Cross Fields and Thin Filaments

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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.