@phdthesis{19630,
  abstract     = {This thesis consists of three chapters, each corresponding to one publication. While each of these projects tackles a topic in a different area of research, they all share a common thread in the type of topological structure they handle - a partition of space into volumes separated by interfaces that meet in non-manifold junctions.

In Chapter 2, we study clusters of soap bubbles from a simulation perspective. In particular, we develop a surface-only algorithm that couples large scale motion and shape deformation of soap bubble clusters with the small scale evolution of the thin film's thickness, which is responsible for visual phenomena like surface vortices, Newton's interference patterns, capillary waves, and deformation-dependent rupturing of films in a foam. We model film thickness as a reduced degree of freedom in the Navier-Stokes equations and from them derive three sets of equations governing normal and tangential motion of the soap film surface, as well as the evolution of the thin film thickness. We discretize these equations on a non-manifold triangle mesh, extending and adapting operators to handle complex topology. We also present an incompressible fluid solver for 2.5D films and an advection algorithm for convecting fields across non-manifold surface junctions. Our simulations enhance bubble solvers with additional effects caused by convection, rippling, draining, and evaporation of the thin film.

In Chapter 3, we introduce a multi-material non-manifold mesh-based surface tracking algorithm that converts mesh defects, such as overlaps, self-intersections, and inversions into topological changes. Our algorithm generalizes prior work on manifold surface tracking with topological changes: it preserves surface features like mesh-based methods, and it robustly handles topological changes like level set methods. Our method also offers improved efficiency and robustness over the state of the art. We demonstrate the effectiveness of the approach on a range of examples, including complex soap film simulations, such as those presented in Chapter 2, but with an order of magnitude more interacting bubbles than what we could achieve before, and Boolean unions of non-manifold meshes consisting of millions of triangles.

Lastly, in Chapter 4, we utilize developments in the theory of random geometric complexes facilitated by observations from Discrete Morse theory. We survey the methods and results obtained with this new approach, and discuss some of its shortcomings. We use simulations to illustrate the results and to form conjectures, getting numerical estimates for combinatorial, topological, and geometric properties of weighted and unweighted Delaunay mosaics, their dual Voronoi tessellations, and the Alpha and Wrap complexes contained in the mosaics.},
  author       = {Synak, Peter},
  issn         = {2663-337X},
  pages        = {106},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{Methods for fluid simulation, surface tracking, and statistics of non-manifold structures}},
  doi          = {10.15479/AT-ISTA-19630},
  year         = {2025},
}

@article{17219,
  abstract     = {We introduce a multi-material non-manifold mesh-based surface tracking algorithm that converts self-intersections into topological changes. Our algorithm generalizes prior work on manifold surface tracking with topological changes: it preserves surface features like mesh-based methods, and it robustly handles topological changes like level set methods. Our method also offers improved efficiency and robustness over the state of the art. We demonstrate the effectiveness of the approach on a range of examples, including complex soap film simulations with thousands of interacting bubbles, and boolean unions of non-manifold meshes consisting of millions of triangles.},
  author       = {Synak, Peter and Kalinov, Aleksei and Strugaru, Irina-Malina and Etemadihaghighi, Arian and Yang, Huidong and Wojtan, Christopher J},
  issn         = {1557-7368},
  journal      = {ACM Transactions on Graphics},
  keywords     = {surface tracking, topology change, non- manifold meshes, multi-material flows, solid modeling},
  number       = {4},
  publisher    = {Association for Computing Machinery},
  title        = {{Multi-material mesh-based surface tracking with implicit topology changes}},
  doi          = {10.1145/3658223},
  volume       = {43},
  year         = {2024},
}

@inproceedings{8135,
  abstract     = {Discrete Morse theory has recently lead to new developments in the theory of random geometric complexes. This article surveys the methods and results obtained with this new approach, and discusses some of its shortcomings. It uses simulations to illustrate the results and to form conjectures, getting numerical estimates for combinatorial, topological, and geometric properties of weighted and unweighted Delaunay mosaics, their dual Voronoi tessellations, and the Alpha and Wrap complexes contained in the mosaics.},
  author       = {Edelsbrunner, Herbert and Nikitenko, Anton and Ölsböck, Katharina and Synak, Peter},
  booktitle    = {Topological Data Analysis},
  isbn         = {9783030434076},
  issn         = {2197-8549},
  pages        = {181--218},
  publisher    = {Springer Nature},
  title        = {{Radius functions on Poisson–Delaunay mosaics and related complexes experimentally}},
  doi          = {10.1007/978-3-030-43408-3_8},
  volume       = {15},
  year         = {2020},
}

@article{8384,
  abstract     = {Previous research on animations of soap bubbles, films, and foams largely focuses on the motion and geometric shape of the bubble surface. These works neglect the evolution of the bubble’s thickness, which is normally responsible for visual phenomena like surface vortices, Newton’s interference patterns, capillary waves, and deformation-dependent rupturing of films in a foam. In this paper, we model these natural phenomena by introducing the film thickness as a reduced degree of freedom in the Navier-Stokes equations and deriving their equations of motion. We discretize the equations on a nonmanifold triangle mesh surface and couple it to an existing bubble solver. In doing so, we also introduce an incompressible fluid solver for 2.5D films and a novel advection algorithm for convecting fields across non-manifold surface junctions. Our simulations enhance state-of-the-art bubble solvers with additional effects caused by convection, rippling, draining, and evaporation of the thin film.},
  author       = {Ishida, Sadashige and Synak, Peter and Narita, Fumiya and Hachisuka, Toshiya and Wojtan, Christopher J},
  issn         = {1557-7368},
  journal      = {ACM Transactions on Graphics},
  number       = {4},
  publisher    = {Association for Computing Machinery},
  title        = {{A model for soap film dynamics with evolving thickness}},
  doi          = {10.1145/3386569.3392405},
  volume       = {39},
  year         = {2020},
}