Machine learning acceleration of simulations of Stokesian suspensions

Particulate Stokesian flows describe the hydrodynamics of rigid or deformable particles in Stokes flows. Due to highly nonlinear fluid-structure interaction dynamics, moving interfaces, and multiple scales, numerical simulations of such flows are challenging and expensive. Here, we propose a generic machine-learning-augmented reduced model for these flows. Our model replaces expensive parts of a numerical scheme with regression functions. Given the physical parameters of the particle, our model generalizes to arbitrary geometries and boundary conditions without the need to retrain the regression functions. It is approximately an order of magnitude faster than a state-of-the-art numerical scheme using the same number of degrees of freedom and can reproduce several features of the flow accurately. We illustrate the performance of our model on integral equation formulation of vesicle suspensions in two dimensions.

Figure 1

Problem setup and notation. We denote a vesicle membrane and a point on a membrane with $\gamma$ and $\mathbf{x}$, respectively. (a) shows the evolution of a vesicle ${\gamma }^0$ into ${\gamma }^{\mathrm{\Delta }t}$ after a time step in a free-space flow. (b) shows multiple vesicles in a confined flow. $\mathrm{\Gamma }$ stands for fixed boundaries (i.e., solid walls), and ${\gamma }_i$ and $\gamma$ represent the $i\mathrm{th}$ membrane and the other membranes.

Figure 2

Single vesicle in a parabolic flow. (a) Equilibrium shapes. The MLARM simulations (third row) are more accurate than the same-DOF simulations (second row) and as accurate as the same-cost ones (not shown) in capturing the true equilibrium shapes (first row). (b) Equilibrium lateral positions. We omit the results for the same-DOF simulations since they are erroneous (i.e., the equilibrium lateral positions are above 0.6 for all Ca values).

Figure 3

Vesicle trajectories in a flow with curved flow lines. A vesicle initialized at $10r_0$ migrates towards the center in time $t$. The final radial position is $3r_0$. (a) shows the true simulation. (b) The MLARM simulation accurately captures the migration with a 6% error in migration velocity. See the Supplemental Material for simulations of numerical examples [30].

Figure 4

Equilibrium organization in dilute suspensions in a Taylor-Couette flow (area fractions of 2.2% at the top and 4.4% at the bottom). The MLARM and true solutions are superimposed and shown in red (light gray) and black, respectively. At the equilibrium of such flows, vesicles exhibit a spatial order by moving in a rim with a uniform angular interdistance. We present snapshots after $n_r$ number of rotations of the inner circle. The dashed lines in the third column show the equilibrium rims. The last two columns demonstrate the magnitude of the perturbation in the velocity field induced by vesicles after $n_r=24$ rotations. See the Supplemental Material for simulations of numerical examples [30].

Figure 5

Statistics of vesicles in a dense Taylor-Couette flows at area fractions of (a) 20% and (b) 35%. We plot the probability distribution of the distance of the vesicle center to the origin $\parallel \mathbf{c}\parallel$ throughout the simulation. Vesicles form regions near circles that are free of vesicles. The MLARM simulations accurately capture these regions in both cases. (b) There is no stable same-cost simulation for an area fraction of 35%. See the Supplemental Material for simulations of numerical examples [30].