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

Robust, broadly applicable fluid-structure interaction (FSI) algorithms remain a challenge for computational mechanics. Efforts in this area are driven by the need to enhance predictive accuracy and efficiency in FSI simulations, align with experimental observations, and unravel complex multiscale and multiphysics phenomena, while addressing challenges in developing more robust and efficient methodologies. 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 enables it to model complex three-dimensional geometries in diverse engineering applications. Although the IIM we previously introduced offers many advantages compared to other FSI algorithms, it also imposes a restrictive mesh factor ratio, requiring the surface mesh to be coarser than the background fluid grid to ensure stability. This is because if the mesh factor ratio constraint is not satisfied, parts of the structure motion are not controlled by the discrete FSI coupling operators. 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 interpolation operator inspired by Tikhonov regularization. The effectiveness of the stabilization scheme is evaluated using benchmark problems with stationary interfaces and FSI models involving both rigid-body dynamics and elastodynamic structures models. This study demonstrates that using a stabilized velocity interpolation operator in the IIM enables a broader range of structure-to-fluid grid-size ratios without compromising accuracy or altering the flow dynamics. Our approach significantly broadens the applicability of the method to real-world FSI problems involving complex geometries and dynamic conditions, offering a robust and practical solution.