A monolithic localized high-order ALE finite element method for multi-scale fluid-structure interaction problems

This paper presents MLH-ALE, a monolithic localized high-order arbitrary Lagrangian-Eulerian finite element method for multi-scale fluid-structure interaction (FSI). The framework employs isoparametric $\mathcal{P}_2$ elements for geometric fidelity and an implicit-explicit partitioned Runge-Kutta (IMEX-PRK) scheme for temporal discretization. To address scale disparity, a localized updating strategy is integrated to focus computational resolution on the moving structure. Numerical benchmarks confirm the optimal high-order convergence of the underlying ALE scheme. Furthermore, simulations of particle focusing in spiral microchannels demonstrate that the MLH-ALE approach provides reliable numerical results in good agreement with experimental observations, confirming its feasibility for complex multi-scale applications.

💡 Research Summary

This paper introduces MLH‑ALE, a monolithic localized high‑order Arbitrary Lagrangian‑Eulerian (ALE) finite element method designed for multi‑scale fluid‑structure interaction (FSI) problems. The authors start by highlighting the limitations of existing FSI approaches—immersed boundary methods, diffuse‑interface techniques, and traditional ALE/DSD‑SST schemes—especially when the fluid domain and solid structures differ by several orders of magnitude, as in microfluidic devices. To overcome these challenges, MLH‑ALE combines several innovative components.

First, the governing equations consist of incompressible Navier‑Stokes for the fluid and a neo‑Hookean incompressible solid model expressed through the left Cauchy‑Green deformation tensor B. Interface conditions enforce velocity continuity and stress equilibrium, allowing the whole system to be written as a single coupled variational problem.

Second, spatial discretization uses isoparametric P₂/P₁ Taylor‑Hood elements for velocity and pressure, and P₁ elements for B. This choice guarantees second‑order geometric fidelity while preserving the optimal convergence properties of the mixed formulation.

Third, the ALE mapping A(x,t) is introduced to pull the moving fluid‑solid domains back to a fixed reference configuration. The mapping satisfies a harmonic extension problem for the mesh velocity w, ensuring that the mesh follows the solid motion without distortion in the fluid region. Jacobian J and deformation gradient F are computed analytically, which enables a rigorous transformation of the governing equations onto the stationary domain.

Fourth, temporal integration employs an Implicit‑Explicit Partitioned Runge‑Kutta (IMEX‑PRK) scheme. The stiff, coupled fluid‑structure terms are treated implicitly, while convective and diffusive terms are advanced explicitly. This partitioned approach yields high‑order temporal accuracy (third order in the presented tests) without sacrificing stability for the strongly coupled system.

The most distinctive feature of MLH‑ALE is the localized mesh‑updating algorithm. Instead of remeshing the entire domain at each time step, the algorithm identifies a narrow band surrounding the moving structure (or particles) and performs mesh refinement and deformation only within this band. The rest of the mesh remains unchanged, dramatically reducing the number of degrees of freedom and preserving mesh quality for long‑range particle migration.

Numerical experiments validate the method. Convergence studies on manufactured solutions in both two‑ and three‑dimensional settings confirm optimal spatial (order 2) and temporal (order 3) convergence rates. A second set of benchmarks simulates particle focusing in a spiral microchannel, a canonical multi‑scale microfluidic problem. The simulated particle trajectories, focusing positions, and shear‑induced lift forces match experimental measurements closely, even when particle diameters are two orders of magnitude smaller than the channel width.

The authors conclude that MLH‑ALE successfully integrates high‑order geometric accuracy, efficient localized mesh handling, and robust high‑order time integration, making it a versatile tool for complex multi‑scale FSI applications. Future work will extend the framework to non‑Newtonian fluids, more sophisticated solid constitutive models, and GPU‑accelerated parallel implementations to tackle industrial‑scale problems.