article published yes Sadashige Ishida author 6F7C4B96-A8E9-11E9-A7CA-09ECE56974250000-0002-3121-3100 Christopher J Wojtan author 3C61F1D2-F248-11E8-B48F-1D18A9856A870000-0001-6646-5546 Albert Chern author ChWo department Computational Discovery of Numerical Algorithms for Animation and Simulation of Natural Phenomena project 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. https://research-explorer.ista.ac.at/download/12431/12433/2022_ACM_Ishida.pdf application/pdfno https://creativecommons.org/licenses/by/4.0/ Association for Computing Machinery2022 eng 0730-0301 1557-7368 00089165190006110.1145/3550454.3555459 416 https://research-explorer.ista.ac.at/record/20551 S. Ishida, C. Wojtan, A. Chern, ACM Transactions on Graphics 41 (2022). Ishida, S., Wojtan, C., &#38; Chern, A. (2022). Hidden degrees of freedom in implicit vortex filaments. <i>ACM Transactions on Graphics</i>. Association for Computing Machinery. <a href="https://doi.org/10.1145/3550454.3555459">https://doi.org/10.1145/3550454.3555459</a> Ishida, Sadashige, et al. “Hidden Degrees of Freedom in Implicit Vortex Filaments.” <i>ACM Transactions on Graphics</i>, vol. 41, no. 6, 241, Association for Computing Machinery, 2022, doi:<a href="https://doi.org/10.1145/3550454.3555459">10.1145/3550454.3555459</a>. Ishida S, Wojtan C, Chern A. Hidden degrees of freedom in implicit vortex filaments. <i>ACM Transactions on Graphics</i>. 2022;41(6). doi:<a href="https://doi.org/10.1145/3550454.3555459">10.1145/3550454.3555459</a> S. Ishida, C. Wojtan, and A. Chern, “Hidden degrees of freedom in implicit vortex filaments,” <i>ACM Transactions on Graphics</i>, vol. 41, no. 6. Association for Computing Machinery, 2022. Ishida, Sadashige, Chris Wojtan, and Albert Chern. “Hidden Degrees of Freedom in Implicit Vortex Filaments.” <i>ACM Transactions on Graphics</i>. Association for Computing Machinery, 2022. <a href="https://doi.org/10.1145/3550454.3555459">https://doi.org/10.1145/3550454.3555459</a>. Ishida S, Wojtan C, Chern A. 2022. Hidden degrees of freedom in implicit vortex filaments. ACM Transactions on Graphics. 41(6), 241. 124312023-01-29T23:00:59Z2026-04-07T12:02:23Z