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In previous studies, particles, such as ellipsoids and super-ellipsoids, are mainly described by implicit functions. However, parametric functions can define more surfaces that can be used to represent a wider variety of particle shapes. In this study, parametric functions are used to construct particles, and the algorithm to determine the multi-point contact between concave particles is also given. Furthermore, the geometric parameters at the contact point, for instance, normal vectors, curvatures, and overlapping, are formulated by the parameters in the parametric function. To verify the proposed method, we use the discrete element method to simulate the systems of torus-shaped particles nested within each other, and analyze the momentum and kinetic energy changes with time. The equilibrium state with approximately zero energy is obtained, which means that the algorithm for multi-point contact of concave particles is suitable and stable. In addition to the torus-shaped particle, another two concave particles defined by parametric function are modeled. The simulation results indicate that the method is universal. Any plane curve can be used to construct surfaces by using the method provided in this work, and computational efficiency of simulations of particles defined by parametric function is higher because calculations of the inverse Jacobian matrix in the Newton-Raphson method are unnecessary. The parametric function method extends the scope of previous studies on particle shape. The falling process of suspended torus-shaped particles are shown in Figs. (a) and (b). Figure (a) presents the initial states of 270 particles. The final equilibrium state is shown in Fig. (b). The final residual velocity is less than 10−4 m/s, which meets the criterion of equilibrium state described by a previous study. [Display omitted] • A general method of constructing particles by parametric functions is provided. • The algorithm to determine the multi-point contact between particles is given. • The geometric parameters at the contact point are formulated. • We simulate the systems of torus-shaped particles nested within each other. • We simulate another two concave particle systems defined by parametric function. [ABSTRACT FROM AUTHOR]

The violent vibration and excessive settlement of trackbeds under high-speed train moving loads are the main problems of ballasted tracks encountered in practice, which limits the rise of operation speed for ballasted railways. To date, the possibility of extending the widely used geogrid in conventional ballasted railways to high-speed railways has not been adequately validated. The stabilization mechanism of geogrids under high-speed train moving loads remains unclear. Herein, a ballasted track DEM model containing five sleepers that reproduced high-speed train moving loads was developed through EDEM software and validated by full-scale model tests. Both cases of ballasted tracks with and without the geogrid were examined. The results showed that the geogrid significantly decreased the total and differential settlement of the ballasted track. After geogrid installation, the coordination number of ballast particles was increased, and the contact forces among ballast particles were reduced, which potentially lowered the ballast breakage. The geogrid effectively constrained the displacement and rotation of ballast particles. The reduced migration of ballast particles led to a smaller and more uniform volumetric strain field of the ballasted trackbed. By investigating the working mechanism of the geogrid, it was shown that the penetration of the ballast particles into the geogrid aperture induced the expansion of the geogrid aperture, and therefore, the lateral force from the geogrid ribs limited the movement of the ballast particles. [ABSTRACT FROM AUTHOR]

Besides normal and tangential forces, rolling and sliding are also important interactions of particles, and should be considered in the discrete element method. However, there are various definitions of rolling and sliding, some of which are quite different and even contradictory. On the premise of trying not to use definitions, the rolling and sliding velocities between non-spherical particles are derived, and satisfy the objectivity. The sliding and rolling velocities between spherical particles are included in the results as a special case. A rolling resistance model calculated by the rolling velocity is applied to simulate direct shear tests of non-spherical particles. In addition to coinciding with experiments, the shear curves also satisfy the rate independent theory. As a demonstration of the model’s universality, the wave propagation in uniaxial compression tests is also simulated. [ABSTRACT FROM AUTHOR]

The rotation of a single particle plays a very important role in the simulation of granular flow through the discrete element method (DEM). Many researchers have noted that the effect of rolling resistance at the contact points should be considered in DEM simulations. The rolling resistance between particles, which is mainly introduced from the force or rolling deformation (velocity) between particles, is indicated in different models. However, various models cause different results. Hence, the most feasible model should be determined. For this purpose, five models are adopted to simulate the simple shear flow by DEM in this work. Simulation indicates that the contraction/dilatancy and force–displacement curves at different volume fractions are in good agreement with the experimental findings. The curves of Models 1 to 3 computed by force and angular velocity separate at different shear speeds, but this result does not satisfy the rate independent theory. The curves of Models 4 and 5, which are calculated by the rolling deformation, are better in keeping with the rate independence than the three previous models. After increasing the particle-size distribution, Model 5 appears to be more reliable. The defects of the four other models are discussed from the physical mechanism perspective. The force–displacement curves of Model 5 at a difference of 20 times in shear speed coincide perfectly. Overall, the findings satisfy both the objectivity and the rate independent theory. [ABSTRACT FROM AUTHOR]

Discrete element method with different shapes has attracted considerable research attentions because particle shapes affect dynamic properties of the system. Although convex particles have been extensively investigated, studies on concave particles are limited. Cardioid particles are constructed in this study using parametric equation. Multi-point contact detection algorithm and calculation methods of geometric parameters, such as normal vector, curvature, and overlapping at the contact point, are provided. Two-dimensional (2D) and three-dimensional (3D) systems are simulated by using the discrete element method. Convergence of energy in falling process under gravity verifies the stability and correctness of the algorithm. On this basis, shear curves of simple shear tests with different shapes are calculated and propagation processes of elastic waves in uniaxial compression systems with periodic boundary conditions are simulated. Results showed that the particle shape affects not only shear strength of the simple shear system but also elastic wave velocity in the uniaxial compression system. The particle shape will cause anisotropies of normal vector of contact force, branch vector, and contact force distributions. Influence of shape on the wave velocity is explained from the perspective of anisotropy. The proposed method in this study is universal and suitable for other shapes defined by different parametric equations and can be used to investigate dynamic properties of particle systems with various shapes. [Display omitted] • Two-dimensional cardioid particles are constructed using parametric equation. • Three-dimensional surfaces are obtained by rotating 2D cardioids around the x -axis. • Multi-point contact detection algorithm is presented. • Calculation methods of geometric parameters are provided. • Influence of shape on the wave velocity is explained by the anisotropy. [ABSTRACT FROM AUTHOR]

Two-dimensional discharge flow of elliptical particles with different aspect ratios from a silo is investigated by using discrete element simulation. The measured flow rate of the particles is well fitted via Beverloo's law. The flow rate shows a monotonic decrease with increasing aspect ratio. The density and velocity profiles of particles at the outlet were systematically measured. At an identical outlet size, the speed of particles through the aperture decreases with increasing aspect ratio. The velocity profiles in all cases follow the same definite self-similar law and are in good agreement with the reported experimental results, thereby showing compatibility with the free fall arch postulation. The density profiles show obvious self-similar feature and can be fitted conveniently by using a fractional power law. Based on these observations and the results of the previous work, the flow rate for elliptical particles is calculated and is in agreement with the simulated results. [Display omitted] • The vertical velocity of particle passing through the exit shows a monotonic decrease with increasing aspect ratio. • The velocity profiles of elliptical particles with different aspect ratios follow the same self-similar law. • The volume fraction profiles display obvious self-similar feature. • The fitting parameters of volume fraction profiles show a dependence on aspect ratios. [ABSTRACT FROM AUTHOR]

Abstract The contact detection between non-spherical particles is an important issue in the discrete element method (DEM). The non-spherical particles commonly used in DEM are ellipsoids or super-ellipsoids. Such particles possess convex shape functions. The determination of contact points can be reduced to a set of nonlinear partial differential equations by the Lagrange method. Then, the Newton-Raphson method is used to solve these equations. The Lagrange method's premise is that the shape function is convex. For heart-shaped particles with concave shape functions, Lagrange method is not applicable. In this study, the contact point between heart-shaped particles is determined by grid method. This method does not need to provide the initial guess point and does not need to solve the Jacobian inverse matrix. There is no divergence in the solution process, and no limit to the concavity or convexity of the shape function. The algorithm is stable and is a general method. In the MATLAB and C compiler environment, the contact points between super-ellipsoid particles are calculated by the grid method and Newton-Raphson method, respectively. The grid method takes 60 and 0.14 ms, respectively, and its computational efficiency is better than that of the Newton-Raphson method. The time consumption for contact point of heart-shaped particles is less than that of super-ellipsoid particles. The number of contact point between two concave particles may be bigger than one; for this case, accurate results can still be obtained via the grid method. The method given in this study is applicable to the simulation of soft or hard particle systems. In addition, the method is used to simulate the evolution of the super-ellipsoid system with boundary constraints under gravity, and the stability and applicability of grid method are verified. Graphical abstract Unlabelled Image Highlights • A method to determine the contact point of heart-shaped particles is provided. • This method does not need to provide the initial guess point. • This method does not need to solve the Jacobian inverse matrix. • The method is applicable to the simulation of soft or hard particle systems. • The stability and applicability of this method are verified. [ABSTRACT FROM AUTHOR]

[Display omitted] • A method to determine the contact point between particles is provided. • Spherical and super-ellipsoidal systems in a spiral tube are simulated by DEM. • The sound velocities in a spiral tube are smaller than those in cylinder confinement. • The system properties in a spiral tube are different from those of other systems. The properties of elastic wave propagation in granular assemblies have become a subject of immense interest in recent years, however, the influence of different confinements on the sound velocity is seldom investigated. This study provides a method to determine the contact point between spherical, super-ellipsoidal particles and complex boundaries, in order to investigate how the anisotropy induced by particle shape or boundary affects velocity. Taking cylinder and spiral tube confinements as examples, the falling process of spherical and super-ellipsoidal assemblies are simulated to verify the validation by the discrete element method (DEM). The convergence of the kinetic energy during the falling process and the equilibrium state with zero residual kinetic energy guarantees the stability and correctness. On the basis, elastic wave propagation of spherical and super-ellipsoidal systems in spiral tube and cylindrical confinements under different pressures are modelled, and sound velocities are calculated. The effective medium theory (EMT), granular solid hydrodynamics (GSH), and elastic stiffness are used to interpret the relationship between velocity and stress in cylindrical confinement. However, the results in the spiral tube deviate from EMT and GSH, which means the boundary affects velocity significantly. The difference of velocity between spiral tube and cylinder is qualitatively explained from the perspective of anisotropy of contact force distribution in the system. The simulation results show that anisotropy introduced by the curved surface affects the acoustic properties greatly. The method used for spiral boundary is also suitable for other complicated confinements. [ABSTRACT FROM AUTHOR]

Discrete element method is used to simulate sound propagation in spherical and superellipsoidal particle systems and calculate the sound velocities under different boundary and stress conditions. Compressional and shear wave velocities at different axial and confining pressures for a conventional triaxial compressed spherical particle system under cylindrical boundary condition are calculated through time-of-flight method and analyzed using stiffness, effective medium theory (EMT), and granular solid hydrodynamics (GSH) theory. Results obtained through the time-of-flight method for a system under fixed boundary condition are consistent with the existing experimental results. The system stiffness calculated by branch vector and contact force can explain the variation of sound velocity with axial stress and confining pressure. For compressional velocities, the simulation results of spherical particle system at different axial stresses under isotropic compression in radial direction are consistent with the theoretical results provided by EMT and GSH theories. Compared with compressional velocities, shear velocities predicted by EMT are bigger than experiments, and shear modulus of theories improving upon EMT gives velocities in accordance with experiment. Moreover, the sound velocities of spherical and superellipsoidal particle systems are calculated under the same boundary and normal stress conditions to investigate the influence of particle shape on the sound velocity. The difference between compressional wave velocities in spherical and superellipsoidal particle systems is evident and is qualitatively explained from the perspective of anisotropy of contact force distribution in the system. The proposed methods of sound propagation simulation and sound velocity calculation can be used to investigate the elastic wave propagation in particle system, and can obtain the microscopic quantities that are difficult to obtain experimentally and theoretically. They can provide references for the study of EMT and GSH theories and further discuss the influence of shape-induced anisotropy on sound velocity. [Display omitted] • Sound propagations in spherical and superellipsoidal systems are simulated by DEM. • Sound velocities under different boundary and stress conditions are calculated. • The influence of particle shape on the sound velocity is investigated. • The difference of sound velocity is explained by the anisotropy of contact force. [ABSTRACT FROM AUTHOR]