Singularly Perturbed and Multiscale Problems

建议：DMS-0405423 PI：Giovanni Leoni机构：美国梅隆大学标题：奇异摄动和多尺度系统摘要本提案的中心主题是将现代数学应用于材料科学，以研究材料在小长度尺度下的宏观行为。 PI建议进行以下具体研究：脱位。计划在这里是一个变分制定的平衡问题的有限数量的位错在一个平面域，和表征的能量含量的一个孤立的缺陷的一个经常性的功能的缺陷配置，重整化能量。 在体弹性存在下第二相粒子和空穴的广义伍尔夫形状。 PI计划研究弹性，表面能和角正则化之间的相互作用对角的影响。弹性固态相变。计划在这里是一个高阶变分模型的精细尺度结构的孪晶马氏体的两相变体中存在的一个马氏体孪晶界面的研究。在铁磁材料微结构的研究中，通常通过忽略或简化微磁能中的各种项而导出的简化模型已经在特定区域被提出。 PI计划研究完整的微磁能，以确定细化的长度尺度和精细几何形状，以及畴分支，即在边界和中心的畴宽度尺度的分离。晶体中的缺陷，微观结构和相变在确定材料性能方面起着关键作用。因此，了解它们的结构对于实现所需的材料特性和优化性能是必要的。 虽然电子显微镜和实验活动提供了大量的详细信息的结构，数学建模，分析与实验方法的集成，有望大大提高我们的理解相变，铁磁，铁电和马氏体材料，纳米材料，薄膜和自组装结构。

Proposal: DMS-0405423PI: Giovanni LeoniInstitution: Carnegie-Mellon UniversityTitle: Singularly Perturbed and Multiscale SystemsABSTRACTThe central theme of this proposal is the application of modern mathematics to materials science in order to study the macroscopic behavior of materials at small length scale. The PI proposes the following specific studies: Dislocations. Planned here is a variational formulation of the equilibrium problem for a finite number of dislocations in a plane domain, and the characterization of the energy content of a body with isolated defects in terms of a regular function of the defect configuration, the renormalized energy. Generalized Wulff shapes of second-phase particles and voids in the presence of bulk elasticity. The PI plans to study the influence on corner angles of the interaction between elasticity, surface energy and the corner regularization.Elastic solid-state phase transitions. Planned here is the study of a higher order variational model for the fine scale structure of twinning of two-phase variants of martensite in the presence of an austenite-twinning martensite interface.Micromagnetics. In the study of the microstructure of ferromagnetic materials simplified models, usually derived by neglecting or simplifying various terms in the micromagnetic energy, have been proposed in particular regimes. The PI plans to study the complete micromagnetic energy in order to determine the length scale and the fine geometry of the refinement, as well as domain branching, that is, the separation of scales of the domain widths at the boundary and at the center of the body.Defects in crystals, microstructures, and phase transitions play a key role in determining properties of materials. Thus, understanding their structure is necessary in achieving desired material properties and optimized performance. While electron microscopy and experimental activity have provided a great deal of detailed information on structures, the integration of mathematical modeling, analysis with experimental approaches promises to greatly increase our understanding of phase transformations, ferromagnetic, ferroelectric and martensitic materials, nanomaterials, thin films and self-assembled structures.