Fluid–structure interaction with contact poses profound mathematical and numerical challenges, particularly when considering realistic contact scenarios and the influence of surface roughness. Computationally, contact introduces challenges in altering the fluid domain topology and preserving stress balance. This work introduces a new mathematical framework for a unified continuum description of fluid-porous-structure-contact interaction (FPSCI), leveraging the Navier–Stokes–Brinkman (NSB) equations to incorporate porous effects within the surface asperities in the contact region. Our approach maintains mechanical consistency during contact, circumventing issues associated with contact models and complex interface coupling conditions, allowing for the modeling of tangential creeping flows due to surface roughness. The unified continuum and variational multiscale formulation ensure robustness by enabling stable and unified integration of fluid, porous, and solid sub-problems. Computational efficiency and ease of implementation – key advantages of our approach – are demonstrated by solving two benchmark problems of a falling ball and an idealized heart valve. This research has broad implications for fields reliant on accurate fluid–structure interactions and promising advancements in modeling and numerical simulation techniques.

中文翻译：

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流体-多孔结构-接触相互作用的机械一致的统一公式

接触的流固相互作用带来了深刻的数学和数值挑战，特别是在考虑现实接触场景和表面粗糙度的影响时。在计算上，接触给改变流体域拓扑和保持应力平衡带来了挑战。这项工作引入了一种新的数学框架，用于流体-多孔结构-接触相互作用（FPSSCI）的统一连续描述，利用纳维-斯托克斯-布林克曼（NSB）方程将多孔效应纳入接触区域的表面粗糙度内。我们的方法在接触过程中保持机械一致性，避免了与接触模型和复杂界面耦合条件相关的问题，从而可以对由于表面粗糙度而导致的切向蠕变流进行建模。统一的连续体和变分多尺度公式通过实现流体、多孔和固体子问题的稳定和统一集成来确保鲁棒性。通过解决落球和理想化心脏瓣膜这两个基准问题，证明了计算效率和易于实施（我们方法的主要优势）。这项研究对于依赖精确的流体-结构相互作用以及建模和数值模拟技术的有希望的进步的领域具有广泛的影响。