This proposal is awarded using funds made available by the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
The mathematical problems arising from poromechanical models are challenging. Even linear models are mathematically and computationally demanding because they require the simultaneous solution of the equations of elasticity and those of fluid mechanics. More complex models may also account for electromagnetic, chemical, or thermal effects, in which case Maxwells equations, equations describing chemical reactions, or an energy equation must also be solved simultaneously. Difficulties arise from the fact that equations for common models have some degree of degeneracy and more complex models may involve nonlinear equations, multiple spatial scales, complicated boundary conditions, and nonlinear interactions. The investigator and his students will study poromechanical models analytically (existence and uniqueness), develop and rigorously analyze finite element based methods for approximating solutions of various model problems in poromechanics, and derive a-priori and a-posteriori error estimates. They will advance the underlying mathematical theory and the science of computer simulation of large-scale, complex, coupled, multi-scale phenomena.
Poromechanics is the science of energy, motion, and forces and their effect on porous material and in particular the swelling and shrinking of fluid-saturated porous media. Modeling and predicting the mechanical (or the electro-chemo-thermo-mechanical) behavior of fluid-infiltrated porous media is of great importance since many natural substances, for example, rocks, soils, clays, shales, biological tissues, and bones, as well as man-made materials such as foams, gels, concrete, water-solute drug carriers, and ceramics are all elastic porous media. The studies conducted by the investigator and his students have applications in a variety of unrelated fields, from geomechanics (ground failure of water-saturated sediments, electroseismic prospecting for oil and natural gas, or underground storage of hazardous waste) to pharmacology, material science, and biomechanics (in particular, the study of bones, corneal swelling, hydrated tissues, and intervertebral discs). Students involved in the project will be trained in the mathematical modeling and computer simulation of large-scale, complex, coupled, multi-scale phenomena.
