Improving the robustness of the immersed interface method through regularized velocity reconstruction

TL;DR

This work addresses the stability bottleneck of the immersed interface method (IIM) for fluid–structure interaction by introducing a stabilization strategy based on $L^2$-Tikhonov regularization of the interfacial velocity interpolation. The stabilized velocity operator $oldsymbol{J}_h^{oldsymbol{ε}}=oldsymbol{P}_h^{oldsymbol{ε}}oldsymbol{J}_h$ regularizes motions along the interface to prevent null spaces in the force-spreading operator when the mesh-factor ratio $M_{ ext{fac}}=h_L/h_E$ falls below unity. Across 2D and 3D test problems—including flow past cylinders and spheres, vortex-induced vibration, shear flow, and a soft disk in a lid-driven cavity—the method achieves accuracy comparable to the original IIM while enabling $0.05 < M_{ ext{fac}} < 1$, significantly broadening practical applicability to complex geometries and multiscale features. The results illustrate robust coupling, stability, and preserved flow dynamics under varied grid sizing, with guidance on selecting the stabilization parameter $oldsymbol{ε}$ to balance stability and fidelity. This advancement offers a computationally efficient path to accurate FSI simulations in engineering and biomedical contexts where multiscale interfaces are common.

Abstract

Robust, broadly applicable fluid-structure interaction (FSI) algorithms remain a challenge for computational mechanics. In previous work, we introduced an immersed interface method (IIM) for discrete surfaces and an extension based on an immersed Lagrangia-Eulerian (ILE) coupling strategy for modeling FSI involving complex geometries. The ability of the method to sharply resolve stress discontinuities induced by singular immersed boundary forces in the presence of low-regularity geometrical representations makes it a compelling choice for three-dimensional modeling of complex geometries in diverse engineering applications. Although the IIM we previously introduced offers many desirable advantages, it also imposes a restrictive mesh factor ratio, requiring the surface mesh to be coarser than the fluid grid to ensure stability. This is because when the mesh factor ratio constraint is not satisfied, parts of the structure motion are not controlled by the discrete FSI system. This constraint can significantly increase computational costs, particularly in applications involving multiscale geometries with highly localized complexity or fine-scale features. To address this limitation, we devise a stabilization strategy for the velocity restriction operator inspired by Tikhonov regularization. This study demonstrates that using a stabilized velocity restriction operator in IIM enables a broader range of structure-to-fluid grid-size ratios without compromising accuracy or altering the flow dynamics. This advancement significantly broadens the applicability of the method to real-world FSI problems involving complex geometries and dynamic conditions, offering a robust and practical solution.

Figures (18)

• Figure 1: Fluid domains and interface in $\mathbb{R}^2$.
• Figure 2: Illustration of a finite difference scheme for the Navier-Stokes equations and discrete interface with local $M_{\text{fac}}$ value approximately in the range $0.1 \leq M_{\text{fac}} \leq 0.25$. $\{e_i\}$ denote surface elements. Elements $e_4,e_6,e_7,e_{8},e_{10},e_{12},e_{13}$ and $e_{15}$ are not cut by any finite difference stencils.
• Figure 3: Drag coefficient $C_{\text{D}}$ and lift coefficients $C_{\text{L}}$ over time for two-dimensional flow past a cylinder with $M_{\text{fac}}=0.05,0.1,0.25,$ and $2$, generated by the unmodified IIM KOLAHDOUZ2020108854. Simulations with $M_{\text{fac}} < 1$ stopped because of excessive spurious interfacial motions.
• Figure 4: Drag coefficient $C_{\text{D}}$ and lift coefficients $C_{\text{L}}$ over time for two-dimensional flow past a cylinder with $M_{\text{fac}}=0.05,0.1,0.25$, and $2$, generated by the stabilized IIM with stability coefficient $\epsilon=116.5$.
• Figure 5: Drag coefficient $C_{\text{D}}$ and lift coefficients $C_{\text{L}}$ over time for two-dimensional flow past a cylinder with non-uniform interface discretization with $M_{\text{fac}}=0.05-0.27$ generated by the stabilized IIM with stability coefficient $\epsilon=116.5$.