We present a unification of the two main approaches to simulate deformable solids, namely elasticity and constraints. Elasticity accurately handles soft to moderately stiff objects, but becomes numerically hard as stiffness increases. Constraints efficiently handle high stiffness, but when integrated in time they can suffer from instabilities in the nullspace directions, generating spurious transverse vibrations when pulling hard on thin inextensible objects or articulated rigid bodies. We show that geometric stiffness, the tensor encoding the change of force directions (as opposed to intensities) in response to a change of positions, is the missing piece between the two approaches. This previously neglected stiffness term is easy to implement and dramatically improves the stability of inextensible objects and articulated chains, without adding artificial bending forces. This allows time step increases up to several orders of magnitude using standard linear solvers.

In this paper we introduce a new approach for the embedding of linear elastic deformable models. Our technique results in significant improvements in the efficient physically based simulation of highly detailed objects. First, our embedding takes into account topological details, that is, disconnected parts that fall into the same coarse element are simulated independently. Second, we account for the varying material properties by computing stiffness and interpolation functions for coarse elements which accurately approximate the behaviour of the embedded material. Finally, we also take into account empty space in the coarse embeddings, which provides a better simulation of the boundary. The result is a straightforward approach to simulating complex deformable models with the ease and speed associated with a coarse regular embedding, and with a quality of detail that would only be possible at much finer resolution.

International audience