The 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.
Over the course of this project we have developed a method for "skinning," or wrapping surfaces around, particle animations. Particles are often used in computer animations, but artists often wish to produce an image of a surface. Our technique (see supporting image 1) generates such surfaces, working on every frame independently without introducing any temporal artifacts or "flickering." Our technique has already been integrated into the toolset at a major animation studio. Additionally, we have made initial progress on the central problem described in our proposal: the development of a tightly coupled meshing and simulation system. We have identified the problem of fluid flows on surfaces as a good testbed for our approach and have nearly completed the simulation system and made progress with regard to automatic mesh generation. In particular, we have worked on automatic and semi-automatic methods for creating high-quality quad meshes from triangulated surfaces. In our work, we follow the premise that topological methods can be used to create quad meshes from the structure of a function on a given 2-manifold, where the function can be generated by a simulation or by a user. Our quadrangulation framework uses the new notion of Reeb atlas that captures the main features of an input shape, with user prescribed extraordinary vertices and alignment. Fine grain mesh tuning is easily achieved with the notion of connectivity texturing, which allows for additional extraordinary vertices specification and explicit feature alignment. Experiments demonstrate the interactivity and flexibility of our approach, as well as its ability to generate quad meshes of arbitrary resolution with high quality statistics (see supporting image 2). Through this project we have provided training to two graduate students and a post-doc. The graduate students have gained experience in performing research, asking open-ended questions, dealing with brittle research code, presenting papers, building on previous work, and education on advanced topics in simulation and computational topology. The post-doc has gained experience managing the research project and the graduate students. The work has resulted in six publications (four journal articles and two in conference proceedings), and the open-source release of our particle-skinning software. Our skinning software has already been integrated into a Maya plugin by a third party and is potentially in use by artists around the world, while our topological techniques are finding adoption in massive simulations for fundamental science such as clean combustion for energy production. Overall, our automated meshing techniques have the potential for adoption in simulation codes that are pervasive in computational sciences, engineering, and the fine arts.
