Partition of Unity Multivariate Approximation for the Volume of Fluid Method

该项目将促进对许多科学和工程应用中遇到的多相流的理解。多相流问题表现出高度动态的复杂界面，该项目将开发计算工具来预测这种界面的演变;挑战在于这些界面可能会在很大程度上变形，并具有复杂的局部图案。数值跟踪和预测界面动态的方法必须能够以最佳计算成本和高精度正确捕获局部特征。该项目旨在开发和分析快速和高精度界面重建方法的技术，重点是三相（液-气-固）流。这里感兴趣的应用是将滴眼剂沉积到3D逼真的眼睛几何形状上。该项目还涉及在跨学科环境中对学生进行研究培训和综合教育，以及开发支持可重复研究的代码。流体体积（VOF）方法是多相流模拟中最常用的界面跟踪方法之一。本项目的研究涉及开发技术，以提高模拟复杂几何形状多相问题的流体体积法的界面重建方案的准确性。本文主要研究了三个方面的内容：（1）稳健的高精度单位分割多变量近似VOF（Partition of Unity Multivariate Approximation Volume-of-Fluid，PUMA-VOF）方法;（2）时变区域上对流型偏微分方程的时空同步格式;（3）运动几何问题的数学建模与数值模拟。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 advance the understanding of multi-phase flows that are encountered in many scientific and engineering applications. Multi-phase flow problems exhibit highly dynamic, complex interfaces, and the project will develop computational tools to predict the evolution of such interfaces; the challenge is that these interfaces may be largely deformed with intricate localized patterns. Methods for numerically tracking and predicting the dynamics of the interfaces must be able to correctly capture local features with optimal computational costs and high accuracy. This project aims to develop and analyze techniques for fast and highly accurate interface reconstruction methods with emphasis on three-phase (liquid-gas-solid) flow. An application of interest here is deposition of eye drops onto 3D realistic eye geometries. The project also involves research training and integrated education of students in an interdisciplinary setting and the development of codes to support reproducible research.The volume-of-fluid (VOF) method is one of the most commonly used interface tracking methods in multi-phase flow simulations. Research in this project involves the development of techniques to improve the accuracy of the interface reconstruction scheme for the volume of fluid method for simulating multi-phase problems on complex geometries. The three most important aspects we are investigating are (1) robust and highly-accurate partition of unity multivariate approximation volume-of-fluid (PUMA-VOF) method; (2) simultaneous space-time schemes for advection-type partial differential equations (PDEs) on time-varying domains; (3) mathematical modeling and numerical simulation of problems with moving geometries. The research will include theory and practical application of PUMA-VOF on the field of scientific simulations on complex geometries.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.