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

高温超导模型的分析、算法和计算杜强密歇根州立大学数学系48824项目摘要在过去的几年中，研究者和他的同事在超导现象的建模、分析和计算方面取得了重大进展。研究者和他的合作者提出的工作包括进一步发展和完善他们以前在中尺度模型（特别是金兹堡-朗道模型及其变体）上的工作，以及对宏观尺度（均质化）模型的新的计算和分析探索。在中尺度水平上，研究者继续开发和实施有效的算法来处理涉及ii型超导体，特别是高临界温度层状超导体中漩涡运动的各种有趣现象。在宏观尺度上，研究者建议开发现有模型的增强功能，以解释不均匀性和应用电流和电压，并开发和实施离散化和解决算法，以确定这些模型的近似解。研究者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住，由热波动和施加的电流和电压引起的运动和去钉住，由运动引起的电阻率，以及临界电流。超导性被认为是对未来经济繁荣和科学领先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/或希望研究超导物理的科学家、工程师和数学家需要强大而高效的算法和代码来进行超导现象的数值模拟。研究者和他的同事们致力于满足这样的需求。将开发出有用的工具，不仅供物理学家和材料科学家在涡旋水平上研究超导现象，而且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.