Variational Methods for Materials Science and Mechanics

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1714088Leoni The central theme of this project is the application of modern applied mathematics to engineering, materials science, and mechanics. One of the main scientific goals is to address mathematically many of the issues encountered in the charge-discharge dynamics of Lithium-Ion batteries. The understanding of the these phenomena is paramount to overcome the shortcomings of rechargeable Lithium-Ion batteries and to develop novel materials. Other thematic areas of focus are phase transitions in solids and fluids and defects in materials. The investigator continues the training of undergraduate and graduate students and postdoctoral fellows through several activities that integrate research and education, including topics courses, summer schools, working groups, course design workshops, and career-building workshops. Graduate and undergraduate students participate in the work of the project. The main scientific thrusts of the project are:1) Variational Models for Lithium-Ion Batteries. High performance cathode and anode materials undergo phase transformations upon lithiation and delithiation. The goal here is to focus on variational models that are sufficiently rich to capture many of the issues encountered in the charge-discharge dynamics of Lithium-Ion batteries.2) Slow motion of interfaces. Sharp energy estimates obtained by the investigator and his collaborators have recently been used to give the first slow motion estimates of the interface of solutions of the mass-preserving Allen-Cahn equations and of the Cahn-Hilliard equations in higher dimensions and without structural assumptions on the initial data and on the form of the domain. The goal is to extend these results to local minimizers and to other functionals.3) Free boundary problems. The goal here is to focus on variational models for free boundary problems whose solutions exhibit singularities.Graduate and undergraduate students participate in the work of the project.