Position based dynamics is a powerful technique for simulating a variety of materials. Its primary strength is its robustness when run with limited computational budget. We develop a novel approach to address problems with PBD for quasistatic hyperelastic materials. Even though PBD is based on the projection of static constraints, PBD is best suited for dynamic simulations. This is particularly relevant since the efficient creation of large data sets of plausible, but not necessarily accurate elastic equilibria is of increasing importance with the emergence of quasistatic neural networks. Furthermore, PBD projects one constraint at a time. We show that ignoring the effects of neighboring constraints limits its convergence and stability properties. Recent works have shown that PBD can be related to the Gauss-Seidel approximation of a Lagrange multiplier formulation of backward Euler time stepping, where each constraint is solved/projected independently of the others in an iterative fashion. We show that a position-based, rather than constraint-based nonlinear Gauss-Seidel approach solves these problems. Our approach retains the essential PBD feature of stable behavior with constrained computational budgets, but also allows for convergent behavior with expanded budgets. We demonstrate the efficacy of our method on a variety of representative hyperelastic problems and show that both successive over relaxation (SOR) and Chebyshev acceleration can be easily applied.

翻译：位置动力学是一种模拟多种材料的强大技术，其核心优势在于有限计算资源下的鲁棒性。针对准静态超弹性材料的位置动力学问题，我们提出了一种新方法。尽管位置动力学基于静态约束的投影，但其更适用于动态模拟。随着准静态神经网络的发展，高效生成大量合理但未必精确的弹性平衡数据集日益重要，这使得该问题尤为关键。此外，位置动力学每次仅投影一个约束。研究表明，忽略相邻约束效应会限制其收敛性与稳定性。近期工作指出，位置动力学可关联于后向欧拉时间步进拉格朗日乘子形式的高斯-赛德尔近似，其中每个约束独立迭代求解/投影。我们证明，基于位置而非约束的非线性高斯-赛德尔方法可解决上述问题。该方法既保留了位置动力学在有限计算资源下保持稳定行为的核心特性，又能在扩展计算资源时实现收敛行为。通过在多种代表性超弹性问题上的验证，我们展示了该方法的高效性，并证明逐次超松弛加速与切比雪夫加速均可简易应用。