Asynchronous Evolution for Fully-Implicit and Semi-Implicit Time Integration

We propose a series of techniques for hybridizing implicit and semi-implicit time integration methods in a manner that retains much of the speed of the implicit method without sacrificing all of the higher quality vibrations one obtains with methods that handle elastic forces explicitly. We propose our scheme in the context of asynchronous methods, where different parts of the mesh are evolved at different time steps. Whereas traditional asynchronous methods evolve each element independently, we partition all of our elements into two groups: one group evolved at the frame rate using a fully implicit scheme, and another group which takes a number of substeps per frame using a scheme that is implicit on damping forces and explicit on the elastic forces. This allows for a straightforward coupling between the implicit and semi-implicit methods at frame boundaries for added stability. As has been stressed by various authors, asynchronous schemes take some of the pressure off of mesh generation, allowing time evolution to remain efficient even in the face of sliver elements. Finally, we propose a force distributing projection method which allows one to redistribute the forces felt on boundaries between implicit and semi-implicit regions of the mesh in a manner that yields improved visual fidelity.