Mathematical Sciences: Investigations of Phase Transitions Using Relaxation and Nonlocal Regularization

小行星9403844 这个项目将支持一个正在进行的研究家庭的相变问题。 问题来自三个应用领域：力学（液体-蒸汽，固体-液体和多相-固体转变），超导性和磁性（刚性铁磁体以及磁致伸缩材料）。 该研究主要涉及研究能量最小化的静态问题，并试图描述能量的亚稳态（局部极小）集。 我们的目标是描述滞后（这发生在所有这些问题）在这些亚稳态。 主要的技术是使用松弛的非局部能量泛函来模拟正在研究的物理问题。 不动点和/或相对最小值，发现使用积分方程，凸分析，变分法和方法的技术。 数值算法的发展，以近似解的欧拉-拉格朗日方程。 许多高科技材料的最有用的特性（例如， 形状记忆合金、磁性和磁致伸缩材料以及超导体）是由经历“相变”的材料引起的。“通常，这些相变不会平滑地发生，而是在微观水平上表现出剧烈的振荡。 在上个世纪，材料科学家对这些现象进行了大量的实证研究。 但直到过去十年，数学家才开始将这些研究转变为一门更加定量的科学。 目前的项目旨在基于内能的基本模型对大规模材料结构的行为进行定量预测。 该项目结合了数学建模，数学分析以及新数值算法的开发和分析。 建议的应用是液体-蒸汽转变，晶体生长，形状记忆合金，磁性和磁致伸缩材料，和超导体。

9403844 Rogers This project will support an ongoing study of a family of problems in phase transitions. The problems come from three areas of application: mechanics (liquid-vapor, solid-liquid, and multiphase-solid transitions), superconductivity, and magnetism (rigid ferromagnets as well as magnetostrictive materials). The research primarily involves studying static problems of energy minimization and trying to characterize the set of metastable states (local minimizers) of the energy. The goal is to describe hysteresis (which occurs in all of these problems) in terms of these metastable states. The primary technique is to model the physical problems being studied using relaxed, nonlocal energy functionals. Stationary points and/or relative minimizers are found using techniques of integral equations, convex analysis, and methods from the calculus of variations. Numerical algorithms are developed to approximate solutions of the Euler-Lagrange equations. The most useful properties of many high-tech materials (e.g. shape memory alloys, magnetic and magnetostrictive materials, and superconductors) are caused by the material undergoing a change of "phase." Often, these phase changes do not occur smoothly, but instead, exhibit wild oscillations at a microscopic level. In the last century, there has been a great deal of empirical study of these phenomena by materials scientists. But it was only in the past decade that mathematicians began to turn these investigations into a more quantitative science. The current project aims to make quantitative predictions of the behavior of large-scale material structures based on fundamental models of the internal energy. The project combines elements of mathematical modeling, mathematical analysis, and the development and analysis of new numerical algorithms. The proposed applications are to liquid-vapor transitions, crystal growth, shape memory alloys, magnetic and magnetostrictive materials, and superc onductors.