Two-Parameter Homogenization Problems in Superconductivity and Related Problems

本项目主要研究均质化理论的理论发展和应用。该理论研究非均相材料（主要是复合材料）的性能，这对现代技术至关重要。在对这些材料进行建模的过程中提出了基本问题——主要是在偏微分方程和变分学领域。这项工作解决了特殊类型的均质问题，其中除了均质参数外，还存在另一个小的材料参数（例如逆金兹堡-朗道参数）。双参数均匀化问题的结果和技术发展基本上取决于这些参数之间的关系，并考虑了几种这样的关系。这些问题在超导复合材料的金兹堡-朗道模型的背景下进行了研究。具体来说，研究者和他的合作者研究了小超导样品中的孔钉。这里的主要数学问题是一个非标准的离散/连续非线性有限维变分问题的均匀化极限，具有整数约束的未知涡旋度族。研究者和他的合作者开发了伽玛收敛技术来解决这些问题。预计均匀涡量在很大程度上取决于几何参数（孔的大小和孔之间的距离）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缩放关系，这导致了多个度的漩涡。此外，研究者和他的合作者开发了新的伽玛收敛技术，用于研究非线性随机均匀化问题。考虑了这些技术在材料科学问题中的应用。其中一个关键问题是，与周期性定位的内含物相比，如何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，为新一代超导微电子技术奠定基础。它利用了超导的金兹堡-朗道模型，描述了它们在没有功率损失的情况下携带电流的能力。在超导的实际应用中，通过的电流产生磁涡流，磁涡流移动，耗散能量，并破坏超导状态。因此，实际应用的中心问题是旋涡的固定问题，专家们称之为旋涡钉住问题。本文研究的方法为解决这一问题提供了一种有效的工具，并为超导导线的人工制造钉住结构开发了实用的配方，从而极大地提高了它们的性能。这个项目对研究生的教育和未来的职业生涯有直接的影响。通过与研究者在阿贡国家实验室的合作者的互动，他们接受跨学科的培训。此外，研究者继续让本科生参与他的nsf支持的研究项目。

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.