Accelerating the force-coupling method for hydrodynamic interactions in periodic domains

TL;DR

A modified kernel is constructed to ensure rapid convergence to the exact hydrodynamic interactions and a positive-splitting of the associated mobility matrix and establishes the optimal choice of the modified kernel width, which is shown plays a similar role to the splitting parameter in Ewald summation.

Abstract

The efficient simulation of fluid-structure interactions at zero Reynolds number requires the use of fast summation techniques in order to rapidly compute the long-ranged hydrodynamic interactions between the structures. One approach for periodic domains involves utilising a compact or exponentially decaying kernel function to spread the force on the structure to a regular grid where the resulting flow and interactions can be computed efficiently using an FFT-based solver. A limitation to this approach is that the grid spacing must be chosen to resolve the kernel and thus, these methods can become inefficient when the separation between the structures is large compared to the kernel width. In this paper, we address this issue for the force-coupling method (FCM) by introducing a modified kernel that can be resolved on a much coarser grid, and subsequently correcting the resulting interactions in a pairwise fashion. The modified kernel is constructed to ensure rapid convergence to the exact hydrodynamic interactions and a positive-splitting of the associated mobility matrix. We provide a detailed computational study of the methodology and establish the optimal choice of the modified kernel width, which we show plays a similar role to the splitting parameter in Ewald summation. Finally, we perform example simulations of rod sedimentation and active filament coordination to demonstrate the performance of fast FCM in application.

Figures (8)

• Figure 1: Contour plots showing the fast FCM error as a function of (a) $\Sigma/\Delta x$ and $M_G$ and (b) $\Sigma/\sigma$ and $R_c/\sigma$ for a random suspension with $N=64457$, $\phi=8\%$ and $a/L=1/150$.
• Figure 2: The computational cost measured in particle timesteps per second (PTPS) as a function of $\Sigma/\sigma$ for random suspensions with different $\phi$. The computations are performed with $\epsilon = 10^{-4}$ and $a/L=1/400$. Data obtained using our a single-precision CUDA implementation of fast FCM run on a single RTX 2080Ti.
• Figure 3: (a) The ratio of PTPS for fast FCM to that for standard FCM as a function of $\phi$ for different $a/L$ and $\epsilon = 10^{-4}$. Timings are obtained by averaging 50 computations after an initial GPU warm-up period. (b) $\phi_c$ as a function of $a/L$. The solid line shows $\phi_c = -9.24(a/L) + 0.22$ which is determined from a linear fit to the data. (c) The same as (a), but with $\epsilon = 10^{-6}$. (d) The same as (b), but with $\epsilon = 10^{-6}$.
• Figure 4: (a) Image showing a snapshot from a simulation of a monolayer of $2304$ sedimenting rods in a domain of size $960a \times 960a \times 60a$. The rods are constructed from FCM particle doublets as shown in the inset image. (b) Average sedimentation velocity, $\langle V_x\rangle = (\sum_n \bm{V}_n\cdot\bm{\hat{x}})/N_R$, over time for simulations with $N_R = 2304$ and $L = 960a$$(\phi=10.50\%)$ (solid line), $L = 1920a$$(\phi=2.62\%)$ (dashed line), and $L = 3840a$$(\phi=0.66\%)$ (dotted line).
• Figure 5: (a) Snapshot of the simulation with 7744 rods at $t = 339.64T$. (b) Image showing the rod speeds at $t = 339.64T$. (c) Snapshot of the simulation with 7744 rods at $t = 4992.81T$. (b) Image showing the rod speeds at $t = 4992.81T$.