不等式约束非光滑哈密顿系统的几何数值积分
Geometric Numerical Integration of Inequality Constrained, Nonsmooth Hamiltonian Systems
AI总结：
针对同时具有等式和“硬”不等式约束的哈密顿系统，提出一族基于离散变分技术的几何数值积分方法，保持辛-动量、近似能量守恒，并处理多个同时不等式约束及光滑/非光滑边界交互。
摘要：
我们考虑受等式和“硬”不等式约束的哈密顿系统的几何数值积分。与标准几何积分设置一样，我们目标是长期结构保持。然而，我们还考虑了在持久、同时和/或频繁边界交互下的不变量保持。由于这些约束带来的固有非光滑性，适当构造包含这些条件的几何方法长期以来一直具有挑战性。为解决这些问题，我们同时关注辛-动量保持行为以及不等式约束设置中特有的额外结构保持。利用离散变分技术，我们构造了一族几何数值积分方法，不仅获得了动量保持、近似能量守恒和等式约束保持等通常期望的性质，而且能够强制执行多个同时不等式约束，沿约束边界获得光滑的单侧运动，并允许非光滑和光滑的边界接近和退出轨迹。通过数值实验，在光滑和非光滑主动约束模式高频持续的困难测试实例上展示了这些方法的性能。
英文摘要：
We consider the geometric numerical integration of Hamiltonian systems subject to both equality and "hard" inequality constraints. As in the standard geometric integration setting, we target long-term structure preservation. We additionally, however, also consider invariant preservation over persistent, simultaneous and/or frequent boundary interactions. Appropriately formulating geometric methods to include such conditions has long-remained challenging due to the inherent nonsmoothness they impose. To resolve these issues we thus focus both on symplectic-momentum preserving behavior and the preservation of additional structures, unique to the inequality constrained setting. Leveraging discrete variational techniques, we construct a family of geometric numerical integration methods that not only obtain the usual desirable properties of momentum preservation, approximate energy conservation and equality constraint preservation, but also enforce multiple simultaneous inequality constraints, obtain smooth unilateral motion along constraint boundaries and allow for both nonsmooth and smooth boundary approach and exit trajectories. Numerical experiments are presented to illustrate the behavior of these methods on difficult test examples where both smooth and nonsmooth active constraint modes persist with high frequency.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
