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.

步进投影是一种在基于物理的动画中模拟非穿透变形体的流行方法。该策略是首先在不考虑接触的情况下及时整合系统，然后解决潜在的交叉点，在合理性和效率之间取得良好的平衡。然而，当使用大时间步长、冒着失败风险或要求重复碰撞测试并解决严重降低性能的问题时，现有方法可能存在缺陷且不安全。在本文中，我们提出了一种新颖的双向方法，用于快速可靠的连续碰撞处理。我们的方法从中间时间积分状态和之前的无交集状态的两端发起优化。它逐步生成分段线性路径，并最终获得下一个时间步的可行解。该算法有效地在前向步骤和后向步骤之间交替，直到结果有条件地收敛。得益于一组统一的基于体积的接触约束，我们的方法可以灵活可靠地处理各种二维可变形体，包括体积体、布料、头发和沙子。实验结果证明了我们的方法的安全性、鲁棒性、物理保真度和数值效率，使其特别适合涉及大变形或大时间步长的场景。和沙子。实验结果证明了我们的方法的安全性、鲁棒性、物理保真度和数值效率，使其特别适合涉及大变形或大时间步长的场景。和沙子。实验结果证明了我们的方法的安全性、鲁棒性、物理保真度和数值效率，使其特别适合涉及大变形或大时间步长的场景。