Collaborative Research: Adaptive Mixed-Dimensional Modeling and Simulation of Porous Media

协作研究：多孔介质的自适应混合维建模与仿真

• 批准号: 2208267
• 负责人: Xiaozhe Hu
• 金额: $ 42.09万
• 依托单位: Tufts University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2208267&HistoricalAwards=false
• 关键词: Collaborative; Research; Adaptive; Mixed; Dimensional

The overall goal of this research project is to develop, analyze, and implement stable numerical methods for the simulation of flow in fractured porous media, based on mixed-dimensional modeling. Fractured porous media flow is multi-physics and multi-scale, and the development of robust and effective numerical tools for such systems represents a class of important challenges in computational mathematics. For example, in many practical applications fractures or other features, such as capillaries in the brain, are lower-dimensional and interact in a complex way through the three-dimensional domains encapsulating them. Important applications of fractured porous media include hydraulic fracturing, waste deposition, and models from biomechanics. The project aims to provide new computational paradigms, promote the usage of mixed-dimensional modeling in these and related fields, and alleviate current limitations in computer simulations. The techniques will be implemented in an open-source software package and will be made available to the scientific community. In this project, three important aspects of mixed-dimensional modeling and simulation will be investigated. The first research objective is to design advanced stable discretizations, which couple stabilized schemes for linear elasticity with structure-preserving discretizations for Darcy flow in a mixed-dimensional setting. Stability and mass conservation will be achieved using a minimal number of degrees of freedom. The second objective is adaptive approximations in the mixed-dimensional framework. Space-time adaptivity will be developed and implemented to improve accuracy with solid theoretical foundations. The third objective is to study robust linear solvers for the resulting discrete linear systems. Based on physical and mathematical properties of the mixed-dimensional models and their numerical discretizations, new block preconditioners and monolithic multigrid methods will be developed. Robustness with respect to physical and discretization parameters will be justified both theoretically and numerically. The main application of this project is linear poromechanics in fractured porous media, but possible generalizations to nonlinear formulations will also be investigated, including several applications in physics and engineering.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.

该研究项目的总体目标是开发，分析和实施稳定的数值方法，用于模拟裂缝性多孔介质中的流动，基于混合维建模。裂缝性多孔介质流动是多物理场和多尺度的，并且针对此类系统的鲁棒且有效的数值工具的发展代表了计算数学中的一类重要挑战。例如，在许多实际应用中，骨折或其他特征，如大脑中的毛细血管，是低维的，并通过封装它们的三维域以复杂的方式相互作用。裂缝多孔介质的重要应用包括水力压裂、废物沉积和生物力学模型。该项目旨在提供新的计算范例，促进混合维建模在这些领域和相关领域的使用，并缓解当前计算机模拟的局限性。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。在这个项目中，混合维建模和仿真的三个重要方面将进行研究。第一个研究目标是设计先进的稳定的离散，耦合稳定计划的线性弹性与结构保持离散达西流在混合维设置。 稳定性和质量守恒将使用最小数量的自由度来实现。第二个目标是混合维框架中的自适应近似。 将开发和实施时空适应性，以提高具有坚实理论基础的准确性。第三个目标是研究所得到的离散线性系统的鲁棒线性求解器。 基于混合维模型的物理和数学性质及其数值离散，将开发新的块预处理器和整体多重网格方法。相对于物理和离散化参数的鲁棒性将在理论和数值上证明。该项目的主要应用是裂缝多孔介质中的线性孔隙力学，但也将研究可能的非线性公式的推广，包括物理和工程中的几个应用。该奖项反映了NSF的法定使命，并已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。

项目成果

Effective Preconditioners for Mixed‐Dimensional Scalar Elliptic Problems

混合维标量椭圆问题的有效预调节器

• DOI: 10.1029/2022wr032985
• 发表时间: 2023
• 期刊: Water Resources Research
• 影响因子: 5.4
• 作者: Hu, Xiaozhe;Keilegavlen, Eirik;Nordbotten, Jan M.
• 通讯作者: Nordbotten, Jan M.

Rational approximation preconditioners for multiphysics problems

多物理场问题的有理逼近预处理器

• 发表时间: 2023
• 期刊: Lecture notes in computer science
• 作者: Ana Budisa, Xiaozhe Hu