### Abstract

We present the mathematical foundation of a fluid animation method for unstructured meshes. Key contributions not previously treated are the extension to include diffusion forces and higher order terms of non-linear force approximations. In our discretization we apply a fractional step method to be able to handle advection in a numerically simple Lagrangian approach. Following this a finite element method is used for the remaining terms of the fractional step method. The key to deriving a discretization for the diffusion forces lies in restating the momentum equations in terms of a Newtonian stress tensor. Rather than applying a straightforward temporal finite difference method followed by a projection method to enforce incompressibility as done in the stable fluids method, the last step of the fractional step method is rewritten as an optimization problem to make it easy to incorporate non-linear force terms such as surface tension.

Original language | English |
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Title of host publication | SCA '11 Proceedings of the 2011 ACM SIGGRAPH/Eurographics Symposium on Computer Animation |

Editors | A. Bargteil, M. van de Panne |

Number of pages | 10 |

Publisher | ACM |

Publication date | 2011 |

Pages | 101-110 |

ISBN (Print) | 978-1-4503-0923-3 |

DOIs | |

Publication status | Published - 2011 |

Event | 34th International Conference and Exhibition on Computer Graphics and Interactive Techniques - San Diego, CA, United States Duration: 5 Aug 2007 → 9 Aug 2007 Conference number: 34 http://www.siggraph.org/s2007/ |

### Conference

Conference | 34th International Conference and Exhibition on Computer Graphics and Interactive Techniques |
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Number | 34 |

Country | United States |

City | San Diego, CA |

Period | 05/08/2007 → 09/08/2007 |

Internet address |

### Keywords

- Optimizationbased Fluid Animation
- Unstructured Meshes
- Finite Element Method
- Deformable Simplicial Complexes
- Diffusion Forces
- Computational Fluid Dynamics

## Cite this

Erleben, K., Misztal, M. K., & Bærentzen, J. A. (2011). Mathematical foundation of the optimization-based fluid animation method. In A. Bargteil, & M. van de Panne (Eds.),

*SCA '11 Proceedings of the 2011 ACM SIGGRAPH/Eurographics Symposium on Computer Animation*(pp. 101-110). ACM. https://doi.org/10.1145/2019406.2019420