The Diffuse Interface Method and Applications to Coupled Systems in Fluid Dynamics

涉及移动域的系统，其中流体与邻近区域相互作用，通常使用具有跨公共界面的耦合条件的偏微分方程来建模。这样的系统出现在医学（肝脏灌注、淋巴循环、心脏瓣膜的关闭和打开）、地质力学（裂缝扩展、地表水和地下水流动之间的耦合）和其他应用中。这类系统的数值模拟是基于对控制方程的离散近似。为了准确地描述动力学，通常使用界面跟踪方法，其中计算网格的节点与界面的参数表示对齐。然而，当区域变形较大时，界面跟踪方法很快变得难以应用。为了防止数值失败，当网格元素高度倾斜时，需要计算昂贵的网格再生或类似的技术。扩散界面法是基于固定网格法的一种替代策略。对于这种方法，使用相场函数将模型从一个区域的零平滑过渡到另一个区域的一。计算网格节点不必与接口对齐，其位置现在使用相场函数捕获。即使当域不随时间变化，或者在两个区域之间的接口难以精确确定的情况下，或者当接口的几何形状很复杂时，这种方法也很有用。然而，漫射界面法在界面处引入了额外的误差，需要仔细控制。本课题旨在为扩散界面法在流体力学中的应用奠定数学基础。在这项工作中开发的技术有望适用于其他涉及流体和孔隙弹性和/或弹性结构的耦合系统。该项目包括通过参与研究来培训研究生。本项目的重点是发展应用于流体动力学耦合系统的扩散界面方法的数学理论和数值方法。这将通过研究一系列耦合流动模型来实现，包括流体-多孔介质相互作用、流体-孔隙-弹性结构相互作用和流体-弹性结构相互作用。对于每个模型，工作需要证明底层扩散界面问题的适定性，显示扩散界面模型在界面层宽度趋于零时收敛到相应的锐界面模型，并计算收敛速度，包括建模误差和基于有限元方法的离散解的近似误差。分析将使用加权Sobolev空间进行。将为每个模型开发和实施数值方法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Systems involving moving domains in which a fluid interacts with a neighboring region are often modeled using partial differential equations with coupling conditions that hold across a common interface. Such systems occur in medicine (liver perfusion, lymphatic circulation, the closing and opening of heart valves), geomechanics (fracture propagation, coupling between the surface and groundwater flows), and other applications. Numerical simulations of such systems are based on discrete approximations to the governing equations. To accurately describe the dynamics, interface tracking methods, in which the nodes of the computational mesh are aligned with the parametric representation of the interface, are often used. However, interface-tracking methods rapidly become difficult to apply when domain deformations are large. To prevent numerical failures, computationally expensive mesh regenerations or similar techniques are necessary when mesh elements become highly skewed. The diffuse interface method is an alternative strategy based on a fixed mesh approach. For this method, the model is reformulated using a phase-field function that smoothly transitions from zero in one region to one in the other region. The computational mesh nodes do not have to be aligned with the interface, whose location is now captured using the phase-field function. This approach is useful even when the domain does not change in time, or in cases where the interface between the two regions is difficult to determine exactly, or when the geometry of the interface is complex. However, the diffuse interface method introduces an additional error at the interface, which needs to be carefully controlled. This project aims to establish mathematical foundations for application of the diffuse interface method in fluid dynamics. The techniques developed in this work are expected to be applicable to other coupled systems involving fluids and poroelastic and/or elastic structures as well. The project includes training of graduate students through involvement in the research.This project focuses on the development of mathematical theory and numerical methods for the diffuse interface method applied to coupled systems in fluid dynamics. This will be achieved by studying a hierarchy of coupled flow models, including the fluid-porous medium interaction, fluid-poroelastic structure interaction, and fluid-elastic structure interaction. For each model, the work entails proving the well-posedness of the underlying diffuse interface problem, showing the convergence of the diffuse interface model to the corresponding sharp interface model as the width of the interfacial layer goes to zero, and calculating the rate of convergence including the modeling error and the approximation error of the discrete solution based on the finite element method. The analysis will be performed using weighted Sobolev spaces. Numerical methods will be developed and implemented for each model.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.