The investigator works on three projects: 1) Mechanical properties of composites and polycrystals. New reliable analytical and numerical methods are developed to predict the strength of a wide range of heterogeneous solids. In addition, detailed results on idealized models are obtained. These serve to gain understanding of real materials and to test existing methods in regimes where their accuracy is uncertain. 2) Solute transport in porous media. New efficient mathematical models are developed and studied. Because this project is relevant to the understanding of the transport of contaminants in soils and the transport of nutrients in bones, the range of potential applications of the results of this project is wide. 3) Small particles migration and clogging. Laboratory experiments that capture the physical mechanisms involved in particles migration and clogging are studied and mathematically modeled. The goal is to improve the current understanding of the physical phenomena that lead to clogging, to provide guidelines for improved methodologies to pump petroleum and water. The phenomena to be studied are also relevant in other contexts including sand filters in water treatment, food grain, pharmaceuticals, bio-mechanical systems (lungs, kidneys, etc.) and filters. The projects are of related nature. In all cases, the macroscopic physical phenomena depend on the microgeometry of the materials. Accordingly, a key component to their study is to identify the relevant different length and time scales to formally obtain or construct the mathematical models.
Guidelines that mathematical analysis can provide in the understanding of properties of heterogeneous materials is of great benefit to society. The current availability of computational power and the maturity that applied mathematics has reached make mathematical modeling and numerical simulation powerful tools. The investigator believes that important discoveries are likely to result from interdisciplinary work; accordingly, he collaborates and interacts with researchers in other disciplines. Potential applications of the project include new guidelines on the disposal of contaminants, the design of improved methodologies to treat osteoporosis and related diseases, a better understanding of the processes that lead to loss of bone experienced in the absence of gravity (during space flights), and improve methodologies for bone implants. The objective of the educational activities is to create interdisciplinary learning and research experiences for students, with mathematics being a central component. To accomplish that goal, a seminar in interdisciplinary mathematics is developed.
