A Barrier Method for Contact Avoiding Particles in Stokes Flow

TL;DR

This work introduces a barrier-based contact-resolution framework for 3D Stokes flow that guarantees non-overlapping configurations of rigid particles by solving for minimal contact force magnitudes on potential-contact pairs. The method represents non-overlap constraints via a barrier energy $b(d,\hat{d})$ and enforces zero barrier energy at the next time-step, using a forward-Euler time update and the rigid multiblob mobility solver. It robustly handles spheres, rods, and boomerangs, including non-convex geometries and complex flows, while keeping the impact of artificial contact forces small and balanced by Newton's third law. The approach achieves collision-free dynamics with adjustable buffer parameters, and a practical solver based on constrained minimisation (via $\texttt{fmincon}$) makes the method accessible for large suspensions, with future directions including mobility-matrix approximations and smoother barrier formulations to further enhance efficiency and symmetry.

Abstract

Rigid particles in a Stokesian fluid can physically not overlap, as a thin layer of fluid always separates a particle pair, exerting increasingly strong repulsive forces on the bodies for decreasing separations. Numerically, resolving these lubrication forces comes at an intractably large cost even for moderate system sizes. Hence, it can typically not be guaranteed that particle collisions and overlaps do not occur in a dynamic simulation, independently of the choice of method to solve the Stokes equations. In this work, non-overlap constraints, in terms of the Euclidean distance between boundary points on the particles, are represented via a barrier energy. We solve for the minimum magnitudes of repelling contact forces between any particle pair in contact to correct for overlaps by enforcing a zero barrier energy at the next time level, given a contact-free configuration at a previous instance in time. The method is tested using a multiblob method to solve the mobility problem in Stokes flow applied to suspensions of spheres, rods and boomerang shaped particles. Collision free configurations are obtained at all instances in time. The effect of the contact forces on the collective order of a set of rods in a background flow that naturally promote particle interactions is also illustrated.

Figures (9)

• Figure 1: Multiblob particles in a Stokesian fluid. Colors indicate the depth in the figures.
• Figure 2: Illustration of the two contact distances $d_i^{\text{p}}$ and $d_{ik}^{\text{s}}$ and the corresponding contact force directions on the particles for two fat rods close to contact in pair $i$.
• Figure 3: Barrier functions as defined in \ref{['b_opt']} and \ref{['b_alt']} are smooth approximations of the indicator function. Here, the barrier function is modified to be defined for negative inter-particle distances, as particles might overlap in the trial time-step and in iterations of the optimisation method before contact forces of the right magnitudes are found.
• Figure 4: Contact forces are computed for all particle pairs that violate $\hat{d}$ among 500 randomly positioned spheres in a cube, where the packing density is $24\%$ and $12\%$ respectively. The spheres are affected by external forces with directions randomly drawn from the unit sphere and $\|\boldsymbol{f}_i\| = 100$ and a single time-step is taken with $\Delta t = 0.01$. At the trial time-step, particles are in contact if they are closer to each other than $\hat{d}$, here picked for each random configuration as the minimum separation distance at the previous, contact-free time-step. Minimum separation distances $\hat{d}$ are respected with the contact forces, and moreover, the forces do not drastically change the inter-particle distances for the closest particles, meaning that the forces are not unnecessarily large.
• Figure 5: Almost all contact forces are small in magnitude relative to the external force, which implies that only the pair or pairs that actually violate the set minimum allowed distance $\hat{d}$ are assigned a significant contact force, even if more particle pairs are flagged to be part of the contact optimisation. For each hyperparameter combination, statistics is collected from all flagged contacts in 200 random configurations of spheres with packing density $24\%$. The same behaviour is noted for all hyperparameter combinations.

Theorems & Definitions (5)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5