Volume-preserving hyperelastic materials are widely used to model near-incompressible materials such as rubber and soft tissues. However, the numerical simulation of volume-preserving hyperelastic materials is notoriously challenging within this regime due to the non-convexity of the energy function. In this work, we identify the pitfalls of the popular eigenvalue clamping strategy for projecting Hessian matrices to positive semi-definiteness during Newton's method. We introduce a novel eigenvalue filtering strategy for projected Newton's method to stabilize the optimization of Neo-Hookean energy and other volume-preserving variants under high Poisson's ratio (near 0.5) and large initial volume change. Our method only requires a single line of code change in the existing projected Newton framework, while achieving significant improvement in both stability and convergence speed. We demonstrate the effectiveness and efficiency of our eigenvalue projection scheme on a variety of challenging examples and over different deformations on a large dataset.

翻译：体积守恒超弹性材料被广泛用于模拟接近不可压缩的材料，如橡胶和软组织。然而，由于能量函数的非凸性，在此范围内对体积守恒超弹性材料进行数值模拟极具挑战性。在本工作中，我们揭示了牛顿法中流行的特征值钳制策略在将Hessian矩阵投影为正半定时的缺陷。我们提出了一种用于投影牛顿法的新型特征值滤波策略，以在高泊松比（接近0.5）和较大初始体积变化下稳定Neo-Hookean能量及其体积守恒变体的优化。我们的方法仅需在现有投影牛顿框架中修改一行代码，即可显著提升稳定性与收敛速度。我们通过一系列具有挑战性的示例以及一个大数据集上的不同变形验证了该特征值投影方案的有效性和高效性。