Computational Methods for Simulating Flows and Structures with Complex Dynamic Geometry

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The intricate splashing of water and the kneading of bread dough are familiar examples of severe deformations undergone by liquids and solids. The ubiquity of such processes, involving rapid, dramatic, and frequent changes in a material's shape, has led to tremendous demand for computer simulation techniques that can seamlessly reproduce them. Such algorithms will be necessary for the creation of truly immersive, physically realistic virtual environments, a longstanding grand challenge of computer graphics research. To be practically useful, these algorithms must be at once robust to difficult inputs, physically accurate, and computationally efficient. Computational meshes, sometimes called wireframe models, are concise representations of digital objects and materials that have been prevalent in computer graphics for decades. Despite their many well-studied benefits, limitations of existing algorithms have prevented their widespread use in animating materials undergoing radical shape changes (i.e., changing geometry and especially topology). For example, mesh-based approaches have often failed to completely handle key sources of error, such as numerical imprecision, tangled intersections, poor quality meshes, and complex mesh edits; this dearth of strong mathematical guarantees leads inevitably to fragile algorithms that can crash unpredictably. Fortunately, recent advances have brought practical mesh-based animation of severe, geometrically complex deformations closer to being realized. The three themes of this research capitalize on this opportunity by: (I) developing improved methods for dynamically evolving triangulated surfaces; (II) investigating fluid and solid simulation techniques that exploit new tools for generating arbitrary polyhedral cut-cell meshes, and (III) deriving novel mesh-based techniques to animate materials composed of complex branching structures or mixed-dimensional components (solids, surfaces, curves), known as non-manifold geometry. Through this research a diverse student team will receive interdisciplinary training spanning fluid and solid mechanics, computational mathematics, and computer graphics. This unique skill set is highly valuable both in digital media (i.e., virtual and augmented reality, visual effects, game development, advertising, etc.) and many areas of engineering and applied mathematics. The proposed agenda will yield open-source software tools to simulate important fluid and solid phenomena with greater accuracy, robustness, and efficiency than is currently available, immediately providing faster and more realistic animation capabilities to the digital media sector in Canada and globally. Furthermore, because of the fundamental nature of the geometric and physical techniques to be developed, it will also provide a firm foundation for future computational investigations in the countless disciplines where highly deforming materials arise, from biophysics to 3D printing to geophysics and more.