3D DDA Research Articles

Investigations of sequential excavation and reinforcement in tunnels based on the extended three-dimensional discontinuous deformation analysis method

The unloading effect of excavation is the key factor leading to rock mass instability, and bolt support is the key means of controlling the stability. However, the original three-dimensional discontinuous deformation analysis method (3D DDA) can only simulate rock mass instability and bolt support after the tunnel is excavated, which makes it difficult to study the effect of tunnel excavation and bolt support on rock mass stability. Therefore, an algorithm to confirm the excavated blocks and reinforced blocks based on the volume judgment method (VJM) was proposed in this paper. Sequential excavation and reinforcement in tunnels can be simulated by introducing the time variables of block excavation and bolt deployment into the 3D DDA method. By embedding these two algorithms in the 3D DDA method, an extended 3D DDA method for sequential excavation and reinforcement in tunnels was developed, named the 3D DDA_ES method. Based on the 3D DDA_ES method, the collapse that occurred during the construction of the Huangjiazhuang Tunnel was simulated, and the displacement of key blocks and the axial force of the bolts were analyzed. According to the simulation results, the axial force on the bolts is strongly related to the position of the key blocks, and the axial force on the bolts at the tunnel roof is much greater than that at the tunnel arch shoulder. In this regard, these bolts with axial forces less than 1 kN are removed during the simulation, and the key blocks remain stable with a maximum displacement of 12 cm. The research results can provide a theoretical basis for tunnel construction and reinforcement parameter optimization for similar projects.

Impact failure and disaster processes associated with rockfalls based on three‐dimensional discontinuous deformation analysis

AbstractRockfalls, a common geohazard in mountainous areas, have destructive impact capacity and may cause failure of dangerous rock masses in their runout range. For slope risk assessment, a thorough understanding of the impact failure processes and dynamic characteristics associated with rockfall movements is necessary. The aim of this study is to investigate the impact of rockfall failure behaviours and disaster processes on the dangerous rock mass along the way through three‐dimensional discontinuous deformation analysis (3D DDA). To validate the reliability and applicability of 3D DDA, numerous laboratory experiments are performed on the impact of downward moving blocks on the unstable block and block system (i.e. single block–single block, single block–block system and block column–block system models) by comparing the displacements, kinetic energies and movement states of blocks. Using the G318 national road in Tibet as an example, 3D DDA simulates the impact and disaster processes associated with upper rockfalls sourced from a complete giant block and multiple discrete blocks on lower dangerous rock mass. Further, rockfall failure modes, movement characteristics, block interactions and impact phenomena are investigated. Results show that 3D DDA can effectively simulate block movement and impact interaction. The upper rockfalls impact the initially stable lower dangerous rock mass, which is the triggering factor for failure of the lower dangerous rock mass. The blocks from the upper rockfalls interact and merge to move downward, increasing the total volume and impact capacity of the rockfalls. It has been discovered that the rockfall disaster caused by the impact of an upper rockfall comprising discrete blocks on a lower dangerous rock mass is more severe than that resulting from the impact of a complete block. Overall, the results of this research can be used to help predict and prevent rockfall disasters.

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.

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.

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.

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.

Motion characteristics of rockfall by combining field experiments and 3D discontinuous deformation analysis

Rockfall is a common geological disaster in rock slope projects. An understanding of the motion processes and characteristics of the rock blocks originating from a rockfall is of profound importance for comprehending the mechanism(s) of rockfall disasters. In this study, a field experimental system comprising binocular stereovision was developed for examining rockfalls, and a series of experiments on rockfall motion characteristics were performed on slopes near an abandoned section of the G318 national highway in Tibet, China. The indices of motion characteristics such as the lateral displacement, staying position, jumping height, and kinetic energy evolution of the blocks under different work conditions (e.g. different masses, shapes, falling heights, falling angles, and different geometrical characteristics in the slopes) were investigated by combining experiments with a three-dimensional discontinuous deformation analysis (3D DDA) method. The results show that the field experimental system has high accuracy in monitoring the block movement processes. The 3D DDA results agree well with those from the experiments, and the evolutions of the motion characteristics of the rockfall under different conditions are quantitatively obtained. Furthermore, the results provide insight into the laws of rockfall movements and dynamic impact processes, and lay a foundation for the design of rockfall protection measures and engineering approaches for disaster prevention and mitigation.

Acceleration of contact detection between arbitrarily shaped polyhedra based on multi-cover methods in three dimensional discontinuous deformation analysis

Three-dimensional discontinuous deformation analysis (3D DDA) can address the large-displacement dynamic responses in rock engineering. However, the efficiency of contact detection is the bottleneck of 3D DDA. This study automatically generates a multi-shell cover (MSC) system and develops a novel multi-cover search (MCS) algorithm for 3D cases. MSC expedites contact detection by dividing the geometric elements of polyhedra into several groups, and MCS accelerates contact detection by efficiently selecting potential contacts using the neighbor relations between faces of polyhedral blocks. These methods increase the accuracy of the rough search in the neighbor search process and improve contact pattern identification. As a result, the number of delicate checks between the geometric elements of two neighbor blocks is greatly reduced. Some numerical tests in rock engineering are used to verify that these two multi-cover methods can accelerate contact detection between arbitrarily shaped polyhedra in 3D DDA.

Exploring inelastic collisions using modified three-dimensional discontinuous deformation analysis incorporating a damped contact model

Abstract Most collisions in reality are partially inelastic. The consideration of energy dissipation is important in block collision phenomena. Therefore, damping should be introduced into three-dimensional discontinuous deformation analysis (3D DDA) to address the energy loss in collisions. A damped contact model based on linear spring-dashpot assumption is adopted and incorporated into 3D DDA. The derivation is first presented to obtain the formulae of the damped contact model in normal and shear directions, and then a single-degree-of-freedom example is modelled to validate the modified 3D DDA. Subsequently, inelastic collisions with unconstrained and fixed targets are carefully investigated. A relationship between viscous damping coefficient and coefficient of restitution (COR) is derived. Finally, viscous damping effect in contact is demonstrated by a freefall example of an assembly of blocks. The results suggest that the modified 3D DDA can model the collision behavior under given COR, and the damping effect may be significant and should be properly addressed in simulations.

A Three-Dimensional Discontinuous Deformation Analysis Method for Investigating the Effect of Slope Geometrical Characteristics on Rockfall Behaviors

In the present study, a three-dimensional discontinuous deformation analysis (3D DDA) method was developed and used to investigate the effect of slope geometrical characteristics on rockfall behaviors. The slope geometrical characteristics of rockfall, such as the slope height, slope angle and slope shape, were summarized, and the corresponding calculation models were proposed. The algebraic computation and contact mechanics of the three-dimensional (3D) global contact theory were illustrated and implemented into the 3D DDA method. The accuracy of the developed 3D DDA method was verified through the four basic movement forms of rockfall: freefalling (oblique projectile), sliding, rolling and bouncing. Numerical examples were performed to study rockfall behaviors under different slope geometrical characteristics by analyzing the kinetic energy or velocity conversion, motion trajectories, and damage degree and range of moving blocks. The results demonstrate that the 3D DDA method can accurately simulate rockfall movement, which is significantly affected by slope geometrical characteristics. Furthermore, some suggestions for slope excavation and protection were put forward.

Research on the Effect of Tree Barriers on Rockfall Using a Three-Dimensional Discontinuous Deformation Analysis Method

On rockfall slopes, trees can play a natural and efficient protective role in preventing the occurrence of rockfall disasters. This paper presents a study on the tree barrier effect on rockfall based on a three-dimensional discontinuous deformation analysis (3D DDA) method. The tree barrier effect on rockfall is studied considering the 3D spatial effect of rockfall movement and the 3D spatial distribution of trees. The basic movement forms and collision modes of rockfall related to the tree barrier are classified, and the impact factors of tree arrangements on rockfall are proposed. The contact mechanism employed in the 3D DDA method is illustrated, and the accuracy of the 3D DDA rockfall simulation is verified by comparing with the results obtained from laboratory experiments. Numerical examples are conducted to study the effect of the tree barrier on rockfall, and general laws are derived by analyzing the variation of kinetic energies and motion trajectories of moving blocks. The results show that the 3D DDA method can effectively predict the 3D motion behaviors of rockfall and assess the tree barrier effect. Furthermore, a reasonable arrangement of trees or a combination of protective facilities and trees can prevent or mitigate the rockfall hazards.

3D DDA Based on Variational Inequality Theory and Its Solution Scheme

The advantage of 3D discontinuous deformation analysis (3D DDA) is the rigorous contact conditions on the interaction of 3D blocks. These conditions are enforced by the penalty function convention; however, inappropriate penalty parameters easily generate numerical instability. To avoid the introduction of the penalty parameters, the contact conditions in 3D DDA are described as variational inequalities in this study, and the extra-gradient method is employed to solve this new formulation of 3D DDA. The proposed computation scheme is more flexible and dispenses with large scale matrix inversion. Some practical examples originally designed by Shi are analysed, verifying the effectiveness and precision of the new scheme.

Tunnel stability assessment by 3D DDA-key block analysis

The key block theory is often the first analysis carried out in assessing potential instability caused by tunneling through jointed rock masses. This study suggests that it is beneficial to include it as the first step for a detailed design analysis by discrete numerical modeling, such as 3D DDA. The procedure is illustrated with an example which is simplified but is general and contains both primary and secondary key blocks. The agreement of the identified key blocks from 3D DDA and the key block theory provides credence for the DDA study, while the DDA, in turn, gives insights on how the failure of the unstable blocks evolves and complements the key block analysis.

A multi-shell cover algorithm for contact detection in the three dimensional discontinuous deformation analysis

In three dimensional discontinuous deformation analysis (3D DDA), the contact detection between blocks is the most expensive part in terms of the total computational cost. The detection normally comprises two stages, namely the search of neighboring blocks and the identification of contact modes. The first stage aims to find out all possible neighboring blocks and the second is to identify the exact contact modes between of neighboring blocks from their vertices, edges and faces. In this paper, an efficient and robust spatial contact detection algorithm is presented linking the above mentioned two stages using a novel multi-shell cover (MSC) system and decomposition of geometrical sub-units. The present MSC method greatly reduces the contact detection volume and iterations. This paper also provides a unified formula of vertex to face and edge to edge contacts. The proposed method is implemented in a 3D DDA computer program. Numerical examples are tested and the results show improved computational efficiency comparing with existing methods.

A Numerical Model Coupling Three-Dimensional Discontinuous Deformation Analysis (3D DDA) with Finite Element Method

A Numerical Model Coupling Three-Dimensional Discontinuous Deformation Analysis (3D DDA) with Finite Element Method

Validation and application of three-dimensional discontinuous deformation analysis with tetrahedron finite element meshed block

In the last decade, three dimensional discontinuous deformation analyses (3D DDA) has attracted more and more attention of researchers and geotechnical engineers worldwide. The original DDA formulation utilizes a linear displacement function to describe the block movement and deformation, which would cause block expansion under rigid body rotation and thus limit its capability to model block deformation. In this paper, 3D DDA is coupled with tetrahedron finite elements to tackle these two problems. Tetrahedron is the simplest in the 3D domain and makes it easy to implement automatic discretization, even for complex topology shape. Furthermore, element faces will remain planar and element edges will remain straight after deformation for tetrahedron finite elements and polyhedral contact detection schemes can be used directly. The matrices of equilibrium equations for this coupled method are given in detail and an effective contact searching algorithm is suggested. Validation is conducted by comparing the results of the proposed coupled method with that of physical model tests using one of the most common failure modes, i.e., wedge failure. Most of the failure modes predicted by the coupled method agree with the physical model results except for 4 cases out of the total 65 cases. Finally, a complex rockslide example demonstrates the robustness and versatility of the coupled method.

Compatible algorithm for integrations on a block domain of any shape for three-dimensional discontinuous deformation analysis

This work calculates the integrations of convex and concave polyhedrons in three-dimensional discontinuous deformation analysis (3D DDA) using a novel method. With this novel method, mathematical topology is applied to the integration, and the numerical mesh used by the Finite Element Method (FEM) is not needed to subdivide polyhedrons into tetrahedrons or hexahedrons. Examples demonstrate the accuracy of the proposed method for integrating convex and concave polyhedrons with planar boundaries.

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

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 Model of Point-to-Face Contact for Three-Dimensional Discontinuous Deformation Analysis

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.

３次元不連続変形法（３Ｄ ＤＤＡ）による岩盤斜面崩落のシミュレーション

３次元不連続変形法（３Ｄ ＤＤＡ）による岩盤斜面崩落のシミュレーション