Hidden Degrees of Freedom in Implicit Vortex Filaments

TL;DR

This paper introduces an implicit level set representation for vortex filaments that enhances numerical stability and handles complex topological changes, leveraging hidden degrees of freedom and a novel regularization approach.

Contribution

It proposes a new implicit curve representation with hidden degrees of freedom, improving stability and robustness in vortex filament simulations compared to traditional explicit methods.

Findings

Stable simulation of large vortex filament systems

Effective handling of topological changes and reconnections

Regularization of twisting modes for numerical stability

Abstract

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.