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、自然的、工程的和社会的系统具有重要意义。该项目旨在通过开发和演示一种低成本、均匀和参数稳健的方案，通过压力稳健有限元方法来实现质量守恒的流体流动。所提出的研究为实际应用在模拟的准确性、效率、有效性、稳健性、灵活性和可靠性方面提供了显著的进步。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，特别强调支持女性和代表性不足的少数民族学生。这些技术将在开放源码软件包中实施，并将向科学界提供。还期望对研究生进行有关该项目主题的培训。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，从而保证了基本质量守恒。主要内容包括：1)设计了一种适用的方案，打破了传统求解器的局限性，实现了压力稳健性和质量守恒性。2)开发了一种计算量最小的统一格式，能够处理混合区域中的各种物理参数。3)。设计一种灵活且负担得起的数值格式，可以处理复杂几何形状的问题，提供高度的网格灵活性。4)。研究了对物理参数和离散化参数都具有鲁棒性的离散线性系统的稳健有效的线性求解器。对于项目的所有方面，将进行严格的数值分析，以证明和验证稳定性、收敛和稳健性。将进行相应的数值实验以进行进一步的验证。这一奖项反映了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。

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.