RUI: Computational Models for Coupled Free/Porous Media Flow

该项目将研究流体流动模型及其在人类健康和环境中的应用。重点放在结合了自由流体部分和多孔介质部分的模型上。例如，在生理学中，这种流体流动模型有助于了解氧气和营养物质在血管和组织之间的传输。在水文学中，这样的模型可以用来研究和预测地下储油罐泄漏或垃圾填埋场与地下水混合产生的有毒化学物质的污染。在工业过滤中，本项目中研究的流动模型可以整合到更复杂的模型中，以优化用于降低车辆排放水平的三效催化转化器的设计。由于系统的多物理性质，对这些自由/多孔耦合流动的有效计算机模拟仍然具有挑战性。本研究旨在为此类模型开发准确、高效的数值模拟技术。该项目通过本科生的研究经验提供培训。该项目使用自由域中的不可压缩Stokes方程和多孔域中的Darcy方程来研究耦合流动系统的模型。在系统中加入了一个对流扩散方程来模拟自由/多孔界面上的输运现象。这项研究旨在开发、分析和实现这一方程组的三维综合数值模型。主要目标是(I)解决由于自由和多孔子区域中的微分算子不兼容而导致的直接解的严重局限性，(Ii)开发健壮、高精度和条件良好的积分方程式，(Iii)对迭代方法进行严格的分析以推导出最优收敛，以及(Iv)增加组合不同的离散化方法以有效地模拟非均质多孔介质的能力。该项目将开发一种新的边界积分公式，具有正则化和高精度的校正，以及基于曲面隐式表示的简单求积。基于非局部算子的新传输条件下的高效区域分解方法将被用来加快迭代过程。将实施高性能计算技术和独立于内核的树码算法来提高计算速度。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project will study fluid flow models with applications in human health and the environment. The focus is on models that combine a free fluid part and a porous media part. For example, in physiology, such fluid flow models help to understand transport of oxygen and nutrients between blood vessels and tissue. In hydrology, such models can be used to study and predict contamination by toxic chemicals from leaky underground storage tanks or landfills mixing with groundwater. In industrial filtration, the flow models studied in this project can be incorporated into more complex models to optimize the design of the three-way catalytic converter used to reduce vehicle emission levels. Efficient computer simulation of these coupled free/porous flows remains challenging due to the multi-physics nature of the systems. This research aims to develop accurate and efficient numerical simulation techniques for such models. The project provides training through undergraduate research experiences.This project studies models of coupled flow systems using the incompressible Stokes equations in the free domain and the Darcy equations in the porous domain. An advection-diffusion equation is added to the system to model the transport phenomena at the free/porous interface. The research aims to develop, analyze, and implement an integrated numerical model of this system of equations in three dimensions. The main goals are to (i) address severe limitations of the direct solution, due to incompatibilities in the differential operators in the free and porous subdomains, (ii) develop robust, high accuracy, and well-conditioned integral equation formulations, (iii) apply rigorous analysis on the iterative methods to deduce optimal convergence, and (iv) add the ability to combine different methods of discretization to effectively model heterogeneous porous media. The project will develop a new boundary integral formulation, with regularization and correction for high accuracy, and a simple quadrature based on implicit representation of the surface. Efficient domain decomposition methods with new transmission conditions based on non-local operators will be used to speed up the iteration process. High-performance computing techniques and a kernel-independent treecode algorithm will be implemented to increase the speed of computations.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.