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

Biot固结模型的混合有限元框架及其界面问题

• 批准号: 1217123
• 负责人: Son-Young Yi
• 金额: $ 26.36万
• 依托单位: University of Texas at El Paso
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1217123&HistoricalAwards=false
• 关键词: mixed; finite; element; framework; Biot

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.

本项目的目的是开发和分析高效和稳健的线性孔弹性数值方法。多孔弹性理论解决了多孔材料的变形和内部流体流动之间的时间依赖耦合。流体饱和多孔介质的力学行为建模在包括油藏工程、土力学、环境工程、材料科学以及最近的生物力学工程在内的广泛的科学和工程领域具有重要的意义。由于孔弹性控制方程的复杂性，很少能找到解析解，因此数值模拟在孔弹性建模中起着重要的作用。PI解决了线性多孔弹性数值方法中的几个重要问题：(I)锁定效应，(Ii)材料性质的非均质性，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，标准的Galerkin有限元方法会产生流体压力的不稳定和振荡的数值行为，这就是所谓的锁定孔隙弹性。克服孔道弹性中的锁定效应一直是一个广泛研究的主题。在孔隙弹性数值模拟中的另一个挑战是，当多孔材料中存在非均质性，或者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。该项目的主要特点是为流动和力学问题开发一种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。发展了各种混合有限元方法，并得到了先验误差估计。这个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。PI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。本项目所开发的数值方法被实现并应用于几个基准问题，以验证其精度和效率。因此，这一研究活动增强了对各种情况下多孔弹性材料中流动和输运过程的预测计算能力和理解。发展有效和稳健的数值方法来研究多孔弹性以及多孔弹性系统与自由流体或其他力学系统的相互作用具有跨学科的意义。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处理，废水处理，动脉壁等软生物组织的建模，以及使用多层面板开发隔音音箱在很大程度上依赖于预测计算模拟。因此，这个项目对计算数学和工程界都有很大的好处。在这个项目中，一名博士生被训练来获得科学计算和有限元方法的数学分析方面的知识和实践技能。