Canopy-frame interactions for umbrella simulation

We resolve the canopy-frame interactions for umbrella simulation in this paper. An umbrella is a heterogeneous articulated system that consists of a frame composed of elastic ribs and other rigid components, and a fabric-made canopy bound to the frame. The canopy interacts with the frame through the binding constraint and collisions. Since the flexible canopy and elastic ribs have different stiffness, we employ a bi-time-step scheme to obtain tractable performance. It evolves the frame with a smaller time step but the canopy with a larger time step. We resolve the canopy-frame interactions by changing the canopy positions and velocities at an interval of the larger time step, which is limited to be an integer multiple of the smaller time step. These displacements and velocity changes are transferred into opposite impulses and forces, and applied to the frame in the following smaller time steps. With these transformations, the solution of the canopy-frame interactions is independent of the difference of the two time steps. Moreover, we adopt a projection-based algorithm to accurately detect the canopy-frame collisions. We demonstrate the effectiveness of our method with several umbrella animations. Crown