ground Roughness Based on Three-Dimensional Discontinuous Deformation Analysis

The use of feasible 3-D numerical methods has become essential for addressing problems related to rockfall hazard. Although several models with various degrees of complexity are available, certain trajectories and impact dynamics related to some model inputs could fall in the rockfall observations area but are rarely calibrated against reflecting its range, especially the lateral deviations. A major difficulty exists in the lack of simulating the apparent randomness during the impact-rebound process leading to both ground roughness and block irregularities. The model presented here is based on three-dimensional discontinuous deformation analysis (3-D DDA). Despite similarities to previous simulations using 3-D DDA, the model presented here incorporates several novel concepts: (1) ground roughness is represented as a random change of slope angle by height perturbation at a grid point in DEM terrain; (2) block irregularities are modelled directly using polyhedron data; (3) a scaled velocity restitution relationship is introduced to consider incident velocity and its angle. Lateral deviations of rebound velocity, both direction and value, at impact are similarly accounted for by perturbing the ground orientation laterally, thus inducing scatter of run-out directions. With these features, the model is capable to describe the stochastic rockfall dynamics. In this study, 3-D DDA was then conducted to investigate the dynamic behavior of the rockfall and examine the role of sphericity of the rock block travelling on bench slopes with different ground roughness levels. Parametric analyses were carried out in terms of cumulative distribution function (CDF) to investigate for spatial distribution (both runout distance and lateral displacement), velocity and jumping height. The effects of block shape and ground roughness revealed by these factors were discussed. It suggests that ground roughness amplifies the randomness and plays important roles on the dynamic behavior of the system; irregularity from block sphericity will further amplify the randomness especially when the size of the rock is relatively small compared to the roughness level. Both irregularities should be taken into consideration in simulating rockfall problems. Further calibration of the new model against a range of field datasets is essential.

Reproduction Method of Rockfall Geologic Hazards Based on Oblique Photography and Three-Dimensional Discontinuous Deformation Analysis

Rockfall geologic hazards are widely distributed. Due to their concealed nature, rockfalls are difficult to investigate using traditional contact survey methods, and the hazards they pose affect major projects and people’s safety. Reproducing methods, including scene survey and movement process analysis, are primary tasks used to prevent these hazards; however, few reconstruction methods can directly apply the parameters of the rockfall geologic hazards obtained by the scene survey to evaluate the movement process. To address this problem, a method of reproduction based on oblique photography and three-dimensional discontinuous deformation analysis (3D-DDA) is proposed; the method consists of three key techniques (oblique photography, 3D rock block system modeling, and 3D rock block system analysis). First, geometric characteristic parameters of the terrain, rockfall, and discontinuities are extracted based on oblique photography using an unmanned aerial vehicle (UAV). Second, the block system model of rockfall is reconstructed by using 3D computational geometry theory and taking these geometric characteristic parameters as an input. Finally, the whole evolution process of rockfall geologic hazard, including initiation, movement, and accumulation, is simulated by the 3D-DDA method. To verify the practicability of this reproduction method, a typical rockfall geologic hazard, located in the K8 + 050 section of the Gaohai expressway, Yunnan, China, is studied. In addition, the characteristics of 19 dangerous rock masses in the survey area are clarified, and the geometric features of the discontinuities in the rock masses are extracted based on oblique photography using an UAV. The block system model of a potential rockfall is reconstructed, the movement trajectory is simulated by the 3D-DDA method, and the evolution process of velocity and kinetic energy of the rockfall verifies that the spatial layout of the current three-level passive protective nets system is reasonable. The case study indicates that the proposed method provides a geological and mechanical model for the risk assessment of rockfall geologic hazards.

Toppling failure analysis of rock slopes based on a three-dimensional discontinuous deformation analysis method

The present study focused on the study of the toppling failure of rock slopes using a three-dimensional discontinuous deformation analysis (3D DDA) method. The failure conditions and their related impact factors of toppling rock slopes were derived by building the corresponding mathematical models based on the limit equilibrium method (LEM), and the deficiency of the existed LEM for analysing the toppling system was pointed out. Several numerical examples, such as a single rock block of different dimensions, classical toppling rock slope model, and multiple rock blocks of different distributions, were adopted to investigate the toppling failure mechanisms under different conditions. The results showed that the 3D DDA is a very powerful numerical tool to study the failure mechanisms of toppling rock slopes and simulate the failure process of various toppling rock slopes with different distributions of structural planes inside rock mass. The controversial safety factor of the slope stability was avoided in the 3D DDA numerical simulation. Through the 3D DDA results, some more real phenomena of slope failure could also be observed and explained. For example, the slope block toppling tended to be toppling-sliding rather than pure toppling. In addition, the general laws of toppling slope failure were presented.

CPU-accelerated explicit three-dimensional discontinuous deformation analysis with cloud computing

Prompt feedback based on a numerical simulation, such as discontinuous deformation analysis, plays an important role in digitalized geotechnical engineering. Cloud computing also prevails owing its flexible configuration and computational power. This study proposes an explicit three-dimensional discontinuous deformation analysis method using explicit contact forces that can circumvent the difficulty of not converging the open–close iteration and the inefficiency of classic implicit discontinuous deformation analysis. The serial code is ready to be parallelized by relocating the postjudgment of the contact status and using LDL decomposition to solve fixed-size linear equations. We propose herein the block-wise parallelization concept, which significantly simplifies the work of parallel computation. No modification under the block level is needed. In addition, one can realize parallel computing by adding just three lines of the new code with the help of OpenMP. A simple virtual cloud machine is used to run the program to validate the improved efficiency of the newly developed algorithms. A 3.5 speedup ratio is achieved with four CPU cores and eight processors.

Three-Dimensional Discontinuous Deformation Analysis of Failure Mechanisms and Movement Characteristics of Slope Rockfalls

Three-Dimensional Discontinuous Deformation Analysis of Failure Mechanisms and Movement Characteristics of Slope Rockfalls

Three-dimensional discontinuous deformation analysis with explicit contact formulation and block-wise multicore CPU acceleration

In this study, an explicit three-dimensional discontinuous deformation analysis (3D-DDA) is parallelized. Using explicit contact force circumvents the open-close iteration and makes the global stiffness matrix block-diagonal. Post-judgment of contacts is split into two parts to unify the data access pattern, constant acceleration Newmark method is adopted to guarantee unconditional stability of time integration, and square-root-free Cholesky decomposition is adopted to solve small linear equations efficiently. To exploit the multicore CPU with few changes in code and achieve good modularity, block-wise parallelization is adopted, which can realize the full-stage parallelization with just several lines of OpenMP directives. There is no need to modify block specific algorithms, such as contact detection and block information update, for the proposed parallelization, which results in the most efficient parallelization scheme to date. Correctness and efficiency are validated using several numerical examples. The speed-up ratio of 5.0–6.5 is achieved on an 8-core CPU.

Experimental Verifications and Applications of 3D-DDA in Movement Characteristics and Disaster Processes of Rockfalls

Rockfalls are a major hazard in mountain areas and along engineering slopes. It is important to fully understand the characteristics and processes of rockfall movements, both for hazard assessment and countermeasure design. This study focused on the movement characteristics and disaster processes of rockfalls by three-dimensional discontinuous deformation analysis (3D-DDA), the effectiveness of which was verified by outdoor experiments. An apparatus specifically built for the experiments was established, which included a releasing device and a binocular camera stereovision system. Blocks of three different shapes and sizes were released on slopes with three different inclinations on the campus of Tibet University. The indices of the movement characteristics of the blocks, such as lateral deviation, resting position, jumping height, and kinetic energy evolution, were investigated. The experimental results were compared with those of the 3D-DDA, so as to study the rockfall movement characteristics. On this basis, the 3D-DDA was applied to simulate a large-scale slope rockfall on the Tibetan Plateau, and the rockfall failure processes and movement characteristics were further analyzed. The results showed that the 3D-DDA method could effectively simulate the entire rockfall process in time and space, including instability and movement. Through the 3D-DDA numerical simulations, rockfall phenomena and behaviors could be fully understood. Furthermore, by analyzing the trajectories and kinetic energies of a rockfall, the prediction of rockfall disasters and the reasonable design of protective countermeasures could be realized.

Modeling Wave Propagation in Rock Masses Using the Contact Potential-Based Three-Dimensional Discontinuous Deformation Analysis Method

The most recently proposed three-dimensional (3D) contact potential-based discontinuous deformation analysis (3D-CPDDA) method is further applied for wave propagation problems in rock masses. A viscous non-reflecting boundary is incorporated into the 3D-CPDDA method to minimize the wave reflections. Furthermore, a force input method is incorporated to eliminate the contamination of scattered waves in the numerical solution and to accurately input the incident wave. Several benchmark problems about P-wave/S-wave propagation in homogeneous rock masses and jointed rock masses are solved to validate the modified 3D-CPDDA method. The numerical results assessed by the modified 3D-CPDDA method agree well with those obtained from analytical methods, which means that the modified 3D-CPDDA method can reliably and correctly simulate wave propagation in rock masses. The modified 3D-CPDDA method warrants further investigation.