This paper presents a new representation of curve dynamics, with applications to vortex filaments in fluid dynamics. Instead of representing these filaments with explicit curve geometry and Lagrangian equations of motion, we represent curves implicitly with a new co-dimensional 2 level set description. Our implicit representation admits several redundant mathematical degrees of freedom in both the configuration and the dynamics of the curves, which can be tailored specifically to improve numerical robustness, in contrast to naive approaches for implicit curve dynamics that suffer from overwhelming numerical stability problems. Furthermore, we note how these hidden degrees of freedom perfectly map to a Clebsch representation in fluid dynamics. Motivated by these observations, we introduce untwisted level set functions and non-swirling dynamics which successfully regularize sources of numerical instability, particularly in the twisting modes around curve filaments. A consequence is a novel simulation method which produces stable dynamics for large numbers of interacting vortex filaments and effortlessly handles topological changes and re-connection events.

翻译：本文展示了曲线动态的新代表, 包括流体动态中的螺旋丝。 我们不是代表这些直曲线几何和拉格朗加方程式的线条, 而是以新的共维2级配置描述来隐含曲线。 我们的隐含代表在曲线的配置和动态中都承认了若干冗余的数学自由度, 这些自由度可以专门用来改善数字的稳健性, 与对受到巨大数量稳定性问题影响的隐性曲线动态的天真的方法形成对比。 此外, 我们注意到这些隐藏的自由度如何完美地映射到流体动态中的克莱布希代表处。 我们受这些观察的激励, 我们引入了非边端设置功能和非旋转的动态, 从而成功地规范了数字不稳定的来源, 特别是在曲线线条周围的扭曲模式中。 一个后果是新颖的模拟方法, 为大量相互作用的涡旋线和不遗余力处理着的地形变化和重新连接事件带来稳定的动态。