Discrete Element Method (DEM) for Granular Materials and Planetary Science

What is a continuum? It is totally hocus-pocus-mathamatocus!

All matter that we see around us can be theoretically understood as continuous, such that differential/integral forms of equations representing their behavior may be possible. This, quite often, allows us to obtain analytical expressions for understanding how they might respond to external agents like forces or moments. However, matter fundamentally, as we all know, is made of elementary units like molecules and atoms i.e. it is discrete. A continuum, thus, is more like a mathematical construct rather than the reality. The interaction law between the elementary units manifests itself in the continuum theory as what we popularly call the 'constitutive relation' which differentiates not only between different states of matter (solid/liquid/gaseous) but also amongst the sub-classes within them, for eg. Newtonian and non-Newtonian fluids. Speaking of continuum, you may look into the Knudsen number, calculating which for a system tells you whether continuum hypothesis is valid or not.

What is the problem of continuum? Breaks down in certain cases!

However useful, the applicability of the popular mass, momentum and energy balances for continuous media, breaks down at discontinuities or when molecules are so arranged that separate constitutive relations are required to describe different regions in the same substance. A popular example of the latter is GRANULAR materials. Why? Granular materials behave like solids when stacked (sugar in a container), flow like liquids when sheared (landslides on mountains) and move randomly like gases when energized significantly (rings of Saturn). Find out more in this video: https://www.youtube.com/watch?v=8xAfzz1HT8s . This requires one to be very careful while dealing with such discrepancies, often complicating the analysis in a way even Boltzmann might find difficult to understand.

What is the solution? DEM to the rescue!

While continuum theories are able to accurately model these different phases, it becomes a tad bit more painful to formulate a unified theory for all of them. This is where DEM or the Discrete Element Method comes in. Well, a lot is obvious from just the nomenclature, right? The treatment is discrete. Granular materials are collections of particles which primarily interact only via collisions. In some cases, there could be other 'contact' interactions involved, for example, cohesion, Coulomb friction etc. In other cases, one particle may interact with all other particles including those which are not in contact, like when electrostatic or gravitational forces exist between particles. All of this is easily, albeit at a much higher computational cost, simulated using DEM. DEM can capture some really interesting phenomena, like jamming in granular materials which is otherwise difficult using continuum models.

Why easy? Can anybody do it? No hi-fi mathy stuff? No? Yaay!

Easy because, all that needs to be solved is, F = ma (well, \tau = I \alpha, if rotations are included, but that's just the same). At every timestep, the total force F can be calculated from a sum total of all the forces acting on a particle. Since the mass m of a particle is known, acceleration a may be computed and numerically integrated once and twice to obtain its velocity v and the displacement s respectively. Collisions are modelled by a pair of normal and tangential springs and dashpots which could either be linear or non-linear in nature depending upon the material being modelled. The spring acts to decelerate relative motion between particles in contact and the dashpot dissipates the energy rendering collisions realistic/inelastic. There are six outputs; three for velocity and three for displacement. Once you have a RK4 or any other time marching scheme you prefer in place, it is basically solving lacs and lacs of ODEs simultaneously.

Is DEM perfect? Well, neither am I.

While DEM works with the least possible assumptions/constraints and captures most possible physics, there are of course some limitations. No one is perfect!

Numero uno: Spherical particles. While people have gone ahead and tried all kinds of weird shapes, spherical particles gets jobs done at a fractional expense compared with the others and even that is very high. You need super computers. And they have an enormous carbon-footprint. Google doesn't care though, or Amazon or Facebook or Exxon Mobil or even Elon Musk for that matter (this has nothing to do with DEM however).

Numero dos: Interaction laws can get super complicated and lead to high computational costs. Once again. Yes. It's the tool for the Ambani's and Bajaj's and the government funded educational institutes in our country. Well, returning our attention to less important issues, for example, 'long range' forces like Coulomb or gravity increase the computational cost quadratically (courtesy Wikipedia) because forces need to be calculated due to every particle on every other particle.

DEM in Planetary Sciences.

In my Masters thesis, I used both continuum theory and DEM to study granular avalanches on rotating and gravitating asteroids. The use of DEM was to primarily evaluate the efficacy of the theory developed. Besides, asteroids which motivated this research are rubble-piles (Itokawa & Bennu) exhibiting all, solid, fluid and gaseous behavior of regolith on the surface. The cores are solid, the landslides occurring on them are fluid and mass shed from the surface is almost gaseous. You could see a simulation in the video above. It shows the formation of a top-shaped asteroid due to surface landslides (many may not agree).

I used the popular open-source DEM software called LAMMPS but had to tweak the source codes to include the gravity field of ellipsoids. Several other people use LIGGGHTs which is better suited for granular materials. For those who like to include long range forces like gravity, PKDGRAV and REBOUND work great. The latter is especially built for large number of particles which may impact each other at really high velocities and hence collision resolution is a little tricky, but nonetheless, the creators are still working on it.

Is there something in between? Well, there's always something in between!

Hybrid methods pack the accuracy of DEM and cost-effectiveness of continuum models. If you are interested you could look into the work of a collaboration between groups at Columbia, MIT and U Tokyo here: https://www.youtube.com/watch?v=Z86Udk2JXeU