The reference map technique for fluid-structure interactions
Fluid-structure interactions (FSI) are ubiquitous in nature, laboratory, and industrial setting. For example, animal locomotion, fluid flowing through porous media, or cheese being stirred and pumped through a cheese making machine. However, FSI problems are challenging to solve analytically or simulate numerically, due to their nonlinear, multi-physics nature.
In particular, it is difficult to reconcile the dilemma of choosing a discretization framework. Solid simulations are typically computed in a Lagrangian framework. whereas fluid dynamics are more conveniently done from the Eulerian perspective. Methods such as Arbitrary Lagrangian-Eulerian and Immersed Boundary methods are proposed to address this challenge, but they require extra computations to bridge the discretization from different perspectives.
My group mate Nick Derr and I have been working with our advisor, Chris, on developing a fully Eulerian method, the reference map technique (Rycroft et al. 2020) along with a 3D implementation of it. We recently submitted our manuscript and published our code on GitHub.
The reference map technique is rooted in large deformation solid mechanics, so the method is particularly suitable for simulating immersed soft, very deformable solids. The idea of the reference map is simple and elegant, and the essence of it is to find a way to calculate the deformation in the material in the current physical space, rather than the undeformed reference space. Being on a fixed regular grid comes with computational time advantages, giving the method a performance edge in many-body interaction problems.
The 3D code is developed with distributed memory parallelism, specialized data structures and methods to ensure efficiency. For instance, a new least squares-regression based extrapolation algorithm is developed to address challenges of field extrapolation in 3D.
A model of Type VI Secretion System in Vibrio fischeri
E. scolopes is a species of small squids living off the coast of Hawaii. Early in its life cycle, it forms a life-long symbiosis with V. fischeri bacteria. The squids need the bioluminescent bacteria for a form of camouflage called counter-illumination, which helps the squid blend in with the moonlit ocean surface at night while out hunting. Scientist also find that the bacteria is crucial in squid’s development – without the bacterial infection, squids do not develop mature light organs (McFall-Ngai, M., Annu. Rev. Microbiol. 2014 68(), 177-94).
What we are interested in is the microbial component of the symbiosis. V. fischeri is very abundant in the ocean, and some strains of it have evolved what’s called Type VI Secretion System (T6SS). When I first heard of this, my mind was blown — what T6SS does is it allows a bacterium to put together a spear, and shoot it at another cell and kill it. I knew bacteria can swim and swarm, but the ability to shoot and kill adds a whole new level of complexity. What’s also fascinating is that these spears, commonly called sheaths, are very similar to that of bacteria phages’. One theory is that at one point in the evolution of bacteria, some species co-opted bacteria phages’ DNA to build T6SS molecular machine – we may have on our hand a classic example of what doesn’t kill you makes you stronger!
As a computational mathematician, I approached the questions in T6SS-dependent interactions using a bit of mathematical modeling and computation. Adapting well-known models for transcription and translation, I developed a simple model for activation of T6SS expression and the dynamics of assembling and firing T6 structures. Then, this model gets plugged in to an agent-based model, so that every single cell is undergoing activation, assembly, and firing independently. But collectively, the contact dependent killing creates an emergent spatial structure (see movies below) that resembles coarsening, or phase separation, in physical systems such as magnetic fluids and liquid crystal.
The story doesn’t just end there though. We are still actively experimenting and developing our theory to account for new observations. Some details and results can be found in a poster I presented for CSGF review.
Diffusion-limited Dissolution and Aggregation
I took this photo on a hike along the Devil’s garden trail in Arches National Park. In that part of the US, fantastic patterns are abundant and breathtaking. They range from the entire canyon of hoodoos in Bryce National Park, to the world famous arches, to smooth cavities of all sizes and shapes in the rocks. The patterns are formed, among others, by slow diffusive and erosive processes that happen over a long time scale. But sometimes, a sudden structure failure can cause rapid changes, for example, the rock falls at the Landscape arch in the 1960s.
In one of my research projects, I look at a simplified computational model of this type of dissolution processes in 2D. In particular, I focus on a type of process called the diffusion-limited dissolution. The matter to be dissolved is modeled as a cluster of uniformed particles on a regular lattice, and the diffusion of a dissolving particle is modeled as a random walk. The conformal mapping technique is used to accurately calculate the probability of contact between a diffusing particle and a particle in the matter cluster. Using this model, we can explore the statistical properties of dissolving interfaces, and the dynamics at the end of the dissolution event, i.e. when a clump of matter is about to be completely dissolved away. I presented a poster on this project at the 2017 DOE CSGF program review, the poster can be found be found here.