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

本研究项目的总体目标是基于混合维模型，开发、分析和实现稳定的数值方法来模拟裂缝性多孔介质中的流动。裂缝性多孔介质流动是多物理场和多尺度的，为此类系统开发强大有效的数值工具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，并且通过封装它们的三维域以复杂的方式相互作用。裂缝性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式，促进混合维建模在这些及相关领域的使用，并缓解当前计算机模拟的局限性。这些技术将在一个开源软件包中实现，并将提供给科学界。在这个项目中，将研究混合维度建模和仿真的三个重要方面。第一个研究目标是设计先进的稳定离散化方法，将线性弹性的稳定方案与混合维环境下达西流动的保持结构的离散化方法结合起来。稳定性和质量守恒可以用最少的自由度来实现。第二个目标是混合维框架中的自适应逼近。开发和实施时空自适应技术，在坚实的理论基础上提高精度。第三个目标是研究得到的离散线性系统的鲁棒线性解。基于混合维模型的物理和数学性质及其数值离散化，将开发新的块预调节器和单片多网格方法。关于物理和离散参数的鲁棒性将在理论上和数值上得到证明。该项目的主要应用是裂缝性多孔介质中的线性孔隙力学，但也将研究非线性公式的可能推广，包括物理和工程中的一些应用。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 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.