Analysis, Algorithms and Computations for Model Problems in Material Sciences

近年来，在现代高性能计算机上使用数值模拟进行科学研究的趋势越来越明显。 在计算材料科学领域取得了相当大的进展。计算工具已被用于新材料的设计以及其性能的研究。该项目的主要目标是：1）发展或改进某些中尺度和大尺度模式，以扩大这些模式适用的物理问题的范围; 2）分析这些模式，以进一步了解它们的性质和解决办法; 3）开发、分析和实现算法，特别是并行和自适应算法，用于这些模型的数值模拟; 4）利用我们的算法和程序研究材料科学中一些有趣的现象。在拟议的工作中，首席研究员将研究模型和开发数值算法，用于一些有趣的材料科学问题，涉及多尺度（中尺度和宏观尺度）和随机效应，例如与超导和磁性中的涡旋和其他缺陷有关的问题。该项目的一个主要部分是旨在扩大中尺度代码的应用范围，并通过使用域和尺度分解/积分和自适应计算技术，进行更多的中尺度和宏观模式之间的比较研究。中尺度模型的代码可以用于获得有关精细结构（如涡旋）与边界、界面、杂质、电流和热波动的物理行为和相互作用的信息和见解。它们可以间接用于设备设计者，特别是当可以识别与宏观属性的连接。基于随机偏微分方程的模型及其数值模拟也将得到重视，以便深入了解超导体和液晶等材料中的热波动和杂质的宏观效应。这项工作的目的是使计算代码强大，高效，灵活，准确，可扩展和用户友好。希望这些代码可以被实验室，大学和工业组织的物理学家，材料科学家和工程师用作研究某些特定材料特性的工具，也可以作为设计器件的工具。

There has been an increasing trend to conduct scientific research using numerical simulations on modern high performance computers in recent years. Considerable progress has been made in the area of computational material sciences. Computational tools have been used in the design of new materials as well as in the study of their properties. The central objectives of this project are: 1) to develop or refine certain mesoscale and macroscale models, so to enlarge the range of physical problems for which such models are valid; 2) to analyze these models in order to gain further understanding of their properties and solutions; 3) to develop, analyze, and implement algorithms, in particular, parallel and adaptive algorithms, for the numerical simulation of these models; and 4) to use our algorithms and codes to study some interesting phenomena in material sciences.In the proposed work, the principal investigator will study models and develop numerical algorithms for some interesting material sciences problems that involve multiscale (mesoscale and macroscale) and stochastic effects, such as problems related to vortices and other defects in superconductivity and magnetism. A major part of the project is aimed at increasing the range of applications for the mesoscale codes and allow more comparative studies between the mesoscale and macroscopic models through the use of domain and scale decomposition/integration and adaptive computation techniques. The codes for mesoscale models can be of use in gaining information and insight about the physical behavior and interaction of the fine structures (such as vortices) with, for example, boundaries, interfaces, impurities, currents, and thermal fluctuations. They can be of indirect use to device designers, in particular, when connections with macroscopic properties can be identified. Models based on the stochastic partial differential equations and their numerical simulations will also be given emphasis, so as to gain insight to the macroscopic effect of thermal fluctuations and impurities in the materials like superconductors and liquid crystals. The work will be aimed at making the computational codes robust, efficient, flexible, accurate, scalable and user-friendly. It is hoped that these codes can be used by physicists, material scientists, and engineers in laboratories, universities, and industrial organizations as a tool for studying some specific material properties and also a tool in designing devices.