Nonlinear hyperelastic energies play a key role in capturing the fleshy appearance of virtual characters. Real-world, volume-preserving biological tissues have Poisson's ratios near 1 /2, but numerical simulation within this regime is notoriously challenging. In order to robustly capture these visual characteristics, we present a novel version of Neo-Hookean elasticity. Our model maintains the fleshy appearance of the Neo-Hookean model, exhibits superior volume preservation, and is robust to extreme kinematic rotations and inversions. We obtain closed-form expressions for the eigenvalues and eigenvectors of all of the system's components, which allows us to directly project the Hessian to semipositive definiteness, and also leads to insights into the numerical behavior of the material. These findings also inform the design of more sophisticated hyperelastic models, which we explore by applying our analysis to Fung and Arruda-Boyce elasticity. We provide extensive comparisons against existing material models.

Many strategies exist for optimizing non-linear distortion energies in geometry and physics applications, but devising an approach that achieves the convergence promised by Newton-type methods remains challenging. In order to guarantee the positive semi-definiteness required by these methods, a numerical eigendecomposition or approximate regularization is usually needed. In this article, we present analytic expressions for the eigensystems at each quadrature point of a wide range of isotropic distortion energies. These systems can then be used to project energy Hessians to positive semi-definiteness analytically. Unlike previous attempts, our formulation provides compact expressions that are valid both in 2D and 3D, and does not introduce spurious degeneracies. At its core, our approach utilizes the invariants of the stretch tensor that arises from the polar decomposition of the deformation gradient. We provide closed-form expressions for the eigensystems for all these invariants, and use them to systematically derive the eigensystems of any isotropic energy. Our results are suitable for geometry optimization over flat surfaces or volumes, and agnostic to both the choice of discretization and basis function. To demonstrate the efficiency of our approach, we include comparisons against existing methods on common graphics tasks such as surface parameterization and volume deformation.

Figure 1. A prescribed particle slowly moves through a set of curtains, then impulsively shifts to a very high velocity. The slow and fast phases highlight the method's ability to handle smooth resting and sliding with deep stacking, and arbitrarily fast penetration-free movements in which collisions are treated when (as opposed to well before or after) they occur. The curtains continue to swing for a long time, even as controlled internal dissipation damps high frequencies. AbstractWe develop a method for reliable simulation of elastica in complex contact scenarios. Our focus is on firmly establishing three parameter-independent guarantees: that simulations of wellposed problems (a) have no interpenetrations, (b) obey causality, momentum-and energy-conservation laws, and (c) complete in finite time. We achieve these guarantees through a novel synthesis of asynchronous variational integrators, kinetic data structures, and a discretization of the contact barrier potential by an infinite sum of nested quadratic potentials. In a series of two-and threedimensional examples, we illustrate that this method more easily handles challenging problems involving complex contact geometries, sharp features, and sliding during extremely tight contact. MotivationEven as computer hardware benefits from Moore's Law, our ability to program, debug, and maintain software advances at a humbler pace. This observation shapes our priorities as we develop physical simulation tools for computer graphics. While making choices that yield up-front simplicity and blazing performance is important today, we prefer that these choices do not obstruct our long-term goals of extending functionality and improving realism. Laying aside ad-hoc models in favor of physical approaches might require a deeper initial investment, but it promises to pay off handsomely in predictability, controllability, and extensibility. From this vantage point, we propose to revisit the long-studied problem of simulating deformable objects in complex contact scenarios.Safety, correctness, progress Robust simulation of complex contact scenarios is critical to applications spanning graphics (training, virtual worlds, entertainment) and engineering (product design, safety analysis, experimental validation). Challenging scenarios involve dynamics with frequent and distributed points of contact, interaction with sharp boundaries, resting and sliding contact, and combinations thereof. Useful resolution of these scenarios requires consideration of the fundamental issues of geometric safety, physical correctness, and computational progress, with the respective meanings that (a) for well-posed problems the simulation does not enter an invalid (interpenetrating) state, (b) collision response obeys physical laws of causality and conservation (of mass, momentum, energy, etc.), and (c) the algorithm completes a simulation in finite, preferrably short, time.An ideal algorithm offers provable guarantees of safety, correctness, and progress. A safety guarantee eliminates the need to ite...

Resolving simultaneous impacts is an open and significant problem in collision response modeling. Existing algorithms in this domain fail to fulfill at least one of five physical desiderata. To address this we present a simple generalized impact model motivated by both the successes and pitfalls of two popular approaches: pair-wise propagation and linear complementarity models. Our algorithm is the first to satisfy all identified desiderata, including simultaneously guaranteeing symmetry preservation, kinetic energy conservation, and allowing break-away. Furthermore, we address the associated problem of inelastic collapse, proposing a complementary generalized restitution model that eliminates this source of nontermination. We then consider the application of our models to the synchronous time-integration of large-scale assemblies of impacting rigid bodies. To enable such simulations we formulate a consistent frictional impact model that continues to satisfy the desiderata. Finally, we validate our proposed algorithm by correctly capturing the observed characteristics of physical experiments including the phenomenon of extended patterns in vertically oscillated granular materials.

We develop an algorithm for the efficient and stable simulation of large-scale elastic rod assemblies. We observe that the time-integration step is severely restricted by a strong nonlinearity in the response of stretching modes to transversal impact, the degree of this nonlinearity varying greatly with the shape of the rod. Building on these observations, we propose a collision response algorithm that adapts its degree of nonlinearity. We illustrate the advantages of the resulting algorithm by analyzing simulations involving elastic rod assemblies of varying density and scale, with up to 1.7 million individual contacts per time step.