Development of an Efficient, Parameter Uniform and Robust Fluid Solver in Porous Media with Complex Geometries

不可压缩流体模型广泛应用于工程和科学的各个领域，其数值解对于理解复杂的自然、工程和社会系统具有重要意义。该项目旨在通过开发和展示一种低成本，统一的，参数稳健的计划，通过压力稳健的有限元方法与质量守恒的流体流动，以推进最先进的技术。所提出的研究提供了显着的进步，在准确性，效率，有效性，鲁棒性，灵活性和可靠性的模拟实际应用。该项目将加强在应用数学、计算机科学、计算流体力学、计算物理和石油工程方面具有不同专业知识的研究人员之间的跨学科合作。本科生和研究生将通过课程开发，研究项目设计和学生培训接受跨学科教育，特别强调支持妇女和代表性不足的少数民族学生。 这些技术将在一个开放源码软件包中实施，并将提供给科学界。研究生的培训项目的主题也有望。提出的数学建模和计算方法解决关键的科学挑战，在流体模拟。新的数学公式的基础上形成的散度保持数值方案，从而保证基本的质量守恒。主要组成部分包括：1）设计一种适用的方案，打破传统求解器的局限性，同时实现压力鲁棒性和质量守恒。2)开发一个统一的计划，以最小的计算成本，能够处理不同的物理参数的混合制度。3）。 设计一种灵活且经济的数值方案，可以处理复杂几何形状的问题，提供高网格灵活性。4）。研究所得到的离散线性系统的鲁棒和有效的线性求解器，这是相对于物理和离散参数的鲁棒性。对于项目的各个方面，将进行严格的数值分析，以证明和验证稳定性，收敛性和鲁棒性。该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

The incompressible fluid model is widely used in various fields in engineering and science and their numerical solutions are of prominent importance in understanding complex, natural, engineered, and societal systems. This project aims to advance the state of the art by developing and demonstrating a low-cost, uniform, and parameter robust scheme for fluid flows with mass conservation via a pressure robust finite element method. The proposed research offers significant advancements in accuracy, efficiency, effectiveness, robustness, flexibility, and reliability of simulations for practical applications. This project will strengthen interdisciplinary collaborations among researchers who have different expertise in applied mathematics, computer science, computational fluid dynamics, computational physics, and petroleum engineering. Undergraduate and graduate students will receive interdisciplinary education through course development, research project design, and student training, with a special emphasis on supporting women and underrepresented minority students. The techniques will be implemented in an open-source software package and will be made available to the scientific community. Training of graduate students on the topics of the project is also expected.The proposed mathematical modeling and computational methods address key scientific challenges in the fluid simulation. Novel mathematical formulations are formed based on a divergence preserving numerical scheme and thus guarantee the fundamental mass conservation. The major components include: 1) Designing an applicable scheme that breaks the limitations of traditional solvers to achieve both pressure robustness and mass conservation. 2) Developing a uniform scheme with minimal computational cost, capable of handling varying physical parameters in the mixed regime. 3). Designing a flexible and affordable numerical scheme that can handle problems with complex geometries, providing high grid flexibility. 4). Investigating the robust and effective linear solvers for the resulting discrete linear systems, which is robust with respect to physical and discretization parameters. For all aspects of the project, rigorous numerical analysis will be carried out to prove and validate stability, convergence, and robustness. Corresponding numerical experiments will be performed for further validation.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.