Understanding the Discrete Element Method Using Sand Particles
On a recent trip to the Outer Banks, I watched as the kids defended their sandcastles from the rising tide and wondered at the physics that make such fun possible.
Granular materials like sand are very interesting in that they don’t constitute a single phase of matter, but can act like a liquid, solid or gas depending upon the circumstance. Everyone is innately familiar with sand behaving as a fluids, as in an hourglass. Less familiar is their behavior as a structural solid after the addition of a little water. What is even more intriguing to me is the subsequent retransition to liquid behavior after a second addition of water, as in a mudslide. However, liquefaction through cyclical compression and fluidized beds are topics for yet another day.
Granular materials like sand are very interesting in that they don’t constitute a single phase of matter, but can act like a liquid, solid or gas depending upon the circumstance.
A little research (#googling) reveals that sandcastles are made possible through the formation of cohesive bonds between adjacent particles. In dry sand, no such bonds exist, and each particle is free to move accordingly. When water is added, the water fills the voids between particles while also adhering to the surfaces of the particles. Due to liquid surface tension, what are known as “capillary bridges” are formed between particles structurally linking them to one another, a phenomenon first studied by Josef Louis Lagrange in 1760. All the CFDers reading this will most likely recognize the last name.
The forces holding the particles together are intermolecular in nature, resulting fundamentally from the electrical charges of the water and sand particles. In the world of CFD, we rarely dabble on the molecular level. Instead, we much prefer to abstract those relationships to corresponding mathematical equations (or models) on the continuum level, which can then be spatially and temporally discretized and solved using computers.
When I got back from the beach (#workfromhome), I was curious enough about the sandcastles I had watched washing away with the tide, that I decided to give simulating this type of sand behavior a try using the latest Discrete Element Method (DEM) tools available in the STAR-CCM+ multiphysics simulation platform. DEM is a method for tracking and simulating the complex interactions of systems consisting of thousands of discrete particles. Such interactions include the forces of friction, contact plasticity, gravity, heat transfer, electrostatics and adhesion/cohesion.
One way to model cohesion in STAR-CCM+ is through the “Parallel Bonds” model. The Parallel Bonds model uses the concept of a massless bar connecting a pair of bonded particles. The bar can transmit force and torque between particles. During the specified formation time period, the Parallel Bonds model acts to form clusters from particles that collide.
Subsequently, one can specify the failure of these bonds through either a simple failure model or through an aggregate damage model. In the simple failure model, bonds that are subject to tensile or shear stresses exceeding a certain threshold are immediately broken. In the aggregate damage model, fracture damage is incurred over time due to repeated stresses until aggregate damage exceeds such a threshold and the bond is broken.
Building our DEM castle would be accomplished by forming bonds between particles that are initially constrained to the shape of the castle; the digital analog of turning a semi-wet sand mold upside down and tapping out the form. Fortunately, because of some pretty sweet methods built into STAR-CCM+, initializing particles in a specific geometry isn’t a big deal, and we do it frequently in our work optimizing industrial equipment such as agricultural sprinklers, packed bed reactors, and rotary oil-sand dryers. This is accomplished by defining two spatial regions; one for the castle and one for the remaining domain.
First, the regions are separated by a solid partition interface. The castle region is then initialized with particles that form bonds, either with a part or lattice type particle injector. After the particles are settled and bonds are formed, this interface is converted into a fluid-fluid interface , allowing both fluids and particles to pass unimpeded.
While the first part of this simulation seemed obvious, how to fracture the bonds with water was more complicated. We work with the volume of fluid (VOF) and wave initialization tools available in STAR-CCM+ on a regular basis, so I knew the creation and simulation of the ocean waves could be handled with ease. The only puzzle left to solvewas linking the bond failure to the “wetness” of the particle. Just to demonstrate the capability, we utilized a simple “if wet, the bond breaks” philosophy implemented through the simple failure model. A field function was written such that threshold stress at which failure would occur would be set to 0 if the particle were submerged.
While admittedly not perfect, the animation from the created using the screenplay feature in STAR-CCM+, is pretty slick, especially for the time and effort put into it.
Most notably, to keep this simulation quick and dirty (aka run on a laptop in a few hours’ time), I reduced the number of sand-grains by making them much larger than real particles. First, you’ll notice in the animation that the towers of the model wobble a bit. This is due to the tower having only a limited number of bonds and not being as rigidly constrained as a tower packed with tens of thousands of such bonds would be. Second, you may notice that some clumps had not bonded before I released the interfaces, allowing chunks to fall off without getting wet. I guess I should have let them settle a little longer. I’d love to hear your thoughts on the effort as well.
I’ll be running an interactive webinar demonstration with a 15 minute Q&A to demonstrate these STAR-CCM+ capabilities on July 29th , so be sure to tune in and learn. Register using the link below.
