Discrete geometric dynamics and artistic control of curves and surfaces
Eitan Grinspun, Miklós Bergou · 2010
Physical simulations are capable of highly realistic and often very accurate reproductions of the true systems that they are modeling. However, the computational cost of a simulation can often limit its usefulness in contexts where not just the accuracy of the results for a single set of parameters but also the promptness of displaying the result to the user is of interest. This type of scenario can often arise not only in real-time applications, such as in virtual environments, training simulators, or games for example, but also in interactive design contexts in which the user is trying to guide the system toward some desired goal. We explore two avenues toward achieving efficient, controllable simulations. We first consider how to ensure that a physical simulation matches a user-prescribed input for the problem of simulating thin, flexible objects such as cloth. Within an iterative design framework, this lets the user run lower resolution preview simulations with a much faster turn-around time than a full simulation and guarantees that the final, high-resolution output matches the previews. In addition, we demonstrate that this system can be applied not just to physically simulated motion, but also to hand-animated motion. This allows the artist to focus on animating the expressive elements of the motion without having to worry about manually creating details such as characteristic wrinkles and folds of cloth. In the second avenue, we apply ideas from Discrete Differential Geometry (DDG) to the problem of deriving principled, efficient computational models for simulating the dynamics of physical systems. The main mandate of DDG is to identify the key geometric structures underlying a system and to preserve these structures when building a discrete model. We build on this foundation within the context of physical simulation to arrive at discrete geometric models for dynamics, and we show the benefit of constructing the discrete models in this geometric manner. We first show that leveraging the assumption of isometric deformations leads naturally to an expression for the bending energy of thin plates that is quadratic in the position of the surface. The associated linear gradient and constant Hessian provide an efficient model for simulating cloth, which we compare to the full, nonlinear model. We next apply this line of thinking to simulating long, thin objects such as elastic rods or viscous threads. By focusing on the geometry of the model and drawing on the parallels to the smooth theory, we arrive at all efficient, unified model for simulating these types of objects, which we validate against a number of analytic and experimental results.