Abstract-We propose a novel technique that allows one to conserve energy using the time integration scheme of one's choice. Traditionally, the time integration methods that deal with energy conservation, such as symplectic, geometric, and variational integrators, have aimed to include damping in a manner independent of the size of the time step, stating that this gives more control over the look and feel of the simulation. Generally speaking, damping adds to the overall aesthetics and appeal of a numerical simulation, especially since it damps out the high frequency oscillations that occur on the level of the discretization mesh. We propose an alternative technique that allows one to use damping as a material parameter to obtain the desired look and feel of a numerical simulation, while still exactly conserving the total energy -in stark contrast to previous methods in which adding damping effects necessarily removes energy from the mesh. This allows, for example, a deformable bouncing ball with aesthetically pleasing damping (and even undergoing collision) to collide with the ground and return to its original height exactly conserving energy, as shown in Figure 2. Furthermore, since our method works with any time integration scheme, the user can choose their favorite time integration method with regards to aesthetics and simply apply our method as a post-process to conserve all or as much of the energy as desired.

We propose a novel framework for simulating reduced deformable bodies that fully accounts for linear and angular momentum conservation even in the presence of collision, contact, articulation, and other desirable effects. This was motivated by the observation that the mere excitation of a single mode in a reduced degree of freedom model can adversely change the linear and angular momentum. Although unexpected changes in linear momentum can be avoided during basis construction, adverse changes in angular momentum appear unavoidable, and thus we propose a robust framework that includes the ability to compensate for them. Enabled by this ability to fully account for linear and angular momentum, we introduce an impulse-based formulation that allows us to precisely control the velocity of any node in spite of the fact that we only have access to a lower-dimensional set of degrees of freedom. This allows us to model collision, contact, and articulation in a robust and high visual fidelity manner, especially when compared to penalty-based forces that merely aim to coerce local velocities. In addition, we propose a new "deformable bones" framework wherein we leverage standard skinning technology for "bones," "bone" placement, blending operations, etc. even though each of our "deformable bones" is a fully simulated reduced deformable model.