Mathematical Sciences: Analysis, Algorithms, and Computations for Models of High-Temperature Superconductivity

高温超导模型的分析、算法和计算密歇根州立大学密歇根州立大学E.Lansing数学系，MI 48824项目摘要在过去的几年里，这位研究人员和他的同事们在超导现象的建模、分析和计算方面取得了重大进展。调查员及其合作者的拟议工作包括进一步发展和完善他们先前关于中尺度模型(特别是金兹堡-朗道模型及其变种)的工作，以及对大尺度(均化)模型进行新的计算和分析探索。在中尺度水平上，研究人员继续开发和实施有效的算法来处理涉及第二类超导体中涡旋运动的各种有趣现象，特别是高临界温度的层状超导体。在宏观尺度上，研究人员建议对现有模型进行改进，以解释不均匀性以及施加的电流和电压，并开发和实现离散化和求解算法，以确定这些模型的近似解。研究人员和他的同事们希望模拟的现象类型包括涡旋液体和涡旋玻璃结构，由正常杂质和厚度变化引起的钉扎，由于热波动和外加电流和电压而产生的运动和脱钉，由运动引起的电阻率，以及临界电流。超导是被认为对未来经济繁荣和科学领导地位至关重要的重大挑战之一。随着现代计算技术的出现，希望设计超导器件和/或研究超导物理的科学家、工程师和数学家需要健壮和高效的算法和代码来对超导现象进行数值模拟。这位调查员和他的同事们的目标就是满足这种需求。将开发有用的工具，不仅供对研究涡旋级别超导现象感兴趣的物理学家和材料科学家使用，也供那些参与超导装置设计的人使用。

ANALYSES, ALGORITHMS, AND COMPUTATIONS FOR MODELS OF HIGH-TEMPERATURE SUPERCONDUCTIVITY Qiang Du Department of Mathematics Michigan State University E. Lansing, MI 48824 Project Abstract During the past few years, the investigator and his colleagues have achieved significant progress in the modeling, analysis, and computation of superconducting phenomena. The proposed work of the investigator and his collaborators includes the further development and refinement of their previous work on mezoscale models (in particular, the Ginzburg-Landau model and its variants), as well as new computational and analytical explorations into the macroscale (homogenized) models. At the mezoscale level, the investigator continues the development and implementation of efficient algorithms for treating a variety of interesting phenomena involving the motion of vortices in type-II superconductors, especially the high critical temperature layered superconductors. At the macroscale, the investigator proposes to develop enhancements of existing models that can account for inhomogenieties and for applied currents and voltages and to develop and implement discretization and solution algorithms for determining approximate solutions for these models. The types of phenomena the investigator and his colleagues wish to simulate include vortex liquid and vortex glass structures, pinning by normal impurities and thickness variations, motion and de-pinning due to thermal fluctuations and applied currents and voltages, resistivity caused by motion, and critical currents. Superconductivity is one of the grand challenges identified as being crucial to future economic prosperity and scientific leadership. With the advent of modern computational technology, scientists, engineers, and mathematicians who wish to design superconducting devices and/or who wish to study the physics of superconductivity are in need robust and efficient algorithms and codes for the numerical simulation of superconduc ting phenomena. The investigator and his colleagues are aimed at meeting such a need. Useful tools will be developed not only for the use of physicists and material scientists interested in studying superconducting phenomena at the vortex level, but also for those involved in the design of superconducting devices.