Higher‐Order Time Integration for Deformable Solids

Abstract

Visually appealing and vivid simulations of deformable solids represent an important aspect of physically based computer animation. For the temporal discretization, it is customary in computer animation to use first‐order accurate integration methods, such as Backward Euler, due to their simplicity and robustness. Although there is notable research on second‐order methods, their use is not widespread. Many of these well‐known methods have significant drawbacks such as severe numerical damping or scene‐dependent time step restrictions to ensure stability. In this paper, we discuss the most relevant requirements on such methods in computer animation and motivate the interest beyond first‐order accuracy. Keeping these requirements in mind, we investigate several promising methods from the families of diagonally implicit Runge‐Kutta (DIRK) and Rosenbrock methods which currently do not appear to have considerable popularity in this field. We show that the usage of such methods improves the visual quality of physical animations. In addition, we demonstrate that they allow distinctly more control over damping at lower computational cost than classical methods. As part of our theoretical contribution, we review aspects of simulations that are often considered more intricate with higher‐order methods, such as contact handling. To this end, we derive an implicit linearized contact model based on a predictor‐corrector approach that leads to consistent behavior with higher‐order integrators as predictors. Our contact model is well suited for the simulation of stiff, nonlinear materials with the integration methods presented in this paper and more common methods such as Backward Euler alike.

Simulating deformable objects for computer animation: A numerical perspective

“…Soft objects, such as the characters that appear in animation movies aimed at young audiences, commonly arise in computer graphics applications. Here, the DIRK methods introduced in Section 3.1, namely TR-BDF2 and SDIRK, have a good record [48,32], even though occasionally an additional stabilization or damping may be required.…”

“…This method performed well in comparative experiments reported in [32]. It has two implicit stages, each requiring the solution of a nonlinear system of size n. S-versions for these are derived directly as before.…”

“…In this section we mention and discuss several favourite time discretization methods that have seen use in practice. See also the recent [32].…”

“…Plot of the smoothing function s = s(x; ) for different values of . It is used in (34) through(32) to approximate the Coulomb friction force.…”

Simulating deformable objects for computer animation: a numerical perspective

“…In this section we mention and discuss several favourite discretization methods that have seen use in practice. See also the recent [30].…”