A mixed finite element framework for Biot's consolidation model and its interface problems

本项目旨在开发和分析有效且稳健的线性孔隙弹性数值方法。孔隙弹性理论解决了多孔材料变形与内部流体流动之间的时间依赖性耦合问题。流体饱和多孔介质的力学行为建模在许多科学和工程领域都非常重要，包括水库工程、土壤力学、环境工程、材料科学，以及最近的生物力学工程。由于孔隙弹性控制方程的复杂性，很少找到解析解；因此，数值模拟在孔隙弹性模拟中起着重要的作用。PI解决了线性孔隙弹性数值方法中的几个重要问题；(i)锁定效应，（ii）材料特性的非均质性，以及（iii）可变形多孔介质与其他自由流体或机械系统的相互作用。众所周知，标准伽辽金有限元方法会产生流体压力的不稳定和振荡数值行为，这被称为孔隙弹性锁定。克服孔隙弹性中的锁紧效应一直是广泛研究的课题。孔隙弹性数值模拟的另一个挑战是，当多孔材料中存在非均质性或孔隙弹性系统与其他流动系统或力学系统相互作用时，如何有效地处理界面条件。本项目的主要特点是为每个流动和力学问题建立一个基于耦合两种混合有限元方法的混合有限元框架，以便它们能够有效地处理上述问题。提出了各种混合有限元方法，并推导了先验误差估计。该项目的另一个重要方面是开发各种流动和力学问题的耦合技术。研究了几种算子分裂方案，并分析了它们的稳定性和收敛性。本课题开发的数值方法在若干基准问题中得到了实现和应用，验证了其准确性和有效性。因此，本研究活动提高了预测计算能力和对孔隙弹性材料在各种情况下的流动和输运过程的理解。研究孔隙弹性以及孔隙弹性系统与自由流体或其他力学系统相互作用的有效、可靠的数值方法具有跨学科的意义。例如，国家能源和自然资源的开发和管理，涡轮叶片和喷墨打印的工业加工，废水处理，动脉壁等软生物组织的建模，以及使用多层板的隔音声包的开发，都严重依赖于预测计算模拟。因此，该项目对计算数学界和工程界都大有裨益。本项目培养博士生在科学计算和有限元方法数学分析方面的知识和实践技能。

This project aims to develop and analyze efficient and robust numerical methods for linear poroelasticity. The theory of poroelasticity addresses the time-dependent coupling between the deformation of porous materials and the fluid flow inside. Modeling the mechanical behavior of fluid-saturated porous media is of great importance in a wide range of science and engineering fields, including reservoir engineering, soil mechanics, environmental engineering, material science, and, more recently, biomechanical engineering. Due to the complicated nature of the governing equations for poroelasticity, analytical solutions have rarely been found; therefore, numerical simulations have played an important role in poroelastic modeling. The PI addresses several important issues in numerical methods for linear poroelasticity; (i) locking effects, (ii) heterogeneity in material properties, and (iii) interaction of a deformable porous medium with other free fluid or mechanical systems. It has been well-known that standard Galerkin finite element methods produce unstable and oscillatory numerical behavior of the fluid pressure, which is known as locking in poroelasticity. Overcoming locking effects in poroelasticity has been a subject of extensive research. Another challenge in numerical modeling of poroelasticity is the effective treatment of interface conditions when heterogeneity is present in porous materials or the poroelastic system is interacting with other flow systems or mechanical systems. The main feature of this proposed project is to develop a mixed finite element framework based on coupling two mixed finite element methods for each of the flow and mechanics problems so that they can efficiently handle the issues addressed above. Various mixed finite element methods are developed and a-priori error estimates are derived. Another important aspect of this project is to develop various coupling techniques for the flow and mechanics problems. The PI investigates several operator-splitting schemes, and analyzes their stability and convergence. The numerical methods developed in this project are implemented and applied to several benchmark problems to demonstrate their accuracy and efficiency. Therefore, this research activity enhances the predictive computational capabilities and understanding of the flow and transport processes in poroelastic materials in various situations.Developing efficient and robust numerical methods for poroelasticity and the interaction of a poroelastic system with a free fluid or with other mechanical systems has cross-disciplinary implications. For example, the development and management of the nation's energy and natural resources, industrial processing of turbine blades and inkjet printing, waste water treatment, modeling of soft biological tissues such as arterial walls, and development of sound packages for acoustic insulation using multilayered panels heavily rely on predictive computational simulations. Therefore, this project greatly benefits both the computational mathematics and engineering communities. A Ph.D. student is trained to gain knowledge and practical skills in scientific computing and the mathematical analysis of finite element methods in this project.