CAREER: Theory and Practice of Space-Time Variational Integrators for Simulation and Animation

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The primary goal of this research project is to develop novel numerical simulation methods offering a wide range of applications, ranging from the simulation of smoke and fire interacting with moving objects, to the evolution of virus surfaces immersed in fluids. The approach leverages geometric understanding of the continuous equations of motion developed over the past century, which frames momentum and energy preservation as consequences of basic symmetries and invariance. However, current numerical techniques for simulation rarely capture the geometric principles that have been shown in mathematics and physics to be fundamental to proper dynamical behavior. To develop simulation algorithms that numerically respect both temporal and spatial structures of complex systems, a major part of this research requires the development of a novel framework to combine Lagrangian and Eulerian methods, two common (but rarely combined) representations of motion. Key applications include thin-shell simulation, collision handling, and fluid-structure interaction. The educational efforts leverage the appeal that graphical simulation demos have to students. The main objective of the educational activities is to motivate students, including underrepresented groups and K-12 students, to train not just in computer science, but also in physics, mathematics, and mechanical engineering to face the increasingly interdisciplinary nature of science.The resulting computational tools and practical applications (including, e.g., deformation model in biometrics, protein folding and docking, and groundwater simulation) will be disseminated through publications in journals and conferences. Project results can be found at the project website (http://www.cse.msu.edu/~ytong/SpaceTimeIntegrator.html).