Nonlinear systems arising from time integrators like Backward Euler can sometimes be reformulated as optimization problems, known as incremental potentials. We show through a comprehensive experimental analysis that the widely used Projected Newton method, which relies on unconditional semidefinite projection of Hessian contributions, typically exhibits a reduced convergence rate compared to classical Newton's method. We demonstrate how factors like resolution, element order, projection method, material model and boundary handling impact convergence of Projected Newton and Newton. Drawing on these findings, we propose the hybrid method Project-on-Demand Newton, which projects only conditionally, and show that it enjoys both the robustness of Projected Newton and convergence rate of Newton. We additionally introduce Kinetic Newton, a regularization-based method that takes advantage of the structure of incremental potentials and avoids projection altogether. We compare the four solvers on hyperelasticity and contact problems. We also present a nuanced discussion of convergence criteria, and propose a new acceleration-based criterion that avoids problems associated with existing residual norm criteria and is easier to interpret. We finally address a fundamental limitation of the Armijo backtracking line search that occasionally blocks convergence, especially for stiff problems. We propose a novel parameter-free, robust line search technique to eliminate this issue.

翻译：由后向欧拉等时间积分方法产生的非线性系统有时可以重构为优化问题，即增量势能。通过系统的实验分析，我们证明广泛使用的投影牛顿法（依赖于Hessian贡献的无条件半定投影）与经典牛顿法相比，通常表现出较低的收敛速率。我们展示了分辨率、单元阶数、投影方法、材料模型和边界处理等因素如何影响投影牛顿法和牛顿法的收敛性。基于这些发现，我们提出混合方法——按需投影牛顿法（仅进行条件投影），并证明该方法兼具投影牛顿法的鲁棒性和牛顿法的收敛速率。我们还引入了动力学牛顿法，这是一种基于正则化的方法，利用增量势能的结构特征并完全避免投影。我们在超弹性和接触问题上比较了四种求解器。此外，我们深入讨论了收敛准则，并提出一种新的基于加速度的准则，该准则避免了现有残差范数准则的问题且更易解释。最后，我们解决了Armijo回溯线搜索的一个基本局限性——该局限性偶尔会阻碍收敛（尤其对刚性问题）。我们提出一种新颖的无参数鲁棒线搜索技术以消除该问题。