# Methods

The computational granular physics is based on various methods for the computation of the equations describing a physical model of a granular system. The physical modeling involves the scale of description and various approximations of shapes, sizes and nature of the particles, and it should be considered as the first step in numerical modeling of a granular system. Specific algorithms are then required for the preparation of the initial configuration according to the targeted application and the constraints imposed by numerical efficiency. Such pre-processing methods and algorithms are a significant part of the simulator’s work in granular physics. The calculation of a single step of evolution according to the equations of motion is the heart of a granular simulation. Several explicit and implicit methods based on different rules and stepping schemes are available and currently used both for research and in application to engineering applications. The main calculation loop consists of a number of consecutive time steps, and the simulation ends when a specified criterion is satisfied. The post-processing methods are then necessary for the treatment of information in view of extracting the useful data for a spatiotemporal analysis of the system. This physical analysis of a discrete system is not straightforward and a feedback to the initial physical model or to the initial configuration might be necessary for an improved simulation of the real physical system. A numerical “method” may refer to either of the above aspects and other algorithmic aspects such as data structures, sorting of nearest neighbors and coupling with allied methods such as hydrodynamic interactions.

## Discrete Element Methods

### Contact dynamics

##### F Radjai and V Richefeu

We present the CD method as a consistent model of nonsmooth and multicontact granular dynamics expressed in contact coordinates. Nonsmoothness refers to various degrees of discontinuity in local or global characteristics of a dynamical system. The mathematical concepts and tools for the treatment of nonsmooth dynamics were developed in relation with mechanical prob- lems involving unilateral constraints and in the context of convex analysis. The multicontact feature is present in static states and in dense flows of granular materials where spatial correlations occur at large length scales and impulse dynamics is mixed with smooth particle motions at different time scales.

### Molecular Dynamics

### Lattice element method

##### V Topin, J-Y Delenne and F Radjai

The effect of a binding matrix occurring in high volume fraction cannot be reduced to DEM pair-wise interaction laws. A sub-particle discretization of both the particles and the matrix is therefore the only viable approach in this limit. We introduce here the lattice element method (LEM), which relies on 1D-element meshing of both the particles and binding matrix. Several simple rheological models can be used to describe the behavior of each phase. Moreover, the behavior of the different interfaces between the phases can be accounted for. In this way, the model gives access to the behavior and failure of cohesive bonds but also to that of particles and matrix. The LEM may be considered to be a generalization of DEM in which the discrete elements are the material points belonging to each phase instead of the particles as rigid bodies. We briefly present this approach in a 2D framework.

### Periodic Boundary Conditions

##### F Radjai and C Voivret

We present here a method for the prescription of periodic boundary conditions in DEM simulations of granular materials. This method is similar in practice to that of Parrinello and Rahman, but since the particle interactions are dissipative in a granular system, the equations of motion for collective dynamic variables can not be based on a Hamiltonian. We consider in detail the particular kinematics of periodic systems, the equations of dynamics and time-stepping schemes for MD-DEM and CD-DEM.

## Assembling Methods

### Polydispersity

##### C Voivret, J-Y Delenne and F Radjai

During a granulometric analysis or a laboratory mechanical test, the sample must be representative of particle properties including the particle size distribution both in volume (mainly for the smallest particles) and in number (mainly for the largest particles). This representativity of PSD is actually one of the conditions for a sample to be a representative volume element (RVE). However, in discrete numerical simulations of polydisperse granular samples this condition can not be always satisfied because of the computational cost imposing a limited an upper limit on the number of particles. For this reason, one needs appropriate methodology for optimal control of the statistical representativity of size classes. The issue is to extract a discrete ensemble of particle sizes from a measurement or an analytical function with a given number of particles and such that the distribution is optimally represented.

### Geometrical Assembling

##### C Voivret, F Radjai and J-Y Delenne

The issue of the assembling methods is to construct configurations of particles as close as possible to a state of mechanical equilibrium with built-in packing properties. This can be, for example, a target packing fraction for a given parti- cle size distribution. In the same way, the average connectivity of the particles (coordination number) and the anisotropy of the contact network are basic geometrical properties that control the mechanical response of a packing and may be built into a packing by an appropriate method. The homogeneity of the particle assembly in terms of packing fraction and connectivity is another important property which depends on the assembling rules. We introduce here several basic assembling methods by simple geometrical rules.

## Particle interactions

### Capillary bonding

##### V Richefeu and F Radjai

We present here the capillary cohesion resulting from a liquid bridge between two particles.

### Solid bonding

##### J-Y Delenne

DEM algorithms can easily be extended to account for cohesive interactions which are often simply supplemented to the repulsive elastic and frictional interactions of cohesionless materials. We present here a general framework for the implementation of cohesion in a discrete-element framework. Generally, simple models of interaction laws are privileged in DEM so as to reduce time- consuming operations arising from too many elementary operations. In any case, one has to check the impact of the simplifications on the resulting global behavior of the material.e distribution is optimally represented.

## Micromechanics and Statistical Analyzing

### Force Transmission

##### F Radjai and S Roux

We analyze below the distributions and correlations of contact forces from numerical simulations. A quantitative description of the force distributions and their link with granular texture reveals the bimodal transmission of stresses in granular media that will also be briefly presented.

### Texture

##### F Radjai, E Azema and J Lanier

The granular texture is disordered with many different variants depending on the composition (particles shapes and sizes), interactions and assembling procedure. The granular disorder is essentially characterized by the fact that, as a result of geometrical exclusions among particles, the local vectors vary discontinuously from one contact to another. In other words, the local environments fluctuate in space. The contact network evolves with loading so that the local environments fluctuate also in time. The highly inhomogeneous distribution of contact forces reflects granular disorder in static equilibrium. In particular, the force chains reveal long-range correlations whereas the presence of a broad population of very weak forces results from the arching effect. The force and fabric anisotropies are two complementary aspects of stress transmission, and they can be employed in local (particle-scale) description of granular media in the quasi-static state.