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.

中文翻译：

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可变形固体的高阶时间积分

可变形实体的视觉吸引力和生动模拟代表了基于物理的计算机动画的一个重要方面。对于时间离散化，由于其简单性和鲁棒性，在计算机动画中通常使用一阶精确积分方法，例如 Backward Euler。尽管对二阶方法有显着的研究，但它们的使用并不广泛。许多这些众所周知的方法都有明显的缺点，例如严重的数值阻尼或场景相关的时间步长限制以确保稳定性。在本文中，我们讨论了计算机动画中对此类方法的最相关要求，并激发了超出一阶精度的兴趣。牢记这些要求，我们研究了来自对角隐式 Runge-Kutta (DIRK) 和 Rosenbrock 方法家族的几种有前途的方法，这些方法目前在该领域似乎并不流行。我们表明使用这种方法可以提高物理动画的视觉质量。此外，我们证明了与经典方法相比，它们可以以更低的计算成本对阻尼进行更多的控制。作为我们理论贡献的一部分，我们回顾了模拟的各个方面，这些方面通常被认为与高阶方法更复杂，例如接触处理。为此，我们基于预测器 - 校正器方法推导出隐式线性化接触模型，该模型导致与作为预测器的高阶积分器一致的行为。我们的接触模型非常适合模拟刚性、