Developing Efficient and Accessible Computational Tools for Poroelasticity Problems

孔隙弹性问题产生于广泛的现实世界问题，在这些问题中，流体流经多孔介质，而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的隔离都涉及固体（位移）和液体（压力）之间的这种相互作用。数学建模和计算机模拟将提供有价值的工具来提高我们理解、预测和控制这些复杂的物理过程的能力。本研究项目旨在开发一套新颖、准确、高效、鲁棒的孔隙弹性问题数值计算方法。这个项目的重点是设计、分析、实现和应用一系列新的有限元方法来解决孔隙弹性问题。基于位移-压力双场公式，将发展出同时逼近固体位移和流体压力的弱伽辽金有限元方法。我们将研究这些子问题解的稳定耦合条件。新方法将应用于膝关节半月板的工程问题。这些新方法也将作为Matlab和c++代码模块实现（待定）。II平台)，对科学计算社区开放访问。该项目中开发的高效且强大的计算工具可以应用于许多涉及孔隙弹性的现实世界问题，如肿瘤部位的药物输送、食品加工、储层工程和组织工程。这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。在这个项目中开发的方法可以应用于处理多个物理过程的大类科学计算任务。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。

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.