Physics-based simulation methods for deformable objects suffer limitations due to the conflicting requirements that are placed on them. The work horse semi-implicit (SI) backward Euler method is very stable and inexpensive, but it is also a blunt instrument. It applies heavy damping, which depends on the timestep, to all solution modes and not just to high-frequency ones. As such, it makes simulations less lively, potentially missing important animation details. At the other end of the scale, exponential methods (exponential Rosenbrock Euler (ERE)) are known to deliver good approximations to all modes, but they get prohibitively expensive and less stable for very stiff material. In this article, we devise a hybrid, semi-implicit method called SIERE that allows the previous methods SI and ERE to each perform what they are good at. To do this, we employ at each timestep a partial spectral decomposition, which picks the lower, leading modes, applying ERE in the corresponding subspace. The rest is handled (i.e., effectively damped out) by SI. No solution of nonlinear algebraic equations is required throughout the algorithm. We show that the resulting method produces simulations that are visually as good as those of the exponential method at a computational price that does not increase with stiffness, while displaying stability and damping with respect to the high-frequency modes. Furthermore, the phenomenon of occasional divergence of SI is avoided.

中文翻译：

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赛尔

用于可变形物体的基于物理的模拟方法由于对其提出的相互冲突的要求而受到限制。工作马半隐式（SI）反向欧拉方法非常稳定且成本低廉，但它也是一种钝器。它将取决于时间步长的重阻尼应用于所有求解模式，而不仅仅是高频模式。因此，它会使模拟变得不那么生动，可能会丢失重要的动画细节。在规模的另一端，已知指数方法（指数罗森布洛克欧拉 (ERE)）可以为所有模式提供良好的近似，但对于非常坚硬的材料，它们变得过于昂贵且不太稳定。在本文中，我们设计了一种称为 SIERE 的混合半隐式方法，它允许之前的方法 SI 和 ERE 分别执行它们擅长的操作。去做这个，我们在每个时间步都采用部分谱分解，它选择较低的领先模式，在相应的子空间中应用 ERE。其余部分由 SI 处理（即有效衰减）。整个算法不需要非线性代数方程的解。我们表明，所得方法产生的模拟在视觉上与指数方法的模拟一样好，计算成本不随刚度增加，同时显示了高频模式的稳定性和阻尼。此外，避免了SI偶尔发散的现象。整个算法不需要非线性代数方程的解。我们表明，所得方法产生的模拟在视觉上与指数方法的模拟一样好，计算成本不随刚度增加，同时显示了高频模式的稳定性和阻尼。此外，避免了SI偶尔发散的现象。整个算法不需要非线性代数方程的解。我们表明，所得方法产生的模拟在视觉上与指数方法的模拟一样好，计算成本不随刚度增加，同时显示了高频模式的稳定性和阻尼。此外，避免了SI偶尔发散的现象。