Step-and-project is a popular method to simulate non-penetrating deformable bodies in physically based animation. The strategy is to first integrate the system in time without considering contacts and then resolve potential intersections, striking a good balance between plausibility and efficiency. However, existing methods can be defective and unsafe when using large time steps, taking risks of failure or demanding repetitive collision testing and resolving that severely degrade performance. In this article, we propose a novel two-way method for fast and reliable continuous collision handling. Our method launches an optimization from both ends of the intermediate time-integrated state and the previous intersection-free state. It progressively generates a piecewise linear path and eventually obtains a feasible solution for the next time step. The algorithm efficiently alternates between a forward step and a backward step until the result is conditionally converged. Thanks to a set of unified volume-based contact constraints, our method offers flexible and reliable handling of various codimensional deformable bodies, including volumetric bodies, cloth, hair, and sand. Experimental results demonstrate the safety, robustness, physical fidelity, and numerical efficiency of our method, making it particularly suitable for scenarios involving large deformations or large time steps.

Abstract Piecewise linear inextensible Cosserat rods are usually represented by Cartesian coordinates of vertices and quaternions on the segments. Such representations use excessive degrees of freedom (DOFs), and need many additional constraints, which causes unnecessary numerical difficulties and computational burden for simulation. We propose a simple yet compact representation that exactly matches the intrinsic DOFs and naturally satisfies all such constraints. Specifically, viewing a rod as a chain of rigid segments, we encode its shape as the Cartesian coordinates of its root vertex, and use axis‐angle representation for the material frame on each segment. Under our representation, the Hessian of the implicit time‐stepping has special non‐zero patterns. Exploiting such specialties, we can solve the associated linear equations in nearly linear complexity. Furthermore, we carefully designed a preconditioner, which is proved to be always symmetric positive‐definite and accelerates the PCG solver in one or two orders of magnitude compared with the widely used block‐diagonal one. Compared with other technical choices including Super‐Helices, a specially designed compact representation for inextensible Cosserat rods, our method achieves better performance and stability, and can simulate an inextensible Cosserat rod with hundreds of vertices and tens of collisions in real time under relatively large time steps.

In this paper, an efficient and scalable approach for simulating inhomogeneous and non-linear elastic objects is introduced. Our numerical coarsening approach consists in optimizing non-conforming and matrix-valued shape functions to allow for predictive simulation of heterogeneous materials with non-linear constitutive laws even on coarse grids, thus saving orders of magnitude in computational time compared to traditional finite element computations. The set of local shape functions over coarse elements is carefully tailored in a preprocessing step to balance geometric continuity and local material stiffness. In particular, we do not impose continuity of our material-aware shape functions between neighboring elements to significantly reduce the fictitious numerical stiffness that conforming bases induce; however, we enforce crucial geometric and physical properties such as partition of unity and exact reproduction of representative fine displacements to eschew the use of discontinuous Galerkin methods. We demonstrate that we can simulate, with no parameter tuning, inhomogeneous and non-linear materials significantly better than previous approaches that traditionally try to homogenize the constitutive model instead.