Many dynamical systems in nature are reasonably well-understood in terms of their underlying equations, but are very slow to simulate with detailed models. For example, the Finite Element Method (FEM) has been used extensively to simulate human tissue, automobile and airplane mechanical components, architectural structures, or characters in a computer game. Real-world systems are, however, very complex, which normally requires large computer processing power for their simulation, control and optimization. This research investigates systematic approaches to approximate complex physics with simple, yet mathematically principled models. The resulting fast simulation and control can make medical training more immersive, computer games more entertaining, and CAD/CAM faster and more reliable. More generally, this work applies to any system governed by differential equations, with broader applications in robotics, aeronautics, and defense systems. In addition to developing new publicly available coursework material, educational activities include releasing a large C++ computer graphics/animation codebase to the world under an open source license, and visits to high schools in under-deserved areas of Los Angeles where students are exposed to the benefits of careers in science and engineering.
The investigators tackle material and geometric complexity using model reduction, an approach where full equations of motion are approximated by a projection to a properly selected low-dimensional space. While model reduction of nonlinear systems has been previously employed in computer graphics and other disciplines, the existing algorithms often lack flexibility, are not adaptive in space or time, and can only provide smooth, low-dimensional output, even if the unreduced output is known to be complex. The investigators study how to overcome these limitations, using techniques from Lagrange mechanics, multi-resolution analysis and nonlinear optimization. The ultimate goal is to accelerate physics to the point where fusion of physics and design becomes possible for large, complex systems of computer graphics and engineering practice.
