Adaptive, Non-stiff, and Stochastic Methods for Phase Field Fluid Models

研究者建议发展多相复杂流体的保守相场模型的创新数值方法。新方法将能够准确有效地分辨拓扑变化和复杂界面形态、微观结构动力学以及相变中随机热波动的影响所涉及的不同时间和长度尺度。这些能力将通过考虑模型及其解的数学结构和多尺度性质的新颖的空间和时间自适应方法获得。这些方法将被设计成不受高阶稳定性约束，并具有最优的计算成本。还提出了一种有效的、创新的随机和确定性公式的耦合。在各种工业应用中发现的复杂流体，如乳液、泡沫和聚合物溶液，其特点是具有多种组分、几种共存的相以及跨越广泛长度范围的结构非均质性。这些流体混合物的宏观行为和性质是流动与微或纳米结构复杂的非线性耦合的函数。因此，对这些系统的动力学的理解具有重要的科学和技术意义。计算机模拟可以在帮助实现这一目标方面发挥重要作用。这些数值方法的发展是本建议的中心目标。建议的研究将在一个多学科的环境中进行，并将服务于一个重要的教育目标：本科生和研究生的跨学科教育和培训。这个项目的一个组成部分也是持续努力促进和扩大代表人数不足的群体的参与，并利用与工业部门有联系的创新教学倡议。

The investigator proposes to develop innovative numerical methods for conservative phase field models of multi-phase and complex fluids. The new methods will becapable of resolving accurately and efficiently the disparate time and length scales involved in topological changes and complex interface morphologies, in the dynamics of the micro-scale structures, and in the effect of stochastic thermal fluctuations in phase transitions. Those capabilities will be obtained with novel space and time adaptive methods that take into account the mathematical structure and multi-scale nature of the models and their solutions. The methods will be designed to be free of high order stability constraints and will be of optimal computational cost. An effective, innovative coupling of stochastic and deterministic formulations is also proposed.Complex fluids found in a wide variety of industrial applications such as emulsions, foams, and polymeric solutions are characterized by multiple components, several coexisting phases, and structural heterogeneities spanning a broad range of length scales. The macroscopic behavior and properties of these fluid mixtures are a function of the intricate nonlinear coupling of the flow with micro- or nano- structures. The understanding of the dynamics of these systems is thus of significant scientific and technological interest. Computer simulation can play an instrumental role toward aiding in achieving this goal. The development of these numerical methods is the central objective of this proposal. The proposed research will be conducted in a multi-disciplinary environment and will also serve an important education goal: the interdisciplinary education and training of undergraduate and graduate students. An integral part of this project is also a sustained effort to promote and broaden the participation of underrepresented groups and the use of innovative pedagogic initiatives with ties with the industrial sector.