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An Embedding Method for Interactive Simulation on Dynamic Surfaces

Stefan Auer
Technische Universitat Munchen
Technische Universitat Munchen, 2013

@article{auer2013embedding,

   title={An Embedding Method for Interactive Simulation on Dynamic Surfaces},

   author={Auer, Stefan},

   year={2013}

}

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Numerical simulation on curved surfaces enables the dynamic texturing of three-dimensional objects. In this thesis, I introduce methods for the realtime simulation and visualization of intrinsic fluid dynamics on deforming surfaces. These novel techniques support arbitrary, including open or nonorientable, surfaces and are universally applicable to a wide range of partial differential equation (PDE) problems involving differential operators up to the second order. The applications within this work focus particularly on interactive visual effects, however. All presented techniques utilize optimized parallel algorithms to employ the computing power and memory bandwidth of current GPUs in the best possible way. First, I show how the closest point method (CPM) can be applied to the real-time simulation of fluid effects on static surfaces. The CPM is an already existing embedding method that solves completely standard PDEs in the embedding space, using common numerical methods on a uniform Cartesian grid. A novel CUDA method enables the efficient conversation of a triangulated surface into the implicit representation required by the CPM. This implicit surface representation is stored together with the simulation attributes in a GPU-friendly multiblock grid, which facilitates the efficient solution of surface PDEs at high resolution. Yet, at the same time, this data structure is also very suitable for raycasting, which enables the visualization of simulation results as dynamic surface displacements. I demonstrate the capabilities of these techniques by simulating the propagation of fluid-like surface deformations, using either the wave equation or the incompressible Navier-Stokes equations. Second, I further develop the presented principles to a semi-Lagrangian closest point method (SLCPM) for the real-time simulation on animated surfaces. The method integrates the semi-Lagrangian scheme directly into the explicit closest point method and rigorously exploits the synergies. The SLCPM can be combined with a wide range of animation techniques for triangulated surfaces. It is unconditionally stable with respect to deformations of the surface and its precision does not depend on the input geometry. To the best of my knowledge, the SLCPM is the first Eulerian method for interactive fluid dynamics on deforming surfaces. Third, I present a novel contouring algorithm, which reconstructs a high quality triangle mesh from an implicit surface representation based on closest points. It selects the vertex positions directly from the set of closest points. Hence all triangle vertices are guaranteed to lie exactly on the zero-contour and no approximations are necessary. Since the vertex selection is guided by a CPM-based Laplacian analysis of the surface, the resulting contour preserves sharp features and small-scale details. A novel table-based triangulation scheme avoids small or degenerated triangles in smooth areas and enables the contouring of open and non-orientable surfaces. The contouring algorithm is fully integrated into the interactive SLCPM pipeline in order to enable future applications in the simulation of dynamic surface deformations.
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