Accelerating droplet-laden Stokes flow simulations with hierarchical surrogate modeling

Technical Summary: Accelerating droplet-laden Stokes flow simulations with hierarchical surrogate modeling

Problem Statement
The paper addresses the computational challenges inherent in simulating incompressible multiphase Stokes flows involving a large number ($M$) of immiscible liquid droplets suspended in a carrier fluid. Accurate high-fidelity simulations of such systems are computationally demanding, particularly when droplets are highly concentrated, approach boundaries, or interact closely. Standard discretization techniques, such as finite-element methods (requiring repeated remeshing) or boundary integral methods (requiring specialized quadrature for near-singular integrals), struggle with the narrow interfacial gaps and near-boundary interactions. While particle-based methods avoid remeshing, they often require fine resolutions for accuracy and lack the precision of high-order methods. The authors aim to develop a surrogate modeling strategy that enables efficient prediction of droplet evolution in densely packed, time-varying configurations without the prohibitive cost of fully resolved multi-fluid simulations.

Methodology
The proposed approach is a multi-fidelity, physics-informed hierarchical surrogate modeling framework. It relies on an iterative error-correction strategy rather than treating the surrogate as a black box. The methodology proceeds as follows:

1. Lowest Fidelity Approximation: The process begins with a baseline Stokes solution ($u_0, p_0$) where droplets are treated as passive tracers. This assumes a homogeneous "nominal" viscosity throughout the domain, neglecting viscosity contrasts and surface tension.
2. Error Equation Derivation: The authors derive a partial differential equation (PDE) governing the error field ($v, q$) between the true solution and the lowest-fidelity approximation. This error equation accounts for the jump in viscosity and surface tension forces across the droplet interfaces.
3. Hierarchical Correction via Iteration: Instead of solving the global error equation directly (which retains the original computational complexity), the method employs an alternating iterative scheme:
   • Droplet Correction: The global error equation is decomposed into $M$ independent local problems, each representing the flow disturbance caused by a single droplet in free space. These local problems are solved approximately.
   • Boundary Correction: The superposition of local corrections violates the global boundary conditions. A subsequent "boundary correction" step solves a global Stokes problem (without droplets) to restore consistency with the domain boundaries.
   • Iteration: These two steps (droplet and boundary corrections) are repeated until a convergence criterion based on boundary residuals is met.
4. Offline-Online Efficiency: For geometrically similar droplets (specifically circular in the presented tests), the method exploits self-similarity. The single-droplet problems are mapped to a reference unit disk. A linear subspace approach is used to approximate the solution operator for these single-droplet problems using a small set of precomputed "fundamental solutions" corresponding to polynomial flow fields. This allows the online evaluation of droplet corrections to be performed via simple linear combinations, significantly reducing computational cost.

Key Contributions

• Physics-Based Error Correction: The paper introduces a multi-fidelity iterative strategy that systematically recovers high-fidelity solutions by solving PDE-based error equations, avoiding the "black box" nature of many machine learning surrogates.
• Replacing Refinement with Iteration: The method avoids the high complexity of solving global error equations in tightly packed domains by constructing corrections locally through the superposition of single-droplet fields. This reduces memory requirements and computational cost.
• Scalability for Many Droplets: The framework is designed to handle systems with up to $10^4$ droplets. By decoupling the global boundary problem from the local droplet interactions and utilizing precomputed fundamental solutions, the method achieves substantial speed-ups compared to fully resolved simulations.
• Handling of Surface Tension: For circular droplets, surface tension effects are accounted for exactly via pressure jumps, allowing the iterative correction to focus on viscosity contrasts and flow interactions.

Results
Numerical experiments in two dimensions validate the accuracy and efficiency of the proposed surrogate:

• Accuracy: The surrogate achieves relative errors below $\sim 0.1\%$ compared to fully resolved high-fidelity CutFEM simulations. Convergence tests show exponential convergence with respect to the polynomial degree of the fundamental solutions and robust convergence of the fixed-point iterations, even for droplets in close proximity to boundaries or each other.
• Efficiency: In a time-domain simulation with 50 droplets, the surrogate achieved an 80-fold speed-up over the reference CutFEM code while maintaining fair accuracy. A "lowest-fidelity" passive-tracer version achieved a speed-up of $3 \times 10^5$ with an order-of-magnitude larger error.
• Scaling: For static problems with up to $10^4$ droplets, the surrogate training and evaluation costs scale favorably. While the training exhibits quadratic scaling ($O(M^2)$) due to pairwise interactions, the evaluation cost is linear ($O(M)$) when the number of evaluation points is moderate. The method remains competitive with CutFEM for $M < 5000$ when including evaluation costs, and significantly outperforms CutFEM when only a limited number of evaluations (e.g., for marker advection in time-domain simulations) are required.
• Robustness: The iterative scheme demonstrated robustness even in non-physical configurations with overlapping droplets, converging where standard methods might fail or require excessive mesh refinement.

Significance and Claims
The authors claim that this hierarchical surrogate modeling framework offers a practical pathway to simulate complex multiphase Stokes flows on standard workstations, a task that typically requires supercomputing resources for high-fidelity methods. The method is particularly significant for scenarios involving many droplets and small interfacial separations, where standard discretization techniques face severe computational bottlenecks. The paper emphasizes that the approach captures droplet-flow, droplet-boundary, and droplet-droplet interactions systematically. While the current implementation assumes circular droplets, the authors note that the framework is sufficiently rich to motivate future extensions to deformable droplets and potentially Navier-Stokes flows. The work demonstrates that by combining physical insights (error PDEs) with efficient numerical linear algebra (subspace approximations), one can achieve a balance of accuracy and computational tractability previously unattainable for large-scale droplet systems.