Mathematical and Computational Studies in Poroelasticity

PI将研究描述流体饱和、弹性、多孔介质（弹性固体）的膨胀和收缩以及流体与弹性多孔结构之间相互作用的数学模型。这项研究的意义在于，许多天然物质，如岩石、土壤，甚至生物组织，以及人造材料，如泡沫、凝胶、混凝土和陶瓷，都是这种弹性多孔介质。因此，研究成果将适用于众多领域（生物力学，药理学，能源技术，地质力学，地球物理学和材料科学）。所开发的模型可用于描述各种人造材料和制造过程。它们还可以用来模拟软组织和生物多孔介质的生物力学，使新疗法的建模和优化，开发新的诊断工具，并推进我们对人体生理学的理解。这项研究还将提高我们模拟、分析和评估水力压裂、废水注入、碳捕获和储存以及与地热能生产相关的活动对环境影响的能力。研究生和本科生将接受培训并参与工作。研究成果和经验将被纳入研究生和本科课程。本研究的重点是在连续介质模型和电孔隙弹性模型中出现的数学和计算问题。这些提供了在科学和工程的不同领域和各种应用中出现的各种多孔材料和工艺的统一和系统的处理。其他物理现象，如介质的机电响应，以及化学和/或热效应，也可能被考虑在内。因此，孔隙弹性、电孔隙弹性甚至电化学-热孔隙弹性都是复杂耦合的、多物理场、多尺度的现象，其中弹性或粘弹性变形多孔介质的膨胀和收缩与介质和饱和流体的机电（和/或热、化学）响应耦合。孔隙弹性也涉及多个尺度；微观尺度对应于分子尺度，连续介质力学的尺度对应于宏观尺度。PI将开发数学和计算工具，并分析和计算地研究孔隙弹性中出现的各种数学问题。其中考虑的问题将是数学偏微分模型的良好定态性，以及推导和分析用于近似这些偏微分方程解的有效和精确的数值算法。

The PI will study mathematical models which describe the swelling and shrinking of fluid-saturated, elastic, porous media (elastic solids) and the interactions between the fluid and the elastic porous structures. The significance of the research is due to the fact that many natural substances, e.g., rocks, soils, and even biological tissues, as well as man-made materials such as foams, gels, concrete, and ceramics are such elastic porous media. Thus, research findings will be applicable in a multitude of areas (biomechanics, pharmacology, energy technology, geomechanics, geophysics, and materials science). The models developed can be used to describe various man-made materials and manufacturing processes. They can also be used to model the biomechanics of soft tissues and biological porous media, enabling the modeling and optimization of new therapies, development of new diagnostics tools, and advancing our understanding of human physiology. This research will also improve our capability to model, analyze, and assess the environmental impact of processes associated with hydraulic fracturing, wastewater injection, and carbon capture and storage, and activities associated with production of geothermal energy. Graduate and undergraduate students will be trained and will participate in the work. Research findings, and experience from it, will be incorporated into the graduate and undergraduate curricula.This research focuses on mathematical and computational issues arising in continuum models of poroelasticity and electroporoelasticity. These provide a unified and systematic treatment of various porous materials and processes which arise in diverse areas of science and engineering and a variety of applications. Additional physical phenomena, such as the electromechanical response of the medium, as well as chemical and/or thermal effects, may also be accounted for. Thus, poroelasticity, electroporoelasticity, or even electro-chemo-thermo-poroelasticity, are all complex coupled, multiphysics, multiscale, phenomena, where the swelling and shrinking of an elastic or viscoelastic deforming porous medium is coupled to the electromechanical (and/or the thermal, and chemical) response of the medium and saturating fluid. Poroelasticity also involves multiple scales; the micro-scale corresponding to the molecular scale, and the scale of continuum mechanics, the macro-scale. The PI will develop mathematical and computational tools and study, analytically and computationally, various mathematical problems arising in poroelasticity. Among the issues considered will be the well posedness of mathematical pde models and the derivation and analysis of efficient and accurate numerical algorithms for approximating solutions of these partial differential equations.