Neural-Network Closures for Complex-Shaped Particles in the Force-Coupling Method

TL;DR

This work tackles the challenge of simulating suspensions with complex-shaped particles in Stokes flow by introducing a data-driven neural-network closure that replaces analytical shape-specific Faxén-type relations in the force-coupling method (FCM). The closure is trained offline on high-accuracy boundary-element-method (BEM) data for spheroids and helicoidal particles, mapping local flow descriptors and orientation to the particle stresslet, rotation, and, for helices, chiral thrust, while leveraging physics-informed features to enforce known tensor structures. Validation shows the BEM solver matches analytical results for spheroids and remains accurate for helicoids via convergence checks, while the neural surrogate achieves median errors below 1% for the stresslet and around 1.3% for rotation in held-out tests, with helicoids exhibiting a few-percent BEM-dominated error. When embedded in quasi-dilute FCM simulations, the surrogate maintains near-linear scaling with particle count and enables large-ensemble rheology studies of three-dimensional particle shapes, facilitating systematic exploration of shape, flow type, and chirality effects on suspension behavior.

Abstract

A data-driven surrogate framework to accelerate particle-resolved modelling of quasi-dilute suspensions of rigid, non-spherical particles in Stokes flow is introduced. A regularized-Stokeslet boundary element method (BEM) is implemented to compute hydrodynamic responses in canonical linear flows, focusing on the particle stresslet and angular velocity for spheroids, and additionally the chiral thrust for helicoidal particles. For spheroids, the BEM solver is validated against available analytical benchmarks (Faxen-type relations for the stresslet and Jeffery's theory for rotation), and parameter choices for surface discretization and regularization are selected through systematic convergence studies. For helicoidal particles, where no analytical solutions exist, accuracy is quantified via Richardson-style self-convergence, complemented by tests of linearity, frame objectivity, and chirality-dependent symmetries. The resulting datasets are used to train a neural-operator surrogate that maps local flow descriptors and particle configuration to the corresponding stresslet, rotation, and thrust at negligible evaluation cost. Across independent test sets spanning random orientations and flow types, the surrogate achieves median relative errors below 1% for the deviatoric stresslet (95th percentile below 3%) and comparable accuracy for angular velocity and thrust. The combination of validated BEM generation and fast inference provides a practical route to coupling complex particle shapes into mesoscale solvers such as the force-coupling method, enabling large-ensemble studies of microstructure and suspension rheology.

Figures (6)

• Figure 1: Discretized particle geometries used to train the surrogate. Left: spheroid surface mesh (boundary-element nodes) for the axisymmetric ellipsoid. Right: helicoidal particle surface discretization (boundary-element nodes), showing the chiral geometry used in the BEM simulations.
• Figure 2: Self-convergence of the spheroidal BEM solver with respect to the number of surface nodes $N$ and the regularisation factor $\varepsilon$ for a spheroid with aspect ratio $r=2$. Top-left: mean relative error in Jeffery angular velocity compared with the analytical solution as a function of $N$ for three $\varepsilon$. Top-right: corresponding mean relative error in the deviatoric stresslet. Bottom: mean relative errors at $\bar{N}=4300$, showing that $\varepsilon=0.4$ minimises the stresslet error while the angular-velocity error remains small and only weakly dependent on $\varepsilon$.
• Figure 3: Convergence of the dimensionless stresslet magnitude $\hat{S}$ for helicoidal particles. The markers show the mean $\hat{S}$ over the validation dataset as a function of the number of surface nodes $N$ for three values of the dimensionless regularisation factor $\varepsilon$ (with $\varepsilon_{\mathrm{reg}} = \varepsilon\sqrt{A_p/N}$). Error bars denote one standard deviation over all orientations, flow types and handednesses: they are large because $\hat{S}$ varies strongly across configurations, not because of numerical noise. The nearly parallel curves and weak dependence on $N$ and $\varepsilon$ indicate that the stresslet level is already close to convergence in the range $N=2000\text{--}6000$.
• Figure 4: Training history of the large feature–augmented FCNN surrogates for spheroids (left) and helicoidal particles (right). The denormalised training and validation MSE rapidly decrease and then plateau, indicating stable convergence without overfitting.
• Figure 5: Parity plots comparing FCNN surrogate predictions with BEM reference values for the best spheroidal model on the test set.