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。该项目旨在通过压力鲁棒有限元法开发和展示低成本、均匀、参数鲁棒的流体流动守恒方案，从而提高技术水平。所提出的研究在准确性、效率、有效性、鲁棒性、灵活性和可靠性方面为实际应用提供了显著的进步。该项目将加强应用数学、计算机科学、计算流体动力学、计算物理和石油工程等不同专业领域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，特别强调对妇女和代表性不足的少数民族学生的支持。这些技术将在一个开源软件包中实现，并将提供给科学界。对研究生的专题训练也是期望的。提出的数学建模和计算方法解决了流体模拟中的关键科学挑战。基于保持散度的数值格式形成了新的数学公式，从而保证了基本的质量守恒。主要内容包括：1)设计一种适用的方案，打破传统求解器的局限性，实现压力鲁棒性和质量守恒。2)以最小的计算成本开发一种统一的方案，能够处理混合状态下不同的物理参数。3)。设计一种灵活且价格合理的数值方案，可以处理复杂几何问题，提供高网格灵活性。4）. 研究得到的离散线性系统的鲁棒和有效的线性求解器，该系统对物理参数和离散参数都具有鲁棒性。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。将进行相应的数值实验以进一步验证。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。

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.