Combined Finite-discrete Element Method Research Articles

HOSS: an implementation of the combined finite-discrete element method

Nearly thirty years since its inception, the combined finite-discrete element method (FDEM) has made remarkable strides in becoming a mainstream analysis tool within the field of Computational Mechanics. FDEM was developed to effectively “bridge the gap” between two disparate Computational Mechanics approaches known as the finite and discrete element methods. At Los Alamos National Laboratory (LANL) researchers developed the Hybrid Optimization Software Suite (HOSS) as a hybrid multi-physics platform, based on FDEM, for the simulation of solid material behavior complemented with the latest technological enhancements for full fluid–solid interaction. In HOSS, several newly developed FDEM algorithms have been implemented that yield more accurate material deformation formulations, inter-particle interaction solvers, and fracture and fragmentation solutions. In addition, an explicit computational fluid dynamics solver and a novel fluid–solid interaction algorithms have been fully integrated (as opposed to coupled) into the HOSS’ solid mechanical solver, allowing for the study of an even wider range of problems. Advancements such as this are leading HOSS to become a tool of choice for multi-physics problems. HOSS has been successfully applied by a myriad of researchers for analysis in rock mechanics, oil and gas industries, engineering application (structural, mechanical and biomedical engineering), mining, blast loading, high velocity impact, as well as seismic and acoustic analysis. This paper intends to summarize the latest development and application efforts for HOSS.

Plate motion in sheared granular fault system

Plate motion near the fault gouge layer, and the elastic interplay between the gouge layer and the plate under stick-slip conditions, is key to understanding the dynamics of sheared granular fault systems. Here, a two-dimensional implementation of the combined finite-discrete element method (FDEM), which merges the finite element method (FEM) and the discrete element method (DEM), is used to explicitly simulate a sheared granular gouge fault system. We focus on investigating the influence of normal load, driving shear velocity and plate stiffness on the velocities and displacements in the direction parallel to the shear direction (x-direction) measured at locations on the upper and lower plates just adjacent to the gouge. The simulations show that during slip phases the magnitudes of the measured velocities on the upper and lower plates are proportional to the normal load and may be inversely proportional to the square root of the plate's shear modulus. Whereas, the driving shear velocity does not show distinct influence on the measured velocities. Additionally, large slip velocities are generally associated with large macroscopic friction coefficient drops. For the models subjected to smaller normal loads, larger shear velocities and with stiffer shear plates, the same magnitude of slip velocity could cause a larger drop of macroscopic friction coefficient. During stick phases, the velocities of the upper and lower plates are respectively slightly greater and slightly smaller than half of the driving shear velocity and are both in the same direction of shear. The shear strain rate of the gouge is calculated from this velocity difference between the upper and lower plate during stick phases and thus the gouge effective shear modulus can be calculated. The results show that the gouge effective shear modulus increases proportionally with normal load, while the influence of shear velocity and plate stiffness on gouge effective shear modulus is minor. The simulations address the dynamics of a laboratory-scale fault gouge system and may aid in revealing the complexities of earthquake frictional dynamics.

Numerical analysis of material modeling rock reinforcement in 2D FDEM and parameter study

This paper introduces a simplified smooth rock reinforcement modeling approach based on the 2D combined finite discrete element method (FDEM). The goal is to facilitate modeling the end grouting process and the full grouting process of rock bolts reinforcement, and to improve further understanding of the interaction between bolt, grouting, and host rock based on material mechanical behavior. The study analyzed the mechanical model and fracture mode extracted from experimental results using the rebar bolt specimen. This facilitated the study of the bond strength component of the rib. Then, the bond force-slip relationship was explored using physical modeling with different combinations of bolt and cover constraints, with different types of grouts and rocks. Corroboration of the laboratory axial pullout test was then conducted. A power law for the stress-displacement of the radial component of bond force was generated, to elaborate simplified modeling method and the data range of parameters in the diverse combination of constraining forces. Then, the action law was implemented into the material rock reinforcement model in FDEM. The verification process, done by comparing the bond force-slip curve with the physical modeling, shows the model can effectively reflect the action rule of rebar bolts. The model simplifies rebar bolt modeling and adds a new ability to model rock reinforcement features into FDEM, making the FDEM technology extend fundamentally.

Numerical Study of Rock Fragmentation Process and Acoustic Emission by FDEM Based on Heterogeneous Model

Rock has the characteristics of natural heterogeneity and discontinuity. Its failure phenomenon induced by external force involves complex processes, including the microcrack initiation, propagation, coalescence, and the macrocrack formation. In this study, the Weibull random distribution based on the rock microstructure characteristics is introduced into the combined finite-discrete element method (FDEM) to establish the heterogeneous rock model, and the mechanical response and damage evolution of rock samples in uniaxial compression test are simulated. The results show that FDEM simulation with loaded heterogeneous rock model can reflect the progressive development of rock damage, fracture, and acoustic emission (AE) activity in real rock well. Meanwhile, the statistical analysis indicates that the number and energy evolution of AE events with different fracture modes in the model are consistent with the macroscopic failure mode of rock. The change of b-value also agrees with the increasing trend of high-energy events in the loading process. This method provides a new tool for the analysis of rock damage and fracture evolution.

Numerical simulation of hydraulic fracturing based on two-dimensional surface fracture morphology reconstruction and combined finite-discrete element method

Hydraulic fracturing is an important technology widely applied in engineering practice for the exploitation of unconventional oil and gas resources. Further research on its mechanism has the potential to advance the development of unconventional reservoir exploitation. In this study, previously acquired experimental data, including surface fracture image and fluid pressure curve, were used for modelling and comparison. Employing digital core reconstruction and the combined finite-discrete element method, A process of modelling crack geological model was proposed based on the rock surface sketch after fracturing. Two-dimensional models were established according to the surface fracture morphology of specimens. The feasibility of these numerical models was simultaneously verified by fluid pressure curves and the surface fracture morphology obtained in physical experiments. The evolution of fracture morphology during hydraulic fracturing was analysed. Simulation results indicate that: (1) the simulation method of this study is more suitable for simulation of coal hydraulic fracturing than for shale hydraulic fracturing; (2) the rapid rise of fluid pressure might be accompanied with the initiation and development of micro-fractures, leading to the occurrence of larger fractures; (3) the hydraulic fracturing process on the experimental scale could induce significantly larger maximum total area and wider fractures in coal specimens than in shale specimens. This indicates that coal hydraulic fracturing forms more complex fractures than in shale; (4) when the rock elastic modulus increases, the maximum fracture width and fracture total area decreases, and fracture length is extended. (5) The initiation and propagation pressures are positively related to the elastic modulus in the fluid pressure evolution. The crack geological model proposed in this study provides direct guidance for crack reconstruction in simulations. Moreover, the research results may provide a guidance for studies on the evolution of hydraulic fracture under the influence of rock heterogeneity.

Time-dependent crack development processes around underground excavations

The time-dependence of rock strength is an issue that cannot be ignored in the rock engineering field. This paper summarizes previous study results on the degradation of the uniaxial compressive strength of rock (including component cohesion and internal friction angle), tensile strength and Young's modulus over time. Based on previous studies, the degeneracy formulas of intact rock cohesion and internal friction angle are derived. Taking circular tunnel excavation under hydrostatic pressure as an example, the surrounding rock response is studied under different degradation degrees of rock strength and stiffness, including the radius of the excavation damage zone, damage degree of surrounding rock, lining support pressure and tunnel boundary displacement. The results show that with the increasing degree of degradation of the surrounding rock compressive strength, the above four parameters increase in the form of an exponential function and that weakening of the tensile strength had little effect on the surrounding rock response. In addition, as the degradation degree of rock stiffness increases, the radius of the excavation damage zone remains unchanged, whereas the other three parameters values are reduced. An actual engineering case simulation using the combined finite-discrete element method shows that results consistent with field observations can be obtained if the rock strength degradation law proposed in this paper is adopted.

A generic computational model for three-dimensional fracture and fragmentation problems of quasi-brittle materials

Fracture and fragmentation in three dimensions are of great importance to understand the mechanical behaviour of quasi-brittle materials in failure stress states. In this paper, a generic computational model has been developed in an in-house C/C++ code using the combined finite-discrete element method, which is capable of modelling the entire three-dimensional fracturing process, including pre-peak hardening deformation, post-peak strain softening, transition from continuum to discontinuum, and explicit interaction between discrete fragments. The computational model is validated by Brazilian tests and polyaxial compression tests, and a realistic multi-layer rock model in an in situ stress condition is presented as an application example. The results show that the computational model can capture both continuum and discontinuum behaviour and therefore it provides an ideal numerical tool for fracture and fragmentation problems.

A novel framework for elastoplastic behaviour of anisotropic solids

A general framework for developing nonlinear hyperelastic/plastic constitutive laws for anisotropic solids experiencing large strains and strain rates has been developed. The proposed framework does not rely on the “a priori” known strain energy function, but instead introduces a physical decomposition of the material element into seven physically independent stress bearing mechanisms, each of which has a constitutive law in terms of internal moments described by a scalar function of a single variable. The model has been encoded into a combined finite-discrete element method and tested against static geomechanical test data. The numerical validation experiments show the model can reproduce plastic anisotropic behaviour in both biaxial and uniaxial loading of a geomaterial.

FDEM simulation of rock damage evolution induced by contour blasting in the bench of tunnel at deep depth

One highly effective approach for tunnelling in rock at deep depth is blasting. The damage and in-situ stress redistribution in the surrounding rock mass induced by the blast loading during this process is, however, unavoidable. In this study, a powerful GPGPU-parallelized combined finite-discrete element method is implemented to study the damage evolution during controlled contour blasting in the bench of a deep-buried tunnel. The proposed method is characterized by the simulation of the blasting-induced pressure variation via the pressure-gas volume history curve and the modelling of the transition from continuum to discontinuum behaviour in the surrounding rock mass. The in-situ stress distribution, the blasting-induced stress wave propagation, and the corresponding rock fracture and fragmentation process are modelled and analysed. The numerical simulation results indicate that the in-situ stresses play a key role in the damage evolution around the tunnel, strongly influencing the stress redistribution pattern and thus the fracture initiation and propagation around the tunnel during blasting. Several different in-situ stress regimes are considered and discussed, revealing the dominating effects of the major principal stress and lateral pressure coefficient on the rock damage behaviour. Moreover, the blasting results obtained from employing the pressure–time history curve with varying decay time ratios and decoupling ratios are also studied. Longer decay time ratios and higher decoupling ratios induce additional rock fracture and fragmentation, indicating that the model can enable the selection of stemming and explosive type for effective breakage with limited damage around a tunnel.

Numerical analysis on tractive performance of off-road tire on gravel road using a calibrated finite element method–discrete element method model and experimental validation

The interaction between off-road tires and granular terrain has a great influence on the tractive performance of off-road vehicles. However, the finite element method or the discrete element method cannot effectively study the interaction between off-road tires and granular terrain. The three-dimensional combined finite element and discrete element method is applied to handle this problem. In this study, a calibrated finite element method–discrete element method model is established, in which the finite element model of off-road tire is validated by stiffness tests, while the discrete element model of gravel particles is validated by triaxial compression tests. The calibrated finite element method–discrete element method model can describe the structural mechanics of the off-road tire and the macroscopic mechanical properties of the gravel road. Tractive performance simulations of the off-road tire on gravel road under different slip conditions are performed with the commercial software LS-DYNA. The simulation results are basically corresponded with the soil-bin test results in terms of granular terrain deformation and tractive performance parameters versus the slip rates. Finally, the effects of tread pattern, wheel load, and tire inflation pressure on tractive performance of off-road tire on granular terrain are investigated. It indicates that the calibrated finite element method–discrete element method can be an effective tool for studying the tire–granular terrain interaction and predicting the tractive performance of off-road tire on granular terrain.

Simulation of mixed-mode fracture using the combined finite–discrete element method

It has been shown experimentally that under mixed tensile and compressive stress states, a corresponding mixed-mode fracture will occur. In this paper, the formation of mixed-mode fractures is investigated using the combined finite–discrete element method. A series of simulations with confining pressures ranging from 7.5 to 150 MPa generate a spectrum of mixed-mode failure conditions. These stress states at failure span the transition from the tensile to compressive failure. The models reproduce previous experiments on Carrara marble using a dog-bone geometry, and a comparison shows good qualitative agreement with the experimental observations. This paper demonstrates that mixed-mode fracture can be captured via numerical simulations and identifies areas where further research is required to better understand mixed-mode fracture processes.

Modeling earthquakes with off-fault damage using the combined finite-discrete element method

When a dynamic earthquake rupture propagates on a fault in the Earth’s crust, the medium around the fault is dynamically damaged due to stress concentrations around the rupture tip. Recent field observations, laboratory experiments and canonical numerical models show the coseismic off-fault damage is essential to describe the coseismic off-fault deformation, rupture dynamics, radiation and overall energy budget. However, the numerical modeling of “localized” off-fault fractures remains a challenge mainly because of computational limitations and model formulation shortcomings. We thus developed a numerical framework for modeling coseismic off-fault fracture networks using the combined finite-discrete element method (FDEM), and we applied it to simulate dynamic ruptures with coseismic off-fault damage on various fault configurations. This paper addresses the role of coseismic off-fault damage on rupture dynamics associated with a planar fault, as a base case, and with a number of first-order geometrical complexities, such as fault kink, step-over and roughness.

Calibration of Microscopic Penalty Parameters in the Combined Finite–Discrete-Element Method

With unique advantages in simulating solid fracture, the combined finite–discrete-element method has become increasingly popular in the rock mechanics field. Nevertheless, a large number of microscopic parameters in the method need to be specified and many of them cannot be directly measured via laboratory tests. Therefore, these parameters must be determined via a calibration procedure. To calibrate appropriate microscopic parameters, we should find the relationship between them and the macroscopic mechanical response. In these microscopic parameters, the normal and tangential penalty parameters play a very important role. Thus, in this paper we study the effect of penalty parameters on macroscopic mechanical parameters in the combined finite–discrete-element method. It is found that as the penalty parameters increase, the elastic modulus and Poisson's ratio gradually approach the experimental values. However, the further increase of penalty parameters has little effect on the elastic modulus and Poisson's ratio when they reach 100 times the elastic modulus. Based on this conclusion, a simple method for the calibration of penalty parameters is proposed, which can be used to quickly determine the penalty parameters according to the uniaxial compression test.

Development of a Numerical Simulator for 3-D Dynamic Fracture Process Analysis of Rocks Based on Hybrid FEM-DEM Using Extrinsic Cohesive Zone Model

The combined finite-discrete element method (FDEM) has attracted significant attention for numerical simulations of complex fracture process of rock-like materials as one of the promising hybrid methods. The mainstream of FDEM simulators developed to date is based on the intrinsic cohesive zone model (ICZM) in which cohesive elements are inserted into all the boundaries of continuum solid elements at the onset of simulations and an artificial elastic behavior must be incorporated to model the intact deformation of rock-like materials. However, the effect of introduction of the artificial elastic behavior on the precision of intact stress wave propagation has not been discussed in previous literatures and this paper discusses this issue. As an alternative for the ICZM-based FDEM, we apply the FDEM based on the extrinsic cohesive zone model (ECZM). An advantage of the ECZM-based FDEM is presented through the 3-dimentional (3-D) numerical modelling of dynamic tension test. In addition, the effect of considering the anisotropy of wave propagation in granite, which has been neglected in all the previous works using FDEM, is investigated through the ECZM-based 3-D FDEM simulation of dynamic Brazilian test with a split Hopkinson pressure bar apparatus. Through the presented numerical simulations, it can be concluded that the ECZM-based FDEM may be an alternative for numerical simulations of complex dynamic fracture process of rock-like materials instead of the ICZM-based FDEM.

A smooth contact algorithm for the combined finite discrete element method

From its inception, the combined finite discrete element method has used a distributed potential contact force algorithm to resolve interaction between finite elements. The contact interaction algorithm relies on evaluation of the contact force potential field. The problem with existing algorithms is that the potential field introduces artificial numerical non-smoothness in the contact force. This work introduces a smooth potential field based on the finite element topology, and a generalized contact interaction law is constructed on top of the smooth potential field. A number of validation cases for the proposed algorithm, considering different shapes of discrete elements, are presented, and detailed aspects of the proposed contact interaction law are tested with numerical examples.

Study on the effect of particle morphology on single particle breakage using a combined finite-discrete element method

Particle morphology is an inherent characteristic that has a significant influence on breakage behaviors of natural sands. Based on X-ray micro-computed tomographic scanning and image-processing, the three-dimensional (3D) particle surfaces of natural sand particles were first reconstructed by the spherical harmonic (SH) analysis. The traditional morphological parameters of the particles, including sphericity, roundness, and aspect ratio were then calculated based on the SH-reconstructed particle surfaces. Furthermore, a 3D local roundness descriptor was used to characterize the local angularity of the contact area between a particle and its loading platens. To model particle breakage, this study developed a combined finite-discrete element method (FDEM) in which the cohesive interface elements were inserted between the finite elements to simulate solid fracture by defining a traction-separation damage law. By using the developed FDEM, a series of single particle crushing simulations were conducted for ellipsoid particles and two types of sand particles. The results show that the developed FDEM model is capable of simulating breakage behaviors of sand particles. Statistical analysis showed that local roundness is the most significant factor in determining the particle fracture pattern, with the strongest positive correlation with the particle crushing strength of sand particles subjected to uniaxial loading.

Combined finite-discrete element modelling of rock fracture and fragmentation induced by contour blasting during tunnelling with high horizontal in-situ stress

A combined finite-discrete element method (FDEM) parallelized on the basis of GPGPU is implemented to model the rock fracture and fragmentation process and the resultant excavation damaged zone (EDZ) development induced by the controlled contour blasting, which was conducted in the TASQ tunnel with high in-situ stresses in the Äspö Hard Rock Laboratory in Sweden. The combination of in-situ stress field, equation-of-state based blast loading, fracturing in tension and shear with gas flow loading of fractures enables the modelling of complex dynamic interactions from multiple blast rounds. For the contour blasting under high horizontal in-situ stresses, blasting-induced fractures initially propagate horizontally, even though the holes are decoupled. Later, these fractures coalesce into larger cracks, which prevents the formation of smooth tunnel walls and increases the EDZ. Smoother surfaces are created at the crown and invert by the propagation of long fractures in the horizontal direction. Due to the combined effect of the free surfaces provided by the adjacent blast-holes and the maximum principal stress induced by in-situ stresses, the fractures at the lower part of the tunnel sidewalls have the tendency to propagate upwards with a diagonal direction about 60° relative to vertical, in accordance with the field test results. Removing the in-situ stresses results in smoother sidewall fracturing with more damage in the crown and invert. Increase in rock heterogeneity, above a threshold, induces more fractures. Increasing the detonation timing between blast-holes induces more damage into the rock mass and fragmentation in the burden. Outcomes of this study show that the GPGPU-parallelized Y-HFDEM IDE provides a powerful tool to replicate the mechanisms of rock fracture and fragmentation induced by blasting.

FSIS: a novel fluid–solid interaction solver for fracturing and fragmenting solids

Transient dynamics of fracturing media has received significant attention in recent years, and a number of simulation approaches have been developed. One of these is the combined finite-discrete element method (FDEM). FDEM produces complex fracture and fragmentation patterns in processes such as blast loads, blasting, impact, mining, and oil exploration. To address a wide range of multi-physics problems, numerous approaches have been undertaken to introduce fluid into the FDEM transient dynamics, ranging from application-specific gas models to more advanced hydraulic fracture processes and biomedical applications. None of these have been satisfactory in terms of robustness, accuracy or computational efficiency. In this work, a completely new explicit fluid solver has been tailor-made and fully integrated (as opposed to coupled) into the combined finite-discrete element method. The solver addresses transient pressure wave propagation in fluid, fluid viscosity, equation of state for the fluid, energy transport, momentum transport, and interaction with the fracturing solid domains, which is done through a novel immersed boundary approach. The solver is based on the governing equations being resolved using different control volume schemes, with discretization errors of either third, second, or first order. The solver is fully explicit and conditionally stable with a time step that is synchronized with the FDEM’s time step.

Simulation of fracture propagation in fibre-reinforced concrete using FDEM: an application to tunnel linings

The application of the combined finite-discrete element method (FDEM) to simulate fracture propagation in fibre-reinforced-concrete (FRC)-lined tunnels has been investigated. This constitutes the first attempt of using FDEM for the simulation of fracture in FRC structures. The mathematical implementations of the new FDEM joint-element constitutive model are first introduced, and the numerical model is then validated comparing the results for plain and FRC beams with three-point bending experimental data. The code has also been applied to two practical tunnel design case studies, showing different behaviours depending on the type of concrete and shape of tunnel section. The FDEM simulations of the linings are also compared with results from a finite element code that is commonly used in the engineering design practise. These results show the capabilities of FDEM for better understanding of the fracture mechanics and crack propagation in FRC tunnels. A methodology for directly inferring the numerical parameters from three-point bending tests is also illustrated. The results of this research can be applied to any FRC structure.

Strength and fragmentation behaviour of complex-shaped catalyst pellets: A numerical and experimental study

The effects of catalyst support shapes on their final strength and fragmentation behaviour are investigated. Uniaxial compression tests by diametrical loading of solid and four-holed discs with high-speed video recordings are employed to investigate strengths and pellet crushing behaviours. The combined finite-discrete element method (FDEM) is employed to simulate the effects of geometrical features and loading orientation on the pre- and post-failure behaviour of catalysts. A comparison with experimental results is also presented and the remarkable agreement in failure evolution and mode is discussed. A methodology to derive representative fragment size distributions from defined pellet shapes and material properties is introduced, providing a further tool to understand the strength and fragmentation behaviour of catalyst supports. The results suggest that fixed-bed reactors made with solid cylindrical catalysts will be more likely to be affected by pressure drops caused by the choking effect of a significant portion of fines than if it was made with catalyst supports with four holes. Two designs of four-hole catalyst supports sintered with different porosities have also been studied, showing different fragment size distributions and fines production. Characterisation of fines production for different catalyst support designs will improve prediction of reactor clogging and pressure drops.