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理解两种应用：微乳液系统和晶体形成模型。第一类应用与油-水-表面活性剂系统相关，这对于石油采收、环保溶剂的开发、消费者和商业清洁产品配方以及药物输送系统将有助于检测晶体材料内的拓扑缺陷，这是材料科学界非常感兴趣的任务，具体例子包括过冷液体、裂纹扩展。阻碍它们被一般数学和科学界使用的一个主要挑战是缺乏对这些复杂系统的理解。该项目将构建高效的算法。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。 PI将开发C0内饰两类应用和数学模型的惩罚有限元方法 C0 内部惩罚有限元方法最初是为了处理力学中出现的四阶椭圆问题而构建的，但其修改已应用于其他四阶和六阶偏微分。该项目的重点是与时间相关的六阶偏微分方程的数值方法，高导数阶数与与时间相关的分量相结合，对创建稳定、收敛和高效的数值方法提出了许多挑战。解决方案要完成的工作包括为所提出的数值方法的独特可解性、稳定性和收敛性建立形式证明，最后，最大的挑战是开发一个建立最佳阶次误差估计的框架。为了提高所提出的数值方法的效率，PI计划使用算子分裂技术和基于通过目标导向的双加权获得的后验误差估计的时空自适应性来开发时空离散系统的高效求解器该项目由计算数学计划和刺激竞争研究既定计划 (EPSCoR) 联合资助。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。