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Cross Fields and Thin Filaments

交叉场和细丝

基本信息

批准号：
2009352
负责人：
Daniel Spirn
金额：
$ 33.17万
依托单位：
University of Minnesota-Twin Cities
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2009352&HistoricalAwards=false
关键词：
Cross Fields Thin Filaments

项目摘要

In scientific computing and computer graphics it is typically important to subdivide complicated objects into small pieces, a process known as mesh generation. Often, it is useful to have these small pieces composed of small cubes, as cubic elements can often prevent jagged boundaries and can often provide better assurances of the accuracy of scientific calculations. This research on n-cross fields has applications to the generation of high quality cubic meshes and to the modelling of crystalline lattice structures in advanced materials design. The project on thin filaments is important to the modeling of microscopic swimmers and microfluidic devices, where numerical methods for these problems typically break down as the filaments get very thin. Such methods can be applied to the modeling of nano-scale swimmers, with potential in biomedical applications. This project includes opportunities for training and human resource development and aims to integrate these research topics with educational components. The overarching goals are to increase the profile of the topics, to make progress on central issues in the field, and to train students at many levels.The principal structures underlying the project are n-cross fields, which are locally defined orthonormal coordinate systems that are invariant with respect to the reordering and inversions. This project will study methods that connect notions from algebraic geometry, geometric measure theory and the calculus of variations to develop and analyze efficient methods for computing n-cross fields. Variational methods provide for the construction of smooth n-cross fields, outside of conjectured co-dimension two rectifiable sets. The second part of the project studies problems focused on immersed membranes in viscous fluids. While prior work focused on theoretical methods that help provide an understanding of well-posedness and qualitative behavior of elastic filaments, this project aims to extend the prior work to even more physically important situations.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.

在科学计算和计算机图形学中，将复杂的对象细分为小块是非常重要的，这一过程称为网格生成。通常，将这些小块由小立方体组成是有用的，因为立方体元素通常可以防止锯齿状边界，并且通常可以更好地保证科学计算的准确性。这项研究的n-交叉领域的应用程序生成高品质的立方网格和先进材料设计中的晶格结构的建模。细丝项目对于微观游泳者和微流体设备的建模非常重要，这些问题的数值方法通常会随着细丝变得非常细而崩溃。这种方法可以应用于纳米尺度游泳者的建模，具有生物医学应用的潜力。该项目包括培训和人力资源开发机会，旨在将这些研究课题与教育内容结合起来。总体目标是增加主题的轮廓，在该领域的中心问题上取得进展，并在多个级别上培养学生。该项目的主要结构是n-交叉场，这是局部定义的正交坐标系，对于重排和反演是不变的。本项目将研究连接代数几何，几何测度理论和变分法的概念的方法，以开发和分析计算n-交叉场的有效方法。变分方法提供了光滑的n-交叉领域的建设，外的约束余维两个可求长集。该项目的第二部分研究了粘性流体中浸没膜的问题。虽然以前的工作集中在理论方法，帮助提供了一个良好的定性行为的弹性长丝的理解，这个项目的目的是扩展以前的工作，甚至更重要的物理情况。这个奖项反映了NSF的法定使命，并已被认为是值得支持的评估使用基金会的智力价值和更广泛的影响审查标准。