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

University Of Texas At El Paso, El Paso TX

Investigators

Abstract

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.