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

本计画将使用计算数学模型来进一步了解微乳系统与晶体形成模型的两个应用。第一类应用与油-水-表面活性剂体系有关，这在石油回收、环境友好溶剂的开发、消费者和商业清洁产品配方以及药物递送体系中是重要的。待研究的晶体模型将有助于检测晶体材料中的拓扑缺陷，这是材料科学界非常感兴趣的任务。具体的例子包括过冷液体、韧性材料中的裂纹扩展以及与光子学和半导体、细胞结构基底和MRI造影剂相关的应用。阻碍一般数学和科学界使用它们的一个主要挑战是缺乏对这些复杂系统的理解。该项目将建立有效的模拟算法，以支持这些过程的研究和先进材料的设计。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将为两类应用程序和数学模型开发C 0内部惩罚有限元方法。C 0内部罚有限元法最初是为了处理力学中的四阶椭圆问题而构造的，但其适应性已被应用于其他四阶和六阶偏微分方程。这个项目的重点是时间相关的六阶偏微分方程的数值方法。高导数阶与时间相关的组件相结合，提出了许多挑战，以创建稳定，收敛和有效的数值方法逼近这些模型的解决方案。要完成的工作包括建立正式证明的唯一的可解性，稳定性和收敛性的数值方法。最大的挑战将是开发一个框架，建立最佳顺序误差估计。最后，为了提高数值方法的效率，研究人员计划利用算子分裂技术和基于目标导向的对偶加权方法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有效求解器。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.