EAGER (G&V): Exploring Morse Theoretic Tools for Automatic Mesh Generation and Simulation on Surfaces

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AbstractThe simulation of realistic physical phenomena, such as fluid interactions and deformable bodies, has become an indispensable part of both computational physics and computer animation. Such simulations produce stunning visual effects for the entertainment industry but also lead to new discoveries in diverse fields, such as astrophysics, energy production, or understanding the global climate. However, prior to running these simulations on a computer, the mathematical representation of the domain must be discretized in order to minimize computational errors (i.e., to obtain accurate physical results). The increased resolution of modern simulations is making this an increasingly important issue, especially for simulations requiring periodic remeshing or necessitating a fully automated approach.Current practice in the coupling of discretization and computation has significant weaknesses. The computational tools often demand specific element shapes, e.g. hexahedra, over-constraining the discretization. On the other hand, meshing quality is generally measured by geometric quantities that provide only a limited connection to overall simulation performance. This research is demonstrating a new approach. Theoretical mathematics is used to develop, for the first time, a discretization scheme that is explicitly dependent on the structure of scalar fields generated by the simulation. The key insight is that a topological structure, the Morse-Smale (MS) complex, acts as a natural quadrilateral decomposition of a domain based on a given scalar field, called a background function. The background function behaves as a mechanism for encoding key information from a simulation. The MS complex then acts as a coarse mesh that coincides geometrically with the input domain while aligning itself with simulation properties. Finally, through optimization and subdivision, fine-grained meshes are generated that adapt locally to the resolution needed. This produces a discretization that more accurately follows the target simulations using fewer elements.