Analysis, Algorithms and Computation of Some Model Problems in Interface and Defect Dynamics

该项目涉及对物理和生物系统中普遍存在的界面和缺陷的研究，这些界面和缺陷对材料性能和生物功能至关重要。在以前的研究工作的基础上，PI将开发和应用分析和数值模拟工具来研究与各种材料界面和缺陷相关的确定性和随机性效应，特别强调超导体和玻色-爱因斯坦凝聚体中的缺陷，其特征在于量子化涡旋，以及以模型仿生细胞膜为特征的软界面。调查将在很大程度上遵循一般金斯堡-朗道（扩散界面，相场）形式主义与其他多尺度和随机建模方法的连接。将进行系统的模型推导、分析和简化。自适应算法和统计检索算法将被设计和分析。这些研究活动将提高复杂系统的模拟能力和大规模数值计算的预测能力。在专注于特定应用的同时，该项目的核心算法开发工作将与集成，自适应和智能科学计算的现代主题保持一致。该项目位于计算数学，物理学，材料和生物科学的接口。通过数学分析和数值模拟研究的物理和生物对象包括量子化的漩涡，这是众所周知的超流性特征，以及生物膜，这是自然界设计的软界面，是生命的基本组成部分。PI将开发关于复杂非线性模型的新分析理论和新工具，用于从大规模模拟中提取有用的统计数据和探索隐藏的结构。这些研究活动将为数学带来新的进展，并使人们更好地理解各种基本的物理和生物过程。它们反过来可以帮助发现基于超导性和超流性的新技术，以及基于仿生膜的药物和药物递送载体的新设计，这两者都具有重要的经济价值和国家优先事项。 此外，该项目将为未来一代劳动力和年轻研究人员提供宝贵的培训环境和跨学科研究经验，展示TEAMS（实验，分析，建模和仿真培训）精神。

This project is concerned with the study of interfaces and defects which are ubiquitous in physical and biological systems and are essential to the materials properties and biological functions. Building on the previous research work, the PI will develop and apply analytical and numerical simulation tools to study both deterministic and stochastic effects associated with various material interfaces and defects with particular emphasis on defects in superconductors and Bose-Einstein condensates as characterized by quantized vortices, and soft interfaces as characterized by model biomimetic cell membranes. The investigation will be largely following the general Ginzburg-Landau (diffuse interface, phase field) formalism with connections to other multiscale and stochastic modeling approaches. Systematic model derivations, analysis and simplifications will be conducted. Adaptive algorithms and statistics retrieval algorithms will be designed and analyzed. These research activities will enhance the simulation capability of complex systems and the predictive power of large scale numerical computation. While focusing on specific applications, the central algorithmic development work in this project will be in tune with the modern theme of integrated, adaptive and intelligent scientific computation. This project lies at the interface of computational mathematics, physics, materials and biological sciences. The physical and biological objects to be studied through mathematical analysis and numerical simulations include quantized vortices which are well-known signatures of superfluidity, and biological membranes which are soft interfaces designed by nature as the fundamental building blocks of life. The PI will develop new analytical theory on the complex nonlinear models and new tools for extracting useful statistics and exploring hidden structures from massive simulations. The research activities will bring new advances to mathematics and provide better understanding of various basic physical and biological processes. They in turn may aid the efforts in discovering new technology based on the superconductivity and superfluidity, and new design of drug and drug delivery vehicles based on biomimetic membranes, both are of significant economic value and national priority. In addition, this project will provide valuable training environment and interdisciplinary research experience to the future generation of workforce and young researchers that showcases the TEAMS (Training in Experiments, Analysis, Modeling and Simulations) spirit.