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

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.

💡 Research Summary

The authors present a data‑driven surrogate framework that enables fast, accurate incorporation of complex‑shaped rigid particles into the force‑coupling method (FCM) for Stokes‑flow suspensions. The workflow consists of two main stages. First, a regularized‑Stokeslet boundary element method (BEM) is implemented to compute single‑particle hydrodynamic responses—stresslet, angular velocity, and, for helicoidal particles, a chiral thrust—in a set of canonical linear flows (simple shear, uniaxial, planar, and biaxial extensions). For spheroids, the BEM results are benchmarked against Faxén‑type relations for the stresslet and Jeffery’s theory for rotation, confirming sub‑percent errors when the regularization parameter ε≈0.02 R and the surface discretization uses 10‑20 k triangular elements. For helices, where analytical solutions are unavailable, Richardson extrapolation demonstrates second‑order convergence, and the authors verify linearity, frame objectivity, and parity‑odd/even symmetries of the computed quantities.

Using these high‑fidelity simulations, a comprehensive dataset of roughly 500 000 samples is generated, each mapping the local strain‑rate tensor E, vorticity tensor Ω, and particle orientation p to the output tuple (stresslet S, angular velocity ω, thrust Fₚ). The dataset spans random orientations and all four flow types, ensuring coverage of the relevant configuration space. The authors then train neural‑operator surrogates—fully‑connected feed‑forward networks with 3–5 hidden layers (200–400 neurons per layer). Input features are enriched with physics‑informed combinations such as E·p·p, Ω·p, and normalized magnitudes, while the loss function combines mean‑squared error with penalty terms that enforce known symmetries (e.g., stresslet symmetry, thrust parity). Training employs the Adam optimizer with a learning‑rate schedule and converges within 200 epochs.

Extensive ablation studies show that physics‑informed features and chirality‑mirrored data augmentation markedly improve accuracy. The best models achieve median relative errors of 0.7 % for deviatoric stresslet components (95 % percentile ≤ 2.8 %), ≈ 1.2 % for angular velocity, and ≤ 1.5 % for helicoidal thrust on held‑out test sets. Model size reduction has only a modest impact on error, indicating robustness. When embedded directly into an FCM solver, the surrogate requires a single forward pass (~10⁻⁶ s) per particle, delivering speed‑ups of three to four orders of magnitude compared with on‑the‑fly BEM calculations. Large‑scale simulations with thousands of helices reproduce macroscopic rheological quantities—effective viscosity, normal‑stress differences, and orientation distributions—with statistical agreement to reference BEM‑based results, confirming that the surrogate does not introduce systematic bias.

The paper discusses limitations: the current framework is restricted to Stokes flow, non‑Brownian, neutrally buoyant particles at quasi‑dilute volume fractions (ϕ≈0.01–0.15). Extensions to higher concentrations would require lubrication and contact models; finite Reynolds number effects, active propulsion, deformable particles, and continuous shape parametrization (e.g., varying helix pitch) are identified as future research directions. Nonetheless, the presented surrogate provides a practical, accurate, and computationally cheap closure for FCM, opening the door to systematic rheological studies of suspensions containing arbitrarily complex three‑dimensional particles.