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

TL;DR

The paper addresses multi-scale FSI where a small moving structure within a large domain challenges resolution and efficiency. It introduces a monolithic localized high-order ALE framework (MLH-ALE) that combines isoparametric $\mathcal{P}_2$ geometry with an IMEX-PRK time integrator and a local updating strategy to concentrate computation where needed. Key contributions include a sharp-interface ALE formulation, high-order spatial discretization via Taylor–Hood elements and isoparametric mapping, and a localized updating algorithm validated through 2D convergence tests, 3D free-fall sphere experiments, and spiral-channel microfluidics showing close agreement with experiments. The method offers a scalable, accurate tool for complex multi-scale FSI, enabling reliable particle-tracking and device optimization in aerospace, naval, and biomedical microfluidic contexts.

Abstract

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.

Figures (11)

• Figure 1: The figure illustrates the dynamic evolution of local mesh configurations across time steps, as well as the process of generating new meshes and interpolating field variables from the outdated mesh to the newly refined mesh when mesh quality metrics exceed predefined thresholds, thereby enabling continuous computational progression
• Figure 2: The second-ordered mesh fitting the boundary of the solid. The green elements represent the fluid phase while the yellow ones stand for solid phase. Notably, due to the application of high-order meshes, the elements at the fluid-solid interface adopt curved edges rather than straight edges. Unlike visualization representations, during finite element integration procedures, these curved edges are approximated using high-order polynomials, thereby better capturing curvature-induced geometric characteristics and achieving enhanced convergence rates in velocity fields.
• Figure 3: The domain set up for convergence study.
• Figure 4: Magnitude of the resulted velocity field of FSI simulation. The particle of large modulus could affect the velocity field near the pillars when the distance is small.
• Figure 5: The figure presents the numerically computed trajectory of the center of the rigid particle. The enlarged view shows the detail of the trajectories obtained with a mesh size of 4/100 and time-step sizes of 3/200 (red), 3/400 (blue), 3/800 (green), 3/1600 (brown) and 3/3200 (black) respectively. The particle positions at different time steps are marked by points along the curves. In particular, the point enclosed by the circle indicates the particle’s position at time t=0.375. For the convergence study, the particle position at this time is used to compute the numerical errors, where the solution obtained with the smallest time step ($dt = 3/3200$) is taken as the reference.

Theorems & Definitions (2)

• Remark 1
• Remark 2