Two-Parameter Homogenization Problems in Superconductivity and Related Problems

超导中的二参数均匀化问题及相关问题

• 批准号: 1106666
• 负责人: Leonid Berlyand
• 金额: $ 28.45万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-10-01 至 2016-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1106666&HistoricalAwards=false
• 关键词: Two; Parameter; Homogenization; Problems; Superconductivity

BerlyandDMS-1106666 The project is concerned with the theoretical development and applications of homogenization theory. This theory studies the properties of heterogeneous materials (primarily composites), which are of critical importance for modern technology. Fundamental questions are raised in the process of modeling such materials -- primarily in the fields of PDEs and Calculus of Variations. The work addresses special classes of homogenization problems where, in addition to the homogenization parameter, another small material parameter (e.g., the inverse Ginzburg-Landau parameter) is present. The results and techniques developed for two-parameter homogenization problems essentially depend on the relation between these parameters, and several such relations are considered. These problems are studied in the context of Ginzburg-Landau models for superconducting composites. Specifically, the investigator and his collaborators study pinning by holes in small superconducting samples. Here the main mathematical issue is the homogenization limit in a nonstandard discrete/continuum nonlinear finite-dimensional variational problem with integer constraints for the unknown family of degrees of vortices. The investigator and his collaborators develop Gamma convergence techniques for such problems. It is expected that the homogenization vorticity drastically depends on the relation between the geometrical parameters (size of the holes and distances between them) and the material parameter. In particular, the investigator and his collaborators look for special scaling relations, which result in vortices with multiple degrees. Also, the investigator and his collaborators develop novel techniques of Gamma-convergence for the study of nonlinear random homogenization problems. Applications of these techniques for material science problems are considered. One of the key issues is the identification of new effects due to randomness when compared to periodically located inclusions. The project is primarily motivated by the quest for energy-efficient materials that comprise a foundation for a new generation of superconductivity-based microelectronics. It utilizes Ginzburg-Landau models of superconductivity, describing their ability to carry electric current without power loss. In practical applications of superconductivity the passing current creates magnetic vortices that move, dissipate the energy, and destroy the superconducting state. Thus the central problem for practical application is the problem of immobilization of vortices, known among specialists as the problem of vortex pinning. The approach studied here offers an efficient tool to tackle this problem and to develop practical recipes for an artificial manufacturing pinning configuration in superconducting wires to crucially improve their performance. The project has a direct impact on the education and future careers of graduate students. They receive an interdisciplinary training through interactions with the investigator's collaborators at Argonne National Laboratory. Additionally, the investigator continues involving undergraduate students in his NSF-supported research projects.

BerlyandDMS-1106666 该项目涉及均质化理论的理论发展和应用。 该理论研究异质材料（主要是复合材料）的特性，这对于现代技术至关重要。 在对此类材料建模的过程中提出了一些基本问题——主要是偏微分方程和变分微积分领域。 这项工作解决了特殊类别的均质化问题，其中除了均质化参数之外，还存在另一个小材料参数（例如逆 Ginzburg-Landau 参数）。 针对双参数均质化问题开发的结果和技术本质上取决于这些参数之间的关系，并且考虑了几种这样的关系。 这些问题是在超导复合材料的金兹堡-朗道模型的背景下研究的。 具体来说，研究人员和他的合作者研究了小型超导样品中的孔钉扎现象。 这里的主要数学问题是非标准离散/连续非线性有限维变分问题中的均质化极限，该问题具有未知涡度族的整数约束。 研究人员和他的合作者开发了针对此类问题的伽玛收敛技术。 预计均匀化涡度很大程度上取决于几何参数（孔的尺寸和孔之间的距离）和材料参数之间的关系。 特别是，研究者和他的合作者寻找特殊的尺度关系，这会导致多级涡流。 此外，研究者和他的合作者开发了伽玛收敛的新技术来研究非线性随机均匀化问题。 考虑将这些技术应用于材料科学问题。 关键问题之一是与周期性定位的内含物相比，识别由于随机性而产生的新效应。 该项目的主要动机是寻求节能材料，为新一代基于超导的微电子学奠定基础。 它利用超导的金兹堡-朗道模型，描述了它们在不损失功率的情况下承载电流的能力。 在超导的实际应用中，通过的电流会产生磁涡流，磁涡流会移动、耗散能量并破坏超导状态。 因此，实际应用的核心问题是涡流固定问题，专家称为涡流钉扎问题。 这里研究的方法提供了一种有效的工具来解决这个问题，并为超导线材中的人工制造钉扎配置开发实用的配方，以显着提高其性能。 该项目对研究生的教育和未来职业生涯有直接影响。 他们通过与阿贡国家实验室的研究人员合作者的互动接受跨学科培训。 此外，研究人员继续让本科生参与他的国家科学基金会支持的研究项目。