Discrete Particle Systems Research Articles

Fluid–rigid body coupling simulations with the passively moving solid model based on a physically consistent particle method

Fluid–rigid body interaction is a significant topic in research on particle methods. This study developed a fluid–rigid body coupling method based on a physically consistent particle method, i.e., the moving particle hydrodynamics (MPH) method, incorporating the passively moving solid (PMS) model. When the discrete particle system satisfies the fundamental laws of physics, i.e., mass conservation, linear and angular momentum conservation, and the second law of thermodynamics, the method is asserted physically consistent, and this feature is important for robust dynamic calculations. The PMS model is a pioneering approach that is practical for particle methods in which fluid and rigid-body particles are initially calculated as a fluid. Then, only rigid-body particles are modified to restore the initial shape by applying rigid-body constraints. Thus, combining the MPH method and the PMS model realizes a fluid–rigid body coupling method that satisfies fundamental physical laws. The proposed method was first verified via the fundamental rigid body and fluid–rigid body coupling problems: the Dzhanibekov effect on a T-shaped rigid body, a floating rectangular solid, a floating cylinder, and water entry of a two-dimensional cylinder. Second, the proposed method was validated via calculating a cylinder rolling on a liquid film as a fluid–rigid body coupling problem with rotation. By using a potential-based surface tension model, the computed results showed reasonable agreement with the experimental data obtained in this study. Overall, it was confirmed that the proposed method is a promising fluid–rigid body coupling approach, in which the surface tension and wettability can be considered as well.

Uphill in Reaction-Diffusion Multi-species Interacting Particles Systems

We study reaction-diffusion processes with multi-species particles and hard-core interaction. We add boundary driving to the system by means of external reservoirs which inject and remove particles, thus creating stationary currents. We consider the condition that the time evolution of the average occupation evolves as the discretized version of a system of coupled diffusive equations with linear reactions. In particular, we identify a specific one-parameter family of such linear reaction-diffusion systems where the hydrodynamic limit behaviour can obtained by means of a dual process. We show that partial uphill diffusion is possible for the discrete particle systems on the lattice, whereas it is lost in the hydrodynamic limit.

A thermodynamics-based micro-macro elastoplastic micropolar continuum model for granular materials

A micro–macro elastoplastic micropolar continuum model for granular materials in the dry and pendular states is proposed on the basis of thermodynamics. The independent rotational degrees of freedom are considered besides the translational degrees of freedom for particles among the discrete particle system, which is described by the equivalent micropolar continuum. In the framework of the micropolar continuum, the elastic part and yield locus at macroscale are firstly rigorously deduced from the general elastic constitutive relation and Coulomb-type yield criterion at inter-particle contacts, respectively. Furthermore, the micro–macro elastoplastic micropolar continuum model is proposed, in which the model parameters have clear physical meanings. Subsequently, a return mapping algorithm for the integration of the micropolar elastoplastic model is presented. Finally, the proposed model, coded into ABAQUS software, is validated by comparisons of test and discrete element method results in terms of the stress–strain relation and strain localization phenomenon. The stress–strain relation and shear band modes are well reproduced, and the effects of the confining pressure and the partially saturated state on granular materials can also be correctly predicted by the proposed model.

Mean stress tensor of discrete particle systems in submerged conditions

The mean stress tensor is essential to investigate the dynamics of granular material. In this paper, we use Hamilton’s principle of least action to derive the averaged stress tensor of discrete granular assemblies subjected to hydraulic force fields, as well as rigorous conditions for a proper definition of the Representative Volume Element (RVE). The main goal behind our efforts is to upscale particle physics into a sound stress tensor for systems involving the complex interaction between grains and water. We identify the contributions from the unbalanced forces, hydraulic forces, gravity, external forces, and particle fluctuation to the mean stress tensor. In doing so, it is convenient to separate the influence of different force fields when the granular system is subjected to complex environments, e.g., subaqueous conditions. The obtained formula is then validated by triaxial test simulations of dry and saturated granular systems using the Discrete Element Method (DEM) and the Lattice-Boltzmann Method (LBM). The results show that the deduced formula can accurately calculate the stress tensor of discrete assemblies with various body-force fields. We used validated DEM-LBM simulations of submerged granular column collapses to explore the physics happening at the grain scale with this mathematical formalism and showcase its potential. We provide a new perspective based on the granular assembly scale to pursue the fluid–solid interaction. Due to the importance of stress analysis in the constitutive modeling of granular materials, this work could help to better obtain the stress–strain relationship of saturated or submerged granular systems.

Discrete element modeling of selective laser sintering additive manufacturing processes

This work presents a simple methodology for the simulation of selective laser sintering processes in advanced manufacturing. Our approach is based on the Discrete Element Method (DEM) to represent the powder bed, combined with lumped heat transfer equations to describe the various thermal phenomena that may take place for thermo-mechanically-active particles forming discrete particle systems. Phase transformation, which may be critical in this type of application, is also considered. Numerical examples are provided to illustrate the practical use of the methodology.

Coexistence for a population model with forest fire epidemics

We investigate the effect on survival and coexistence of introducing forest fire epidemics to a certain two-species competition model. The model is an extension of the one introduced by Durrett and Remenik (Ann. Appl. Probab. 19 (2009) 1656–1685), who studied a discrete time particle system running on a random 3-regular graph where occupied sites grow until they become sufficiently dense so that an epidemic wipes out large clusters. In our extension we let two species affected by independent epidemics compete for space, and we allow the epidemic to attack not only giant clusters, but also clusters of smaller order. Our main results show that there are explicit parameter regions where either one species dominates or there is coexistence; this contrasts with the behavior of the model without epidemics, where the fitter species always dominates. We also discuss the survival and extinction regimes for the model with a single species. In both cases we prove convergence to explicit dynamical systems; simulations suggest that their orbits present chaotic behavior.

Genealogy and spatial distribution of the N-particle branching random walk with polynomial tails

The N-particle branching random walk is a discrete time branching particle system with selection. We have N particles located on the real line at all times. At every time step each particle is replaced by two offspring, and each offspring particle makes a jump of non-negative size from its parent’s location, independently from the other jumps, according to a given jump distribution. Then only the N rightmost particles survive; the other particles are removed from the system to keep the population size constant. Inspired by work of J. Bérard and P. Maillard, we examine the long term behaviour of this particle system in the case where the jump distribution has regularly varying tails and the number of particles is large. We prove that at a typical large time the genealogy of the population is given by a star-shaped coalescent, and that almost the whole population is near the leftmost particle on the relevant space scale.

Many-particle limit for a system of interaction equations driven by Newtonian potentials

We consider a one-dimensional discrete particle system of two species coupled through nonlocal interactions driven by the Newtonian potential, with repulsive self-interaction and attractive cross-interaction. After providing a suitable existence theory in a finite-dimensional framework, we explore the behaviour of the particle system in case of collisions and analyse the behaviour of the solutions with initial data featuring particle clusters. Subsequently, we prove that the empirical measure associated to the particle system converges to the unique 2-Wasserstein gradient flow solution of a system of two partial differential equations with nonlocal interaction terms in a proper measure sense. The latter result uses uniform estimates of the \(L^m\)-norms of a piecewise constant reconstruction of the density using the particle trajectories.

Law of large numbers and central limit theorems through Jack generating functions

In a series of papers [23–25] by Bufetov and Gorin, Schur generating functions as the Fourier transforms on the unitary group U(N), are introduced to study the asymptotic behaviors of random N-particle systems. We introduce and study the Jack generating functions of random N-particle systems. In special cases, this can be viewed as the Fourier transforms on the Gelfand pairs (GLN(R),O(N)), (GLN(C),U(N)) and (GLN(H),Sp(N)). Our main results state that the law of large numbers and the central limit theorems for such particle systems, is equivalent to certain conditions on the germ at unity of their Jack generating functions. Our results provide a new toolbox for the analysis of the global behavior of stochastic discrete particle systems via their Jack generating functions, and the limiting behaviors of Jack symmetric functions via stochastic discrete dynamics. Some applications of our main results include:1.Using the law of large numbers and the central limit theorems for β-nonintersecting Poisson random walks in [33] as input, our theorem gives the asymptotics of Jack characters at unity.2.We prove law of large numbers and central limit theorems for the Littlewood-Richardson coefficients of zonal polynomials.3.We show that the fluctuations of the height functions of a general family of nonintersecting random walks are asymptotically equal to those of the pullback of the Gaussian free field on the upper half plane. As far as we know, the results are new even for Poisson/Bernoulli nonintersecting random walks.

Design of a high precision mechanical linear system for the study of the variables involved in the gas metal arc welding process

The gas metal arc welding process is studied because of its high productivity and low cost. One of the drawbacks of this process is achieving repeatability of the welding seams. This article shows the design of a linear mechanical system that provided a high precision in the trajectory of the base platform. Three important factors were investigated such as platform speed, the main stresses present and the manufacturing material. A descriptive type methodology was used which consisted of conducting the study as a discrete particle system. The optimum conditions for the efficiency of its operation were a linear velocity of 1.44 mm/s, maximum shear stress in the power screw of 283.57 kPa, which was calculated by means of static analysis to determine as the material of medium carbon steel of type AISI 1040, which due to its ductility, facilitated the machining of the power screw. This provided a mechanical efficiency of 30.9%. The kinematic calculations of the system were made with standardized elements found on the market.

A coupled thermo-mechanical model for the simulation of discrete particle systems

This work presents a computational model for the simulation of problems involving thermo-mechanically active particles forming discrete particle systems. Our approach is based on the discrete element method for description of the particles’ dynamics, combined with simple heat transfer equations to describe the various thermal effects that may take place when the system is excited by temperature gradients and external heat sources. We are able to track the motion of the particles and their thermal states over time under the influence of body (e.g., gravitational) forces, contact and friction forces (and the related moments w.r.t. the particles’ centers), as well as applied heat from external devices, heat transfer through conduction (at the particles’ interfaces upon contact with other particles and objects), convective cooling and radiative effects. Numerical examples are provided to validate our scheme and illustrate its applicability to the simulation of a wide range of engineering applications. We believe that simple, consistent particle models of the type as shown here may be a useful tool to the modeling of discrete particle systems that are consisted of thermo-mechanically active particles and, in a broader sense, many other multiphysical discrete systems.

Capturing heat transfer for complex-shaped multibody contact problems, a new FDEM approach

This paper presents a new approach for the modelling of heat transfer in 3D discrete particle systems. Using a combined finite–discrete element (FDEM) method, the surface of contact is numerically computed when two discrete meshes of two solids experience a small overlap. Incoming heat flux and heat conduction inside and between solid bodies are linked. In traditional FEM (finite element method) or DEM (discrete element method) approaches, to model heat transfer across contacting bodies, the surface of contact is not directly reconstructed. The approach adopted here uses the number of surface elements from the penetrating boundary meshes to form a polygon of the intersection, resulting in a significant decrease in the mesh dependency of the method. Moreover, this new method is suitable for any sizes or shapes making up the particle system, and heat distribution across particles is an inherent feature of the model. This FDEM approach is validated against two models: a FEM model and a DEM pipe network model. In addition, a multi-particle heat transfer contact problem of complex-shaped particles is presented.

Fast, flexible particle simulations — An introduction to MercuryDPM

We introduce the open-source package MercuryDPM, which we have been developing over the last few years. MercuryDPM is a code for discrete particle simulations. It simulates the motion of particles by applying forces and torques that stem either from external body forces, (gravity, magnetic fields, etc.) or particle interactions. The code has been developed extensively for granular applications, and in this case these are typically (elastic, plastic, viscous, frictional) contact forces or (adhesive) short-range forces. However, it could be adapted to include long-range (molecular, self-gravity) interactions as well.MercuryDPM is an object-oriented algorithm with an easy-to-use user interface and a flexible core, allowing developers to quickly add new features. It is parallelised using MPI and released under the BSD 3-clause licence. Its open-source developers’ community has developed many features, including moving and curved walls; state-of-the-art granular contact models; specialised classes for common geometries; non-spherical particles; general interfaces; restarting; visualisation; a large self-test suite; extensive documentation; and numerous tutorials and demos. In addition, MercuryDPM has three major components that were originally invented and developed by its team: an advanced contact detection method, which allows for the first time large simulations with wide size distributions; curved (non-triangulated) walls; and multicomponent, spatial and temporal coarse-graining, a novel way to extract continuum fields from discrete particle systems. We illustrate these tools and a selection of other MercuryDPM features via various applications, including size-driven segregation down inclined planes, rotating drums, and dosing silos. Program summaryProgram Title: MercuryDPMProgram Files doi: http://dx.doi.org/10.17632/n7jmdrdc52.1Licensing provisions: BSD 3-ClauseProgramming language: C++, FortranSupplementary material: http://mercurydpm.orgNature of problem: Simulation of granular materials, i.e. conglomerations of discrete, macroscopic particles. The interaction between individual grains is characterised by a loss of energy, making the behaviour of granular materials distinct from atomistic materials, i.e. solids, liquids and gases.Solution method: MercuryDPM (Thornton et al., 2013, 2019; Weinhart et al., 2016, 2017, 2019) is an implementation of the Discrete Particle Method (DPM), also known as the Discrete Element Method (DEM) (Cundall and Strack, 1979). It simulates the motion of individual particles by applying forces and torques that stem either from external forces (gravity, magnetic fields, etc.) or from particle-pair and particle–wall interactions (typically elastic, plastic, dissipative, frictional, and adhesive contact forces). DPM simulations have been successfully used to understand the many unique granular phenomena – sudden phase transitions, jamming, force localisation, etc. – that cannot be explained without considering the granular microstructure.Unusual features: MercuryDPM was designed ab initio with the aim of allowing the simulation of realistic geometries and materials found in industrial and geotechnical applications. It thus contains several bespoke features invented by the MercuryDPM team: (i) a neighbourhood detection algorithm (Krijgsman et al., 2014) that can efficiently simulate highly polydisperse packings, which are common in industry; (ii) curved walls (Weinhart et al., 2016) making it possible to model real industrial geometries exactly, without triangulation errors; and (iii) MercuryCG (Weinhart et al., 2012, 2013, 2016; Tunuguntla et al., 2016), a state-of-the-art analysis tool that extracts local continuum fields, providing accurate analytical/rheological information often not available from experiments or pilot plants. It further contains a large range of contact models to simulate complex interactions such as elasto-plastic deformation (Luding, 2008), sintering (Fuchs et al., 2017), melting (Weinhart et al., 2019), breaking, wet and dry cohesion (Roy et al., 2016, 2017), and liquid migration (Roy et al., 2018), all of which have important industrial applications.

Particle near-neighbour separation index for quantification of segregation of granular material

In this study, an alternative approach named near-neighbour separation index (NNSI) is proposed which defines mixing and segregation in terms of the nearest-neighbour particle scale methodology to characterise the evolution of particle-particle relationships in granular material flow systems so as to inform the extent of mixing and segregation (homogeneity) in space and time. Use is made of the geometrical particle information (position) as a function of time from the DEM simulations data to provide an instantaneous description and evaluation of the mixing and segregation process without due regard of particle history and initial configuration. The NNSI was found to give useful inferences on segregation and dispersion as it provides an unprecedented quantification of the particle-particle scale distributions of particles and fully provides a quantitative description of the particle mixture homogeneity. Furthermore, the NNSI is quite promising with added advantage and capability of handling discrete multi-size particle systems.

Fluctuations of particle systems determined by Schur generating functions

We develop a new toolbox for the analysis of the global behavior of stochastic discrete particle systems. We introduce and study the notion of the Schur generating function of a random discrete configuration. Our main result provides a Central Limit Theorem (CLT) for such a configuration given certain conditions on the Schur generating function. As applications of this approach, we prove CLT's for several probabilistic models coming from asymptotic representation theory and statistical physics, including random lozenge and domino tilings, non-intersecting random walks, decompositions of tensor products of representations of unitary groups.

Asymptotics of random domino tilings of rectangular Aztec diamonds

We consider asymptotics of a domino tiling model on a class of domains which we call rectangular Aztec diamonds. We prove the Law of Large Numbers for the corresponding height functions and provide explicit formulas for the limit. For a special class of examples, the explicit parametrization of the frozen boundary is given. It turns out to be an algebraic curve with very special properties. Moreover, we establish the convergence of the fluctuations of the height functions to the Gaussian Free Field in appropriate coordinates. Our main tool is a recently developed moment method for discrete particle systems.

Influence of inertia and material properties on discrete simulation of dynamic fracture of concrete

It is well known that the material resistance to fracture increases with increasing strain rate. For relatively slow strain rates, the response is strongly dependent on the material fracture properties such as material strength or fracture energy, whereas with further increase of loading rate, the loading force is getting influenced predominantly by the material inertia. The response of the material changes with strain rate in the sense of loading force as well as character of the crack pattern. The contribution is focused on simulations of fracture experiments at various strain rates. Dynamical concrete fracture is simulated using meso-scale discrete model, i.e. a system of interconnected discrete particles. Material properties in the model are randomly distributed within the volume domain. Similar to quasi-static loading rates, the material randomness influences the dynamic fracture as well. Deterministic and probabilistic simulations are compared and the influence of material parameters and their strain rate dependence are discussed.

Universal edge fluctuations of discrete interlaced particle systems

We impose the uniform probability measure on the set of all discrete Gelfand-Tsetlin patterns of depth $n$ with the particles on row $n$ in deterministic positions. These systems equivalently describe a broad class of random tilings models, and are closely related to the eigenvalue minor processes of a broad class of random Hermitian matrices. They have a determinantal structure, with a known correlation kernel. We rescale the systems by $\frac1n$, and examine the asymptotic behaviour, as $n \to \infty$, under weak asymptotic assumptions for the (rescaled) particles on row $n$: The empirical distribution of these converges weakly to a probability measure with compact support, and they otherwise satisfy mild regulatory restrictions. We prove that the correlation kernel of particles in the neighbourhood of `typical edge points' convergences to the extended Airy kernel. To do this, we first find an appropriate scaling for the fluctuations of the particles. We give an explicit parameterisation of the asymptotic edge, define an analogous non-asymptotic edge curve (or finite $n$-deterministic equivalent), and choose our scaling such that that the particles fluctuate around this with fluctuations of order $O(n^{-\frac13})$ and $O(n^{-\frac23})$ in the tangent and normal directions respectively. While the final results are quite natural, the technicalities involved in studying such a broad class of models under such weak asymptotic assumptions are unavoidable and extensive.

Discussion of Stokes' hypothesis through the smoothed particle hydrodynamics model

Stokes' hypothesis, the zeroing of the bulk viscosity in a Newtonian fluid, is discussed in this paper. To this aim, a continuum macroscopic fluid domain is initially modeled as a Hamiltonian system of discrete particles, for which the interparticle dissipative forces are required to be radial in order to conserve the angular momentum. The resulting system of particles is then reconverted to the continuum domain via the framework of the smoothed particle hydrodynamics (SPH) model. Since an SPH-consistent approximation of the Newtonian viscous term in the momentum equation incorporates interparticle radial as well as nonradial terms, it is postulated that the latter must be null. In the present work it is shown that this constraint implies that first and second viscosities are equal, resulting in a positive value for the bulk viscosity, in contradiction to the cited Stokes' hypothesis. Moreover, it is found that this postulate leads to bulk viscosity coefficients close to values found in the experimental literature for monoatomic gases and common liquids such as water.

Dependent scattering and absorption by densely packed discrete spherical particles: Effects of complex refractive index

Due to the dependent scattering and absorption effects, the radiative transfer equation (RTE) may not be suitable for dealing with radiative transfer in dense discrete random media. This paper continues previous research on multiple and dependent scattering in densely packed discrete particle systems, and puts emphasis on the effects of particle complex refractive index. The Mueller matrix elements of the scattering system with different complex refractive indexes are obtained by both electromagnetic method and radiative transfer method. The Maxwell equations are directly solved based on the superposition T-matrix method, while the RTE is solved by the Monte Carlo method combined with the hard sphere model in the Percus-Yevick approximation (HSPYA) to consider the dependent scattering effects. The results show that for densely packed discrete random media composed of medium size parameter particles (equals 6.964 in this study), the demarcation line between independent and dependent scattering has remarkable connections with the particle complex refractive index. With the particle volume fraction increase to a certain value, densely packed discrete particles with higher refractive index contrasts between the particles and host medium and higher particle absorption indexes are more likely to show stronger dependent characteristics. Due to the failure of the extended Rayleigh-Debye scattering condition, the HSPYA has weak effect on the dependent scattering correction at large phase shift parameters.