Finite-discrete Element Method Research Articles

A novel mesoscale modelling method for steel fibre-reinforced concrete with the combined finite-discrete element method

This paper proposes a novel mesoscale modelling method to study the effect of steel fibres on the fracture properties of steel fibre reinforced concrete (SFRC) based on an integrated finite-discrete element method (FDEM) framework. The present method models the steel fibre of SFRC as multiple bar elements, while simulating the cement matrix using triangle elements. A comprehensive bond-slip model is also introduced between the triangle element and bar element, enabling accurate reproduction of the interaction behaviour between the concrete matrix and steel fibre. Additionally, the progressive failure process of cohesive elements in FDEM is considered as crack propagation, confirming the superiority of FDEM from continuum to discontinuum. The results of the Brazilian split test and uniaxial compression test validate the capability of the FDEM framework for mode Ι and mode II fractures, while single fibre pull-out tests and 3-point-bending beam tests provide validation of the effectiveness of the integrated FDEM framework for the mesoscale modelling of SFRC. The discussion also covers employing the mesoscale modelling method to investigate the effect of steel fibre on the fracture properties of SFRC. Owing to its merits in quantitative analysis, the present method can serve as a feasible potential tool for mesoscale modelling analysis and SFRC design.

Macro-micro analysis of ground response to circular tunnel excavation in granular media

Ground reaction in terms of stress and displacement analysis is one of the major challenges in tunnel design and is helpful for understanding the stability around the excavation zone. In this paper, the analysis of stress, strain responses and radial displacement around circular tunnel excavated in cohesive-frictional granular materials is discussed via an innovative multi-scale approach (FEM × DEM) that combined finite element (FEM) and discrete element (DEM) methods and compared to other solutions obtained with conventional constitutive laws. Thanks to simulations, (i) the Ground response curve is then established and provides a better understanding of the complex behaviour in the vicinity of the tunnel; (ii) the prediction by FEM × DEM approach differs considerably from conventional constitutive laws; and (iii) in a unique manner, macroscopic and microscopic analyses are simultaneously performed showing that at the macroscopic level (FE mesh), diffuse failure mode are observed, while at the microscopic level (DEM level), microscopic evolutions are observed to be strictly related to the macroscopic behaviour.

Evaluation of Damage Stress Thresholds and Mechanical Properties of Granite: New Insights from Digital Image Correlation and GB-FDEM

We employed a novel combination of digital image correlation (DIC) and grain-based hybrid finite–discrete element method (GB-FDEM) to improve the comprehension of the relationships between microstructural features and the mechanical properties of granitic rocks. DIC and numerical results showed that macrocracks initiated and propagated along grain boundaries among different minerals driven by the high stiffness contrast between the compliant biotite and the stiffer feldspar/quartz grains. Surface deformation analyses revealed that tensile-dominated macrocracks open at monotonically increased rates before the crack damage threshold, and the opening accelerated afterwards with the increased shear component. The onset of the acceleration of the opening rate of macrocracks can be used to infer the crack damage threshold. Both strain and acoustic emission were used to infer damage stress thresholds in the synthetic numerical samples. Numerical results showed that the damage stress thresholds and uniaxial compressive strength decrease with increasing grain size following log-linear relations. Coarse-grained samples tend to fail by axial splitting, while fine-grained samples fail by shear zone formation. Biotite and quartz contents significantly affect mechanical properties, while quartz to feldspar ratio is positively related to the mechanical properties. Our study demonstrates the capacities of DIC and GB-FDEM in inferring damage conditions in granitic rocks and clarifies the microstructural control of the macroscopic mechanical behaviors. Our results also provide a comprehensive understanding of the systematics of strain localization, crack development, and acoustic emission during the rock progressive failure process.

A two-level finite element method with grad-div stabilizations for the incompressible Navier–Stokes equations

This article presents and studies a two-level grad-div stabilized finite element discretization method for solving numerically the steady incompressible Navier–Stokes equations. The method consists of two steps. In the first step, we compute a rough solution by solving a nonlinear Navier–Stokes system on a coarse grid. And then, in the second step, we pass the coarse grid solution to a fine grid to linearize the nonlinear term, update the solution by solving a linearized problem based on Newton iterations. In both steps, a grad-div stabilization term is incorporated into the system to reduce the influence of pressure on the approximate velocity. We analyze stability and asymptotic convergence of the approximate solutions, derive explicit dependence of the solution errors on the grad-div stabilization parameter and viscosity. We perform also some numerical tests to validate the theoretical analysis and illustrate the efficiency of the proposed method. Compared with the standard two-level method without stabilizations, the grad-div stabilization term added in present method improves the accuracy of the approximate velocity, accelerates the convergence of the nonlinear iterations for the coarse mesh nonlinear system, and reduces the computational time.

Effect of fully-grouted bolts on the failure behaviors of mine pillars: Insights from block-based FDEM modeling

In this study, the Voronoi-based breakable block model (VBBM) based on the combined finite-discrete element method (FDEM) is used to investigate the failure mechanism of mine pillars and the rock-rockbolt interactions. The mechanical parameters of grout-rock and grout-bolt interfaces of a fully-grouted bolt are calibrated via numerical pull-out tests. Then, uniaxial compression tests are carried out on the reinforced pillars. The kinetic energy release rate (KERR) and crack aperture (CA) are used to evaluate the reinforcement effect. The results revealed that the VBBM can effectively characterize the damage failure mechanisms of spalling on the pillar surface and conjugate shear failure in the core zone of the pillar and reproduce the pre-peak nonlinear deformation and post-peak pseudo-ductile behaviors. The passive confining pressure generated by rockbolts is activated only when the rock mass produces sufficient deformation, and the constraints generated by different numbers of rockbolts can only affect the post-peak behavior of the pillar. There is a gradient feature in the internal deformation of the pillar from the shallow surface to the core. If the rockbolt density meets the strain demand, then rockbolts can delay the inward expansion of cracks by generating local constraints and can effectively fix the spalling rock blocks. With increasing support pressure, the CA and KERR exhibit a decreasing trend and strong power-exponential relationships with support pressure, indicating that there is a transition interval of support pressure and an optimal value for setting the rockbolt density.

The Second-Order Numerical Approximation for a Modified Ericksen–Leslie Model

In this study, two numerical schemes with second-order accuracy in time for a modified Ericksen–Leslie model are constructed. The highlight is based on a novel convex splitting method for dealing with the nonlinear potentials, which is integrated with the second-order backward differentiation formula (BDF2) and leap frog method for temporal discretization and the finite element method for spatial discretization. The unconditional energy stability of both schemes is further demonstrated. Finally, several numerical examples are presented to demonstrate the efficiency and accuracy of the proposed schemes.

Towards automated image-based cohesive zone modeling of cracking in irregular masonry

The models developed for masonry and historical structures in the literature are usually classified into macromodels considering masonry as an equivalent continuum; and micromodels in which brick, blocks, or stones and mortar joints are represented explicitly. In this second category, many contributions dealt with regular bond masonry for which the geometrical description and the discretization are rather straightforward. For irregular masonry structures however, even though both finite element method (FEM) and discrete element method (DEM) approaches have been developed, obtaining a versatile geometry representation and its discretization remains much less straightforward. This becomes an important issue nowadays with the availability of image acquisition techniques, based on which computational models could be derived. The present contribution proposes an automated methodology to produce a line description of the mortar joints of an irregular masonry blocks/stones and mortar assembly, which can subsequently be used in modeling approaches. The starting point of the development is a generation technique based on inclusions packings which uses distance fields to the nearest neighboring inclusions to describe an assembly of blocks or stones geometrically. It is shown that such an assembly and the corresponding distance fields can be used to extract efficiently and in an automated way a line or lumped description of the corresponding mortar joint based on the concept of a medial axis. This line description can subsequently be used to define computational models. This is illustrated by the automated generation of FEM models that represent mortar joints by interface elements equipped with a cohesive law. Simulations on representative volume elements (RVEs) and on a wall are shown to illustrate the methodology that paves the way towards the image-based modeling of irregular masonry structures through the automated generation of cohesive zone-based models.

OpenMP Parallel Finite-Discrete Element Method for Modeling Excavation Support with Rockbolt and Grouting

OpenMP Parallel Finite-Discrete Element Method for Modeling Excavation Support with Rockbolt and Grouting

Predicting Stick-Slips in Sheared Granular Fault Using Machine Learning Optimized Dense Fault Dynamics Data

Predicting earthquakes through reasonable methods can significantly reduce the damage caused by secondary disasters such as tsunamis. Recently, machine learning (ML) approaches have been employed to predict laboratory earthquakes using stick-slip dynamics data obtained from sheared granular fault experiments. Here, we adopt the combined finite-discrete element method (FDEM) to simulate a two-dimensional sheared granular fault system, from which abundant fault dynamics data (i.e., displacement and velocity) during stick-slip cycles are collected at 2203 “sensor” points densely placed along and inside the gouge. We use the simulated data to train LightGBM (Light Gradient Boosting Machine) models and predict the gouge-plate friction coefficient (an indicator of stick-slips and the friction state of the fault). To optimize the data, we build the importance ranking of input features and select those with top feature importance for prediction. We then use the optimized data and their statistics for training and finally reach a LightGBM model with an acceptable prediction accuracy (R2 = 0.94). The SHAP (SHapley Additive exPlanations) values of input features are also calculated to quantify their contributions to the prediction. We show that when sufficient fault dynamics data are available, LightGBM, together with the SHAP value approach, is capable of accurately predicting the friction state of laboratory faults and can also help pinpoint the most critical input features for laboratory earthquake prediction. This work may shed light on natural earthquake prediction and open new possibilities to explore useful earthquake precursors using artificial intelligence.

Fracture characteristic and support effect around deep lined tunnels using CGP-FDEM simulation and field investigation analysis

The lining is one of the main methods of support and is chosen as the primary support for composite and reinforced roadways during mining excavation. Systematic investigations into the supportive effect of lined roadways in deep weak rock masses are highly desirable to improve support design and enhance roadway stability. The fracture characteristics and deformation are more prominent in the lining failure and damage to the surrounding rock. For investigations into the fracture mode and failure characteristics of the lining support, this study proposed a lining element support model in the graphics processing units (GPU) parallel combined finite-discrete element method (FDEM) program using compute unified device architecture (CUDA), referred to as CGP-FDEM. The lining element model and excavation simulation method are implemented in the CGP-FDEM aim of simulation large-scale rock fractures in underground excavations. On this basis, the simulation of deep roadway excavation is performed under various support conditions, including low-strength and high-strength shotcrete, steel arch, and U-profile yieldable steel. Further, a comparative analysis is conducted between the field investigation and simulation results, focusing on fracture-fragmentation and large deformation of the surrounding rock, as well as the failure and fracture mode of the lining elements. Results of individual support effects indicate that the reinforcement conditions mainly affect the excavation damaged zone (EDZ) and the deformation of the surrounding rock. Rock fragmentation and buckling intensify the large deformation and instability of the roadway. Fracture evolution and instability of the surrounding rock can easily occur in weak areas due to severe stress concentration and the accumulation and release of energy. Based on the analysis, the principle of "uniform reinforcement of reactive force throughout the entire section" is proposed to alleviate stress concentrations and maintain the overall stability of the excavation space.

Evaluating the impact of calcite and heterogeneity on the mechanical behavior of coal: A numerical study with grain-based finite-discrete element method

Understanding the mechanical properties and fracture behaviors of coal is essential for the safety of underground mining engineering. This research aims to evaluate the impact of calcite and heterogeneity on the mechanical properties of coal. Neper, a software package for polycrystal generation and meshing, is used to create numerical models with a non-uniform grain size distribution. A random parameter assignment method is proposed to simulate the mechanical properties and failure behavior of coal using the combined finite-discrete element method (FDEM). A comprehensive parametric investigation of the properties of triangular and joint elements is conducted, and calibration procedures are suggested for numerical modeling. The model was calibrated to experimental results and then used to understand the impact of calcite grains on the mechanical behavior of coal. After calibration, a triaxial compression was performed on coal sampleswith and withoutcalcite grains at differentconfining pressures. It is found that the presence of calcite in a coal sample increases Young's modulus and decreases the peak strength of coal. This research uses Voronoi tessellation technology to present a Weibull statistical grain-based model (GBM) to investigate the microstructural heterogeneity induced by grain morphologies. The deformation response, strength characteristics and cracking behavior are investigated by varying the heterogeneity of numerical models under compressive tests with different confining pressures. The simulation results of triaxial compression indicate that the compressive strength increases linearly as the heterogeneity m value increases. The consistency of the numerical results with the experimental observationssuggests that the GBM approachcan be used to better understand the mechanics of brittle rock failure at the micro-structure scale.

Insights into the Effect of Water Content on Mudstone Fragmentation and Cutter Force during TBM Cutter Indentation via the Combined Finite-Discrete Element Method

Insights into the Effect of Water Content on Mudstone Fragmentation and Cutter Force during TBM Cutter Indentation via the Combined Finite-Discrete Element Method

Study on the influence of perforation parameters on hydraulic fracture initiation and propagation based on CDEM

Perforation hydraulic fracturing plays a crucial role in the development of oil and gas reservoirs, and perforation layout can significantly affect the initiation and propagation of complex reservoir rock fractures. In this paper, we used a mixed finite-discrete element method with an explicit iterative scheme called the continuous–discontinuous element method (CDEM) to study the process of hydraulic fracture propagation through perforation. The coupling effects of fluid flow and reservoir mechanics are fully considered, and the accuracy of the model in simulating the directional propagation of perforation hydraulic fractures is validated by comparison with physical experimental results. We also discussed in detail the impact of perforation pattern, density, depth, and aperture on crack initiation, propagation, and complexity, and used the new index of fracture degree to quantitatively evaluate the fracturing effects under the influence of multiple perforation layout parameters. The results indicate that: (i) hydraulic fractures near the wellbore compete for initiation, and the length distribution of cracks is uneven and intricate; (ii) linear perforation patterns produce the maximum initiation pressure, which is negatively correlated with the perforation density and positively correlated with the perforation depth and aperture; (iii) the fracture degree is mainly affected by perforation density and depth, and is less affected by aperture. Perforation can effectively control the fracture morphology and initiation pressure, and has a substantial influence on the crack propagation layout near the wellbore. This study provides a basis for the rational selection and optimization of perforation parameters, which can effectively improve the fracturing effects of tight oil and gas reservoirs.

An efficient Kriging-based calibration framework for FDEM

The combined finite-discrete element method (FDEM) is a powerful numerical tool to simulate the transition of materials from continuous deformation to discontinuous fracturing and fragmentation. Calibrating parameters using laboratory data is crucial for applying FDEM to engineering-scale analysis. Considering the high computational costs in existing calibration schemes, this study proposes an effective FDEM calibration framework employing the Kriging surrogate model rooted in Uniaxial Compressive Strength (UCS) and Brazilian Tensile Strength (BTS) tests. The sensitivity analysis is first conducted on mesh size, loading rate, and numerical parameters to find a suitable set that balances computational efficiency and accuracy. Subsequently, an efficient mapping relationship between input fracture parameters and simulated strengths is established based on the Kriging surrogate model, replacing the expensive FDEM program for forward calculations. With the fracture parameters treated as unknown variables and experimental strengths available, the objective function is formulated and then solved to estimate fracture parameters. The effectiveness of the proposed method is validated through a numerical example. Compared to the conventional methods, the proposed approach requires fewer simulations of UCS and BTS, which highlights a significant computational efficiency advantage.

Model order reduction with novel discrete empirical interpolation methods in space–time

This work proposes novel techniques for the efficient numerical simulation of parameterized, unsteady partial differential equations. Projection-based reduced order models (ROMs) such as the reduced basis method employ a (Petrov-)Galerkin projection onto a linear low-dimensional subspace. In unsteady applications, space–time reduced basis (ST-RB) methods have been developed to achieve a dimension reduction both in space and time, eliminating the computational burden of time marching schemes. However, nonaffine parameterizations dilute any computational speedup achievable by traditional ROMs. Computational efficiency can be recovered by linearizing the nonaffine operators via hyper-reduction, such as the empirical interpolation method in matrix form. In this work, we implement new hyper-reduction techniques explicitly tailored to deal with unsteady problems and embed them in a ST-RB framework. For each of the proposed methods, we develop a posteriori error bounds. We run numerical tests to compare the performance of the proposed ROMs against high-fidelity simulations, in which we combine the finite element method for space discretization on 3D geometries and the Backward Euler time integrator. In particular, we consider a heat equation and an unsteady Stokes equation. The numerical experiments demonstrate the accuracy and computational efficiency our methods retain with respect to the high-fidelity simulations.

Application of a Finite-Discrete Element Method Code for Modelling Rock Spalling in Tunnels: The Case of the Lyon-Turin Base Tunnel

Brittle failure, or spalling, occurs around openings excavated in hard rock masses with high in situ stresses. It takes place due to the nucleation and growth of cracks around the excavation boundary, induced by the redistribution of stresses following the excavation. Modelling this failure process is a tough challenge. The hybrid finite-discrete element method (FDEM) can overcome the boundary between continuum and discontinuum, capturing emergent discontinuities associated with brittle fracturing processes. In this study, FDEM is applied using a commercial code to show its applicability to model brittle behaviour around deep underground excavations in the case of the Torino-Lyon Base tunnel in three different stress conditions. Except for the hydrostatic condition, cracking is triggered immediately after the excavation. Spalling occurring around the tunnel is quite extended; therefore, an accurately designed support must be installed to prevent blocks from falling from the tunnel boundary. The obtained results are aligned with previous results existing in the literature. However, in this case, a deeper spalling is caused by the shape change due to the gradual stress redistribution. Such a phenomenon underlines the importance of using a code able to identify crack propagation, opening, and the formation of loose blocks that progressively modify the tunnel contour.

Coupled hydro-mechanical two-phase flow model in fractured porous medium with the combined finite-discrete element method

Coupled hydro-mechanical two-phase flow model in fractured porous medium with the combined finite-discrete element method

Numerical treatment for some abstract degenerate second-order evolutionary problem

This paper addresses the numerical analysis of a class of a degenerate second-order evolution equations. We employ a finite element method for spatial discretization and a family of implicit finite difference schemes for time discretization. Introducing a stabilization parameter, denoted by θ, we propose a well-posed fully-discrete scheme. Sufficient conditions for its well-posedness and for quasi-optimal error estimates are established. The abstract theory is illustrated through the application to the degenerate wave equation, and numerical results validate our theoretical findings.

Research on the impact of pre‐existing geological fractures on hydraulic fracturing in high in situ stress environments

AbstractThe formation of complex fracture network is a difficult problem in the process of deep rock reservoir reconstruction, and the key to the generation of intricate fracture systems in deep reservoirs is to communicate more pre‐existing fractures through hydraulic fracturing, so as to enhance oil recovery. At present, there are few studies on the interactions of hydraulic fractures with pre‐existing fractures under high stress. Therefore, this research studied the influence of the number of pre‐existing fractures on the characteristics of hydraulic fractures by using advanced hydraulic fracturing instruments, and then analyzed the characteristics of hydraulic fractures under different pre‐existing fracture number based on three‐dimensional topography scanning technology and three‐dimensional square box fractal dimension calculation method. Based on the three‐dimensional finite element discrete element method, further research is carried out. The laboratory test findings indicated that the increase of pre‐existing fractures will make the pump pressure curve more stable during the test, and will significantly improve the roughness of hydraulic fractures. The numerical simulation indicated that as the quantity of pre‐existing fractures grows, the fracture area and width also increase, and in instances of a substantial quantity of pre‐existing fractures, the hydraulic fractures are prone to bifurcation, which contributes to the development of intricate fracture networks. With the increase of natural fracture angle, the fracture width decreased, but the fracture area increased. In this study, the influence of the number of pre‐existing fractures on hydraulic fracturing was studied through laboratory tests and numerical simulations, which provided theoretical reference for engineering practice.

Numerical Methods for Optimal Boundary Control of Advection-Diffusion-Reaction Systems

This paper considers the optimal boundary control of chemical systems described by advection-diffusion-reaction (ADR) equations. We use a discontinuous Galerkin finite element method (DG-FEM) for the spatial discretization of the governing partial differential equations, and the optimal control problem is directly discretized using multiple shooting. The temporal discretization and the corresponding sensitivity calculations are achieved by an explicit singly diagonally-implicit Runge Kutta (ESDIRK) method. ADR systems arise in process systems engineering and their operation can potentially be improved by nonlinear model predictive control (NMPC). We demonstrate a numerical approach for the solution to their optimal control problems (OCPs) in a chromatography case study. Preparative liquid chromatography is an important downstream process in biopharmaceutical manufacturing. We show that multi-step elution trajectories for batch processes can be optimized for economic objectives, providing superior performance compared to classical gradient elution trajectories.