Collaborative Research: Numerical Methods and Adaptive Algorithms for Sixth-Order Phase Field Models

合作研究：六阶相场模型的数值方法和自适应算法

• 批准号: 2110774
• 负责人: Natasha Sharma
• 金额: $ 12.5万
• 依托单位: University of Texas at El Paso
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-09-01 至 2024-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2110774&HistoricalAwards=false
• 关键词: Collaborative; Research; Numerical; Methods; Adaptive

This project will use computational mathematics models to further the understanding of two applications, microemulsions systems and crystal formation models. The first class of applications has relevance for oil-water-surfactant systems, which are important in oil recovery, development of environmentally friendly solvents, consumer and commercial cleaning product formulations, and drug delivery systems. The crystal models to be studied will be useful in detecting topological defects within crystalline materials, a task which is of great interest in the material science community. Specific examples include supercooled liquids, crack propagation in a ductile material, and applications relating to photonics and semiconductors, cell structure substrates and MRI contrast agents. A major challenge impeding their use by the general mathematical and scientific community has been a lack of understanding of these complex systems. This project will build efficient algorithms for simulation that will support the study of these processes and the design of advanced materials. The project will provide opportunities to undergraduate and graduate students and introduce them to the theory and implementation of state-of-the-art numerical methods. In this project the PIs will develop C0 interior penalty finite element methods for the two classes of applications and mathematical models. The C0 interior penalty finite element method was originally constructed to handle fourth-order elliptic problems arising in mechanics, but its adaptations have been applied to other fourth- and sixth-order partial differential equations. The focus of this project is on numerical methods for time-dependent sixth-order partial differential equations. The high derivative order in combination with a time-dependent component presents many challenges to the creation of stable, convergent, and efficient numerical methods approximating solutions to these models. The work to be accomplished includes the establishment of formal proofs for the unique solvability, stability, and convergence of the proposed numerical methods. The largest challenge will be to develop a framework which establishes optimal order error estimates. Finally, in order to improve upon the efficiency of the proposed numerical methods, the PIs plan to develop efficient solvers for space-time discretized systems using operator-splitting techniques and space-time adaptivity based on a posteriori error estimates obtained by the goal-oriented dual weighted approach.This project is jointly funded by Computational Mathematics program, and by the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目将使用计算数学模型来进一步了解两个应用程序，微乳液系统和晶体形成模型。第一类应用程序与石油水 - 表面活性剂系统相关，这些系统对于石油回收，环保解决方案的开发，消费者和商业清洁产品配方和药物输送系统很重要。要研究的晶体模型将有助于检测晶体材料中的拓扑缺陷，任务特定的例子包括超冷液体，延性材料中的裂纹繁殖以及与光子学和半导体有关的应用，细胞结构底物和MRI相反的药物。这是一般数学和科学群体的使用，这是一项复杂的群体的使用，这是一项复杂的群体的使用。 ThisProject将建立有效的模拟算法，以支持对这些过程的研究和高级材料的设计。该项目将为本科生和研究生提供机会，并向他们介绍最先进的数值方法的理论和实施。在这个项目中，PI将为两个类别的应用程序和数学模型开发C0内部惩罚有限元方法。 C0内部惩罚有限元方法最初是为处理机械师产生的四阶椭圆问题而构建的，但其适应性已应用于其他四阶和六阶偏微分方程。该项目的重点是用于时间依赖的六阶部分微分方程的数值方法。高导数顺序与时间相关的组件结合使用，对近似于这些模型的解决方案的稳定，收敛性和有效的数值方法的创建提出了许多挑战。要完成的工作包括建立正式的证据，以实现所提出的数值方法的独特可溶性，稳定性和收敛性。最大的挑战是开发一个建立最佳订单误差估算的框架。最后，为了提高所提出的数值方法的效率，PIS计划使用操作员分配技术和时空适应性开发有效的求解器，以基于目标双重加权方法获得的后验误差估计来开发运算符的系统和时空适应性。这些项目由计算机研究计划（由计算机数学计划）共同资助，并启用了该计划（竞争者的竞争力）（激发了竞争力）（激发了竞争力）（激发扬声器），并激发了竞争力的竞争力（启动级别的竞争）。 NSF的法定使命，并使用基金会的知识分子优点和更广泛的审查标准来评估，被认为是宝贵的支持。