Developing Efficient and Accessible Computational Tools for Poroelasticity Problems

孔隙弹性问题源于广泛的现实世界问题，其中流体流过多孔介质，多孔介质会因流体压力而变形。 例如，对流增强的药物输送到肿瘤部位、膝关节半月板的功能和二氧化碳封存都涉及固体（位移）和流体（压力）之间的这种相互作用。 数学建模和计算机模拟将为增强我们理解、预测和控制这些复杂物理过程的能力提供有价值的工具。 该研究项目旨在开发一系列新颖、准确、高效且稳健的数值方法来解决孔隙弹性问题。 该项目的重点是针对孔隙弹性问题的一系列新颖的有限元方法的设计、分析、实施和应用。 基于位移-压力双场公式，将开发弱伽辽金有限元方法来近似固体位移和流体压力。 将研究这些子问题求解器稳定耦合的条件。 新方法将应用于膝关节半月板的工程问题。这些新方法还将作为 Matlab 和 C 代码模块（在 deal.II 平台上）实现，可供科学计算社区公开访问。该项目开发的高效且强大的计算工具可应用于许多涉及孔隙弹性的现实问题，例如将药物输送到肿瘤部位、食品加工、储层工程和组织工程。 这些将对更好地利用自然资源或治疗疾病产生进一步的经济影响。 该项目开发的方法可应用于处理多个物理过程的一大类科学计算任务。 该项目还将为研究生和本科生提供实践培训机会。该奖项反映了 NSF 的法定使命，并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，被认为值得支持。

Poroelasticity problems arise from a broad range of real world problems, in which fluid flows through a porous medium that can deform due to the fluid pressure. For example, convection-enhanced drug delivery to tumor sites, functioning of knee meniscus, and CO2 sequestration all involve this type of interaction between solid (displacement) and fluid (pressure). Mathematical modeling and computer simulations will provide valuable tools to enhance our ability in understanding, predicting, and controlling these complicated physical processes. This research project aims at developing a family of novel accurate, efficient, and robust numerical methods for poroelasticity problems. The focus of this project is on design, analysis, implementation, and applications of a family of novel finite element methods for poroelasticity problems. Based on the displacement-pressure two-field formulation, weak Galerkin finite element methods will be developed for approximating both solid displacement and fluid pressure. Conditions for stable coupling of these sub-problem solvers will be investigated. The new methods will be applied to an engineering problem on knee meniscus. These new methods will also be implemented as Matlab and C++ code modules (on deal.II platform) that are openly accessible to the scientific computing community. The efficient and robust computational tools developed in this project can be applied to many real world problems that involve poroelasticity such as drug delivery to tumor sites, food processing, reservoir engineering, and tissue engineering. These will have further economic impact on better use of natural resources or cures for diseases. The methodology developed in this project can be applied to a large class of scientific computing tasks that deal with multiple physical processes. This project will also provide hands-on training opportunities for both graduate and undergraduate students.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.