Partition of Unity Multivariate Approximation for the Volume of Fluid Method

该项目将促进对在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、复杂的界面，该项目将开发计算工具来预测这些界面的演变；挑战是这些界面可能会以复杂的局部化模式在很大程度上变形。用于数值跟踪和预测界面动态的方法必须能够以最优的计算成本和高精度正确地捕捉局部特征。该项目旨在开发和分析快速和高精度的界面重建方法，重点是三相(液-气-固)流。这里一个有趣的应用是将眼药水沉积到3D逼真的眼睛几何图形上。该项目还包括在跨学科背景下对学生进行研究、培训和综合教育，以及开发支持可重复性研究的代码。流体体积法(VOF)是多相流模拟中最常用的界面跟踪方法之一。该项目的研究涉及提高流体体积法界面重建方案的精度的技术开发，用于模拟复杂几何形状上的多相问题。我们所研究的三个最重要的方面是：(1)稳健和高精度的单位分裂多元近似流体体积(PUMA-VOF)方法；(2)求解时变域上平流型偏微分方程(PDE)的联立时空格式；(3)运动几何问题的数学建模和数值模拟。这项研究将包括PUMA-VOF在复杂几何科学模拟领域的理论和实际应用。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.