Discontinuous Deformation Analysis Research Articles

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.

On Minkowski difference-based contact detection in discrete/discontinuous modelling of convex polygons/polyhedra

PurposeContact detection for convex polygons/polyhedra has been a critical issue in discrete/discontinuous modelling, such as the discrete element method (DEM) and the discontinuous deformation analysis (DDA). The recently developed 3D contact theory for polyhedra in DDA depends on the so-called entrance block of two polyhedra and reduces the contact to evaluate the distance between the reference point to the corresponding entrance block, but effective implementation is still lacking.Design/methodology/approachIn this paper, the equivalence of the entrance block and the Minkowski difference of two polyhedra is emphasised and two well-known Minkowski difference-based contact detection and overlap computation algorithms, GJK and expanding polytope algorithm (EPA), are chosen as the possible numerical approaches to the 3D contact theory for DDA, and also as alternatives for computing polyhedral contact features in DEM. The key algorithmic issues are outlined and their important features are highlighted.FindingsNumerical examples indicate that the average number of updates required in GJK for polyhedral contact is around 6, and only 1 or 2 iterations are needed in EPA to find the overlap and all the relevant contact features when the overlap between polyhedra is small.Originality/valueThe equivalence of the entrance block in DDA and the Minkowski difference of two polyhedra is emphasised; GJK- and EPA-based contact algorithms are applied to convex polyhedra in DEM; energy conservation is guaranteed for the contact theory used; and numerical results demonstrate the effectiveness of the proposed methodologies.

Investigation on failure behavior of collapse column in China's coal mine based on discontinuous deformation numerical method.

Karst collapse column is a serious geological disaster in China’s coal mines. There are various karst collapse columns in coal mine areas, such as Huainan, Huaibei, Xingtai, Lu’an. And they have seriously affected mining safety and geological environment. The present research is focused on analyzing subsidence mechanism and dynamic collapse process. Based on mechanical analysis of thin plate theory, a detailed model of collapse column slipping and bending fracture is constructed to gather the critical conditions of the collapse column roof. The sensitivity parameters analysis shows that both the radius and roof thickness of cave have a great influence on the sliding instability and bending fracture. Meanwhile, the buried depth also affects bending failure. The discontinuous deformation analysis (DDA) method is used to simulate and analyze the collapse process. The numerical results indicate that the stability of inverted funnel collapse column is dominated by the bending stress of roof strata. The movement of columnar collapse column is mainly caused by sliding instability. However, the funnel collapse column is relatively stable, and does not change in the same condition. The displacement field analysis shows that the collapse range of inverted funnel collapse column is obviously larger than that of columnar collapse column, in which its maximum displacement is approximately 1.5 times that of the columnar collapse column. There is no large area collapse on the upper part of the funnel collapse column, and the block system is relatively stable. The principal stress field analysis proves the above results.

A static discrete element method with discontinuous deformation analysis

SummaryFor discrete element methods (DEMs), integrating the equation of motion based on Newton's second law is an integral part of the computation. Accelerations and velocities are involved even for modeling static mechanics problems. As a consequence, the accuracy can be ruined and numerous calculation steps are required to converge. In this study, we propose a static DEM 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, ie, artificial springs between contact blocks, is 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.

Modeling large displacement of rock block and a work face excavation of a coal mine based on discontinuous deformation analysis and finite deformation theory

Large deformations and displacements of rock blocks may occur under high stresses around deep underground rock engineering works such as deep mines. The original discontinuous deformation analysis (DDA) developed by Shi and Goodman was dedicated to analyze large deformations, rotations and displacements of rock blocks by accumulating small components of these quantities in a time-marching scheme. The small rotation angle approximation adopted in the original DDA may induce block expansion with rotation (rotation error). Based on solid mechanics, the free expansion is caused by (1) the error due to the approximation of the real behavior using a displacement function, and (2) the theory used to describe the geometrical relationship between movement (displacement and rotation) and deformation. The original DDA uses the geometrical relationship that is based on small strain theory. Small strain theory assumes a linear relationship between displacement and strain that does not consider the high order components and the decomposition of rotation and displacement. The finite deformation theory, however, decomposes the displacement and rotation and gives a nonlinear relationship between displacement and strain. Therefore, the finite deformation theory can solve the block rotation problem at a higher accuracy. The DDA method is extended by coupling with the finite deformation theory. The implementation of this method is first verified by three illustration examples, and later employed to modelling a work face excavation of a coal mine in China. The stress field and displacement field are investigated. The results show that the occurrence of the first fracture of the key stratum increases the displacement and the stress. The second fracture of the key stratum partially releases the stress. The periodic weighting phenomenon is also reflected in this modelling.

Distributed-Spring Edge-to-Edge Contact Model for Two-Dimensional Discontinuous Deformation Analysis

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.

CPU-accelerated explicit discontinuous deformation analysis and its application to landslide analysis

• Proposed explicit DDA in two-dimensional case. • Verified the accuracy of explicit DDA by comparing with analytical solution. • Simulated failure process of a mixed block slope in an experiment. • Reproduced the failure process of the Donghekou landslide. • Accelerated explicit DDA based on OpenMP and showed computing efficiency. 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.

Discontinuous Deformation Analysis with Potential Contact Forces

AbstractIn 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...

Development of a Coupled DDA–SPH Method and its Application to Dynamic Simulation of Landslides Involving Solid–Fluid Interaction

Landslides involving solid–fluid interaction such as submarine landslides and landslide dams occur frequently around the world, which may bring severe damage to human lives and properties. Investigation of such landslides is thus of significance to hazard prevention and mitigation. To conduct the analysis, there are three key points to be addressed: (a) the landslide failure process, (b) the free surface flow, and (c) the solid–fluid interaction process. Discontinuous deformation analysis (DDA) method is suitable for analyzing discontinuous blocky systems and has outstanding advantages in simulating the landslide failure process. Meanwhile, smoothed particle hydrodynamics (SPH) method is well-suited for modeling the free surface flow. However, the consideration of solid–fluid interaction in these two methods is seldom, which somehow restricts their applications. With the aim to take advantages of these two methods, a coupled DDA–SPH method in two-dimensional case is proposed, in which the solid–fluid interaction is forced using a penalty approach. The SPH formulations are implemented into DDA code. Several numerical examples are presented to check the validity of the proposed method. A dam-break test is first investigated to show the success of implementing SPH into DDA code for modeling the fluid flow in later simulations of fluid–solid systems. Subsequently, the performance of the coupled DDA–SPH method is validated through a submarine rigid landslide, and the simulation results are in good agreement with the experimental data. Further, an extension study on the submarine deformable landslide is performed, in which the landslide mass consists of multiple blocks and a sensitivity analysis on the interface friction angle between blocks is conducted. Finally, a designed landslide dam is simulated to show the applicability and feasibility of the coupled DDA–SPH method.

Evaluation of impact force of rock landslides acting on structures using discontinuous deformation analysis

Impact force evaluation of rock landslides acted on structures is important to the hazard prevention. This paper presents a numerical method to estimate the impact force on structures. An impact force model for the contact force on the given boundary, rather than the resultant force at the center of gravity is presented and incorporated into Discontinuous Deformation Analysis (DDA). The accuracy is verified by comparing simulation results with both analytical and experimental results. A case is then carried out to study the impact force of sliding mass acting on structures, showing the applicability and capability of the presented method.

Variational inequality‐based particle discontinuous deformation analysis

SummaryOrienting 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.

Application of 3D-DDA integrated with unmanned aerial vehicle–laser scanner (UAV-LS) photogrammetry for stability analysis of a blocky rock mass slope

In stability analysis of discontinuity-controlled slopes, the rationality of results is related to the accuracy of three-dimensional (3D) slope morphology and the reliability of the discontinuity survey. With the advent of remote sensing technologies for engineering geological surveys and slope stability analyses, step-change increases have been made in the quality of data available and geometrical characterization of rock slopes. Although these techniques are frequently employed in the characterization of slope geometry and joint surfaces at present, limited research has been undertaken to effectively process the derived data and improve the quality in the reconstruction of slope geometry imported into 3D discontinuous numerical models. In this paper, an integrated system coupling 3D-DDA and UAV-LS photogrammetry is presented as a tool to evaluate the stability of a blocky rock mass slope. The system includes a UAV-LS module, a modeling module, a block-generation module, and a 3D-DDA calculation module. In the UAV-LS module, the use of UAV-LS system integrated with field mapping and site observations allows the acquisition of detailed outputs (point clouds) on both the slope and discontinuity geometry. An effective combination of commercial software Geomagic and Hyperworks is used in the modeling module to process oceans of 3D point cloud data and construct complex 3D geometrical models based on reverse engineering. In the block-generation module, the three-dimensional discontinuous deformation analysis (3D-DDA) method is then carried out in order to simulate the movement of potentially unstable blocks, within which an independent block-cutting algorithm is used to generate the blocks with arbitrary shapes and the finite structural planes similar to the real cases. The 3D-DDA calculation module uses 3D-DDA calculation algorithm to derive the simulation results. The capability of the proposed system for stability analysis of a jointed slope is demonstrated by a practical example.

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.

Insights into the catastrophic Xinmo rock avalanche in Maoxian county, China: Combined effects of historical earthquakes and landslide amplification

A large and catastrophic rock avalanche occurred at the Xinmo village in the Maoxian county of the Sichuan province of China on June 24, 2017. The avalanche destroyed 64 houses and killed 10 people, and 73 people were reported to be missing. In this study, we focused on the contributions of six historical earthquakes to the collapse of rocks in the source area and the landslide amplification caused by the entrainment of pre-existing deposits on the landslide path. The historical earthquakes in this area contributed different levels of damage on the source rocks; particularly, the 1933 Diexi earthquake induced a few initial cracks in rocks at the crest of the steep hillslope. Under the combined time-effects of long-term weathering and gravity, the rock blocks in the source area were detached from the crest along the intersecting cracks and over-dip soft layer (namely, the sliding surface). The avalanche was amplified by the entrainment of colluvium on its path. The influence of the colluvium on the movement process of the Xinmo landslide was modelled by the GPU-based parallel discontinuous deformation analysis (DDA) method, and the effect of the impact and entrainment in the debris flow was rigorously analysed. The results indicate that the impact and entrainment can continuously increase the volume of debris flow, and the thickness of the colluvium was established to be in the range of 25.9–33.9 m based on the analysis of the deposit pattern and movement duration of the avalanche. There were numerous unstable rocks induced by historical earthquakes, at the crests of steep hill slopes in the earthquake-stricken area. With long-term damage caused by environmental factors and gravity, the failure of unstable rocks with high potential energy is likely to cause severe disasters. The excess pore water pressure likely to be present in the colluvium or deposits during impact and entrainment can considerably reduce the effective stress and degrade friction resistance along the runout path. As a result, the landslide can be intensively amplified owing to the effect of impact and entrainment under a lower friction resistance

DDA based grouting prediction and linkage between fracture aperture distribution and grouting characteristics

Apertures of rock fractures is frequently considered as one of the most important factors affecting grouting characteristics. The real fracture distribution, however, is inaccessible due to its complex intrinsic properties, making it cumbersome to directly assess its impact on on-site grouting. In this paper, we argued that investigation of the grouting characteristics of a fracture network with a priori aperture distribution is promising to be an alternative approach for understanding the linkage between fracture distribution and grouting characteristics. A grouting module based on the Discontinuous Deformation Analysis (DDA) was developed to explore the linkage, with transient flow of grout studied. It is found that the grout consumed by the preset fracture networks exhibits a lognormal distribution, in agreement with the statistics of grouting data from 1361 grout holes at the Jurong Rock Cavern (Singapore). The observed agreement can be treated as an indirect verification of the robustness of our proposed grouting module. As anticipated, the grouting characteristics are significantly affected by the distribution of fracture apertures and connectivity among fractures around an injection hole, which are controlled by the distribution parameters. The grouting features of some dominant fractures could represent that of an entire fracture network.

Three-dimensional spherical discontinuous deformation analysis using second-order cone programming

In this paper, a new formulation of three-dimensional spherical discontinuous deformation analysis (DDA) based on second-order cone programming has been proposed. Artificial springs with open-close iteration used in classic DDA have been removed, given that improper stiffness parameters might cause numerical problems. Furthermore, to account for irregular granular shapes, a rolling resistance model is incorporated in the variational formulation. The proposed formulation can be cast into a standard second-order cone programming program, which can be solved using efficient off-the-shelf optimisation solvers. The proposed approach is validated by a series of numerical examples.

A three‐dimensional heat transfer and thermal cracking model considering the effect of cracks on heat transfer

SummaryA simple three‐dimensional heat transfer model is developed to consider the hindering effect of cracks on heat transfer. The 3D heat transfer model can also be applied to numerical methods such as the combined finite‐discrete element method (FDEM), discrete element method (DEM), discontinuous deformation analysis (DDA), the numerical manifold method (NMM), and the finite element method (FEM) to construct thermo‐mechanical coupling models that allow these methods to solve thermal cracking problems and dynamically consider the hindering effect of cracks on heat transfer. In the 3D heat transfer model, the continuous‐discontinuous medium is discretized into independent tetrahedral elements, and joint elements are inserted between adjacent tetrahedral elements. Heat transfer calculations for continuous‐discontinuous media are converted to heat conduction in tetrahedral elements and the heat exchange between the adjacent tetrahedral elements through the joint element. If the joint element between adjacent tetrahedral elements breaks (ie, a crack generates), the heat exchange coefficient of the joint element is reduced to account for the hindering effect of cracks on heat conduction. Then the model and the FDEM are combined to build a thermo‐mechanical coupling model to simulate thermal cracking. The thermally induced deformation, stress, and cracking are investigated by the thermo‐mechanical coupling model, and the numerical results are compared with analytical solutions or experimental results. The 3D heat transfer model and thermo‐mechanical model can provide a powerful tool for simulating heat transfer and thermal cracking in a continuous‐discontinuous medium.

Investigating the effect of external forces on the displacement accuracy of discontinuous deformation analysis (DDA) method

Blocks of different sizes are formed in a typical rock mass and affect its stability. The size difference in between the blocks consisting a rock mass is also important from the rock stability point of view. In this paper, the effects of external loading changes on the rock blocks displacements in the DDA method has been studied. The effects of external forces on the stability of the blocky rock mass not only depends on the magnitude of the applied forces but also depends on the dimensions of the blocks constituting the rock mass. In this research, an attempt has been made to examine the effects of centralized external forces on the stability of a blocky system via the definition of the External Force Index. The intensity of its effects on the displacement accuracy with respect to the time step length was analyzed and its relation with the analysis time is also investigated. It has been shown that accuracy of the displacement analysis of horizontal sliding block has an initial disturbance and is a function of the number of time steps, time interval and the EFI. In addition, the relation between the analysis time and EFI has also been studied.

Failure Mechanisms of Toppling Rock Slopes Using a Three-Dimensional Discontinuous Deformation Analysis Method

Toppling is a representative failure mode of discontinuity-controlled rock slopes. As a discrete numerical method, discontinuous deformation analysis (DDA) is well-suited for simulating the movement process, contact transformation, and large displacement and deformation of rock block systems, which are formed in jointed rock masses. In the present study, a three-dimensional (3D) DDA method was conducted to study the toppling failure mechanisms of rock slopes by introducing the global contact theory. Considering dynamic equilibrium conditions, the analyses of toppling slopes were performed and the corresponding formulations were derived. The algebraic and mechanical computation of the global contact theory was illustrated and implemented into the 3D DDA method. An experimental apparatus for studying the toppling process of blocks was developed and a series of laboratory experiments were carried out for different block distributions under different conditions. By comparing with the results of the analytical methods and laboratory experiments, the accuracy of the 3D DDA method was verified. Numerical examples, including a classical toppling slope, an ideal mountain slope, and a real toppling case were analyzed to further research the toppling failure mechanisms. The results revealed that the 3D DDA method can effectively predict slope instability and simulate the failure process of toppling slopes. The results of the dynamics-based formulations, experiments and 3D DDA enrich the instability conditions of slope-toppling. Moreover, the general phenomena and laws of the toppling failure were presented.

An Improved Discontinuous Deformation Analysis to Solve Both Shear and Tensile Failure Problems

This paper is mainly aimed to present an improved DDA (discontinuous deformation analysis) that can deal well with both shear and tensile failure problems. Firstly, the shear mechanism of DDA is detailed investigated. The results show that when handling the frictional interface, the critical shear resistance can be accurately determined only if the penalty value is carefully selected, however, when handling the cohesive interface, the critical shear resistance is significantly underestimated. The inaccurate prediction is due to the inconsistent distribution of normal force and shear force between the two vertex-to-edge contacts in one edge-to-edge contact. Here an edge-to-edge treatment is introduced into DDA. Secondly, to moderately reflect the tensile failure process of rock masses, a two-phase constitutive model is introduced into the DDA with edge-to-edge treatment, and the improved DDA is obtained. Finally, the improved DDA is used to simulate the failure process of gypsum centrifuge model. The results show the improved DDA can deal well with rock failure problems by shear or tension failure.