Abstract
One canonical practice in the development and application of level set methods is to convect a shape represented by zero level set with a given, reversible, and periodic velocity field and test how well the original shape is recovered after each cycle. In this short letter, we mathematically show that Crank–Nicolson time integration, combined with standard Galerkin finite element, can exactly recover the original shape after each cycle, regardless of spatiotemporal resolution. This surprising finding is also numerically demonstrated by LeVeque’s problem.
References
Zalesak S (1979) Fully multidimensional flux-corrected transport algorithms for fluids. J Comput Phys 31(3):335–362
Leveque R (1996) High-resolution conservative algorithms for advection in incompressible flow. SIAM J Numer Anal 33(2):627–665
Yan J, Lin SS, Bazilevs Y, Wagner G (2019) Isogeometric analysis of multi-phase flows with surface tension and with application to dynamics of rising bubbles. Comput Fluids 179:777–789
Wang Z, Yang J, Stern F (2012) A new volume-of-fluid method with a constructed distance function on general structured grids. J Comput Phys 231(9):3703–3722
Wang Z, Yang J, Koo B, Stern F (2009) A coupled level set and volume-of-fluid method for sharp interface simulation of plunging breaking waves. Int J Multiph Flow 35(3):227–246
Zhao Y, Chen H (2017) A new coupled level set and volume-of-fluid method to capture free surface on an overset grid system. Int J Multiph Flow 90:144–155
Akkerman I, Bazilevs Y, Kees CE, Farthing MW (2011) Isogeometric analysis of free-surface flow. J Comput Phys 230:4137–4152
Kees CE, Akkerman I, Farthing MW, Bazilevs Y (2011) A conservative level set method suitable for variable-order approximations and unstructured meshes. J Comput Phys 230:4536–4558
Zhu Q, Yan J (2021) A mixed interface-capturing/interface-tracking formulation for thermal multi-phase flows with emphasis on metal additive manufacturing processes. Comput Methods Appl Mech Eng 383:113910
Lin S, Yan J, Kats D, Wagner G (2019) A volume-conserving balanced-force level set method on unstructured meshes using a control volume finite element formulation. J Comput Phys 380:119–142
Tezduyar TE (1992) Stabilized finite element formulations for incompressible flow computations. Adv Appl Mech 28:1–44. https://doi.org/10.1016/S0065-2156(08)70153-4
Takizawa K, Tezduyar TE (2011) Multiscale space-time fluid-structure interaction techniques. Comput Mech 48:247–267. https://doi.org/10.1007/s00466-011-0571-z
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Zhao, Z., Yan, J. On the space discretization and time integration for level set convection. Comput Mech 72, 1115–1117 (2023). https://doi.org/10.1007/s00466-023-02338-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00466-023-02338-3