@phdthesis{20551,
  abstract     = {The space of codimension-2 shapes, such as curves in 3D and surfaces in 4D, is an infinite-dimensional manifold. This thesis explores geometric structures and dynamics on this space, with emphasis on their implications for physics, particularly hydrodynamics.

Our investigation ranges from theoretical studies of infinite-dimensional symplectic and prequantum geometry to numerical computation of the time evolution of shapes. The thesis presents four main contributions.

In the first part, we introduce implicit representations of codimension-2 shapes using a class of complex-valued functions, and prove that the space of these implicit representations forms a prequantum bundle over the codimension-2 shape space. This reveals a new geometric interpretation of the canonical symplectic structure on the codimension-2 shape space.

In the second part, we use implicit representations to develop a simulation method for the dynamics of space curves. To handle chaotic systems such as vortex filaments in hydrodynamics, we exploit the infinite degrees of freedom, hidden in both the configuration and dynamics of implicit representations.

In the third part, we introduce new symplectic structures on the space of space curves, which generalize the only previously known symplectic structure on this space, allowing for new Hamiltonian dynamics of space curves.

In the fourth part, we apply a symplectic viewpoint to a differential geometric problem with practical applications. We derive a new area formula for spherical polygons via prequantization. },
  author       = {Ishida, Sadashige},
  isbn         = {978-3-99078-070-1},
  issn         = {2663-337X},
  pages        = {141},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{Symplectic-prequantum structures and dynamics on the codimension-2 shape space}},
  doi          = {10.15479/AT-ISTA-20551},
  year         = {2025},
}

@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},
}

@phdthesis{18301,
  abstract     = {Physics simulation in computer graphics can bring triangle meshes into topologically invalid states. The method in this thesis contributed to Heiss-Synak* and Kalinov* et al. [2024] who devised a non-manifold hybrid surface tracker—a surface tracker that repairs explicit non-manifold triangle meshes with the help of the implicit domain. Specifically, this thesis provides an algorithm for filling the holes that are left after removing problematic parts of the mesh.},
  author       = {Etemadihaghighi, Arian},
  issn         = {2791-4585},
  keywords     = {surface tracking, non-manifold, hole-filling, topology change, multi-material, solid-modeling},
  pages        = {39},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{Filling the holes of non-manifold self-intersecting meshes for implicit topology changes in surface tracking}},
  doi          = {10.15479/at:ista:18301},
  year         = {2024},
}

@phdthesis{12358,
  abstract     = {The complex yarn structure of knitted and woven fabrics gives rise to both a mechanical and
visual complexity. The small-scale interactions of yarns colliding with and pulling on each
other result in drastically different large-scale stretching and bending behavior, introducing
anisotropy, curling, and more. While simulating cloth as individual yarns can reproduce this
complexity and match the quality of real fabric, it may be too computationally expensive for
large fabrics. On the other hand, continuum-based approaches do not need to discretize the
cloth at a stitch-level, but it is non-trivial to find a material model that would replicate the
large-scale behavior of yarn fabrics, and they discard the intricate visual detail. In this thesis,
we discuss three methods to try and bridge the gap between small-scale and large-scale yarn
mechanics using numerical homogenization: fitting a continuum model to periodic yarn simulations, adding mechanics-aware yarn detail onto thin-shell simulations, and quantitatively
fitting yarn parameters to physical measurements of real fabric.
To start, we present a method for animating yarn-level cloth effects using a thin-shell solver.
We first use a large number of periodic yarn-level simulations to build a model of the potential
energy density of the cloth, and then use it to compute forces in a thin-shell simulator. The
resulting simulations faithfully reproduce expected effects like the stiffening of woven fabrics
and the highly deformable nature and anisotropy of knitted fabrics at a fraction of the cost of
full yarn-level simulation.
While our thin-shell simulations are able to capture large-scale yarn mechanics, they lack
the rich visual detail of yarn-level simulations. Therefore, we propose a method to animate
yarn-level cloth geometry on top of an underlying deforming mesh in a mechanics-aware
fashion in real time. Using triangle strains to interpolate precomputed yarn geometry, we are
able to reproduce effects such as knit loops tightening under stretching at negligible cost.
Finally, we introduce a methodology for inverse-modeling of yarn-level mechanics of cloth,
based on the mechanical response of fabrics in the real world. We compile a database from
physical tests of several knitted fabrics used in the textile industry spanning diverse physical
properties like stiffness, nonlinearity, and anisotropy. We then develop a system for approximating these mechanical responses with yarn-level cloth simulation, using homogenized
shell models to speed up computation and adding some small-but-necessary extensions to
yarn-level models used in computer graphics.
},
  author       = {Sperl, Georg},
  isbn         = {978-3-99078-020-6},
  issn         = {2663-337X},
  pages        = {138},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{Homogenizing yarn simulations: Large-scale mechanics, small-scale detail, and quantitative fitting}},
  doi          = {10.15479/at:ista:12103},
  year         = {2022},
}

@phdthesis{839,
  abstract     = {This thesis describes a brittle fracture simulation method for visual effects applications. Building upon a symmetric Galerkin boundary element method, we first compute stress intensity factors following the theory of linear elastic fracture mechanics. We then use these stress intensities to simulate the motion of a propagating crack front at a significantly higher resolution than the overall deformation of the breaking object. Allowing for spatial variations of the material's toughness during crack propagation produces visually realistic, highly-detailed fracture surfaces. Furthermore, we introduce approximations for stress intensities and crack opening displacements, resulting in both practical speed-up and theoretically superior runtime complexity compared to previous methods. While we choose a quasi-static approach to fracture mechanics, ignoring dynamic deformations, we also couple our fracture simulation framework to a standard rigid-body dynamics solver, enabling visual effects artists to simulate both large scale motion, as well as fracturing due to collision forces in a combined system. As fractures inside of an object grow, their geometry must be represented both in the coarse boundary element mesh, as well as at the desired fine output resolution. Using a boundary element method, we avoid complicated volumetric meshing operations. Instead we describe a simple set of surface meshing operations that allow us to progressively add cracks to the mesh of an object and still re-use all previously computed entries of the linear boundary element system matrix. On the high resolution level, we opt for an implicit surface representation. We then describe how to capture fracture surfaces during crack propagation, as well as separate the individual fragments resulting from the fracture process, based on this implicit representation. We show results obtained with our method, either solving the full boundary element system in every time step, or alternatively using our fast approximations. These results demonstrate that both of these methods perform well in basic test cases and produce realistic fracture surfaces. Furthermore we show that our fast approximations substantially out-perform the standard approach in more demanding scenarios. Finally, these two methods naturally combine, using the full solution while the problem size is manageably small and switching to the fast approximations later on. The resulting hybrid method gives the user a direct way to choose between speed and accuracy of the simulation. },
  author       = {Hahn, David},
  issn         = {2663-337X},
  pages        = {124},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{Brittle fracture simulation with boundary elements for computer graphics}},
  doi          = {10.15479/AT:ISTA:th_855},
  year         = {2017},
}

@phdthesis{1122,
  abstract     = {Computer graphics is an extremely exciting field for two reasons. On the one hand,
there is a healthy injection of pragmatism coming from the visual effects industry
that want robust algorithms that work so they can produce results at an increasingly
frantic pace. On the other hand, they must always try to push the envelope and
achieve the impossible to wow their audiences in the next blockbuster, which means
that the industry has not succumb to conservatism, and there is plenty of room to
try out new and crazy ideas if there is a chance that it will pan into something
useful.
Water simulation has been in visual effects for decades, however it still remains
extremely challenging because of its high computational cost and difficult artdirectability.
The work in this thesis tries to address some of these difficulties.
Specifically, we make the following three novel contributions to the state-of-the-art
in water simulation for visual effects.
First, we develop the first algorithm that can convert any sequence of closed
surfaces in time into a moving triangle mesh. State-of-the-art methods at the time
could only handle surfaces with fixed connectivity, but we are the first to be able to
handle surfaces that merge and split apart. This is important for water simulation
practitioners, because it allows them to convert splashy water surfaces extracted
from particles or simulated using grid-based level sets into triangle meshes that can
be either textured and enhanced with extra surface dynamics as a post-process.
We also apply our algorithm to other phenomena that merge and split apart, such
as morphs and noisy reconstructions of human performances.
Second, we formulate a surface-based energy that measures the deviation of a
water surface froma physically valid state. Such discrepancies arise when there is a
mismatch in the degrees of freedom between the water surface and the underlying
physics solver. This commonly happens when practitioners use a moving triangle
mesh with a grid-based physics solver, or when high-resolution grid-based surfaces
are combined with low-resolution physics. Following the direction of steepest
descent on our surface-based energy, we can either smooth these artifacts or turn
them into high-resolution waves by interpreting the energy as a physical potential.
Third, we extend state-of-the-art techniques in non-reflecting boundaries to handle spatially and time-varying background flows. This allows a novel new
workflow where practitioners can re-simulate part of an existing simulation, such
as removing a solid obstacle, adding a new splash or locally changing the resolution.
Such changes can easily lead to new waves in the re-simulated region that would
reflect off of the new simulation boundary, effectively ruining the illusion of a
seamless simulation boundary between the existing and new simulations. Our
non-reflecting boundaries makes sure that such waves are absorbed.},
  author       = {Bojsen-Hansen, Morten},
  issn         = {2663-337X},
  pages        = {114},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{Tracking, correcting and absorbing water surface waves}},
  doi          = {10.15479/AT:ISTA:th_640},
  year         = {2016},
}