Cross Fields and Thin Filaments

在科学计算和计算机图形学中，将复杂对象细分为小块通常很重要，这一过程称为网格生成。通常，让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边界参差不齐，并且通常可以更好地保证科学计算的准确性。这种关于n-交叉场的研究在高质量立方体网格的生成和先进材料设计中的晶格结构模拟方面具有应用价值。细丝项目对微观游泳者和微流体装置的建模很重要，这些问题的数值方法通常随着细丝变得非常细而失效。这种方法可以应用于纳米级游泳者的建模，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，目的是将这些研究主题与教育组成部分结合起来。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，并在多个层面上培训学生。该项目的主要结构是n-交叉场，这是局部定义的正交化坐标系，相对于重新排序和反转是不变的。这个项目将研究将代数几何、几何测度论和变分的概念联系起来的方法，以开发和分析计算n-交叉场的有效方法。变分方法提供了在猜想的余维两个可校正集之外构造光滑n交叉场的方法。该项目的第二部分研究了粘性流体中的浸没膜问题。虽然之前的工作集中在帮助理解弹性细丝的适当性和质量行为的理论方法上，但该项目旨在将先前的工作扩展到更重要的物理情况。该奖项反映了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。

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.