Collaborative Research: Geometric Time Integrators for Mechanical Dynamical Systems

时间积分器是研究非线性动力系统的重要计算工具。多年来已经开发了许多时间步进方法，其中许多方法现在在现成的求解器中可用。然而，即使在高度精确的算法中，能量漂移和数值耗散问题仍然经常困扰工程应用。几何时间积分器最近被证明对阐明和修正固体力学中的这些问题非常有用。然而，这些贡献并没有延续到欧拉环境中，在欧拉环境中，它们可能会影响计算流体动力学中时间积分器的理解和可靠性。因此，本研究项目的目标是为一类动力学由作用原理（可能包括耗散和强迫）描述的问题开发新颖的、基于几何的欧拉时间积分器，这些问题包括规范的欧拉和纳维-斯托克斯方程，以及许多其他模型。我们将探讨哈密顿-庞特里亚金原理的欧拉离散化，并结合数学和数值工具，如离散外微积分、正双随机矩阵的半群和隐函数。结果积分器被期望，就像拉格朗日设置一样，尊重物理结构，也就是说，不引入关键物理量（如能量或循环）的人为数值损失。提出的研究活动旨在开发流体力学系统的预测和高阶精确模拟的基础设施，将现代应用几何的力量与现代计算力学相结合。特别是，它承诺引入新的变分流体模拟算法：这种创新的计算方法依赖于多学科的努力，利用几何力学、离散几何、数值分析和图形学等技术，从而有望产生广泛的理论和实践影响。从统一的几何角度发展这种变分积分器代表了我们解决复杂物理现象的长期目标的踏脚石，例如流动的衣服，游动的鱼或飞溅的水，这些现象的模拟需要当前技术水平的相当大的改进才能变得普遍。在此项目中获得的研究经验将通过在数学、工程和计算机科学期刊、书籍和会议上发表，以及在我们的网站、暑期学校、研讨会和其他教育活动中传播给广泛的受众。我们三个机构的外展工作包括从代表性不足的群体中招募学生来帮助开展这项研究项目，利用现有的努力来加强妇女和少数民族参与科学研究。

Time integrators are crucial computational tools for studying nonlinear dynamical systems. Numerous time stepping methods have been developed over the years, many of which are now available in off-the-shelf solvers. However energy drifts and numerical dissipation problems present even in highly accurate algorithms still routinely plague engineering applications. Geometric time integrators have been recently proven greatly useful to elucidate and fix these issues in solid mechanics. Yet these contributions have not carried over to the Eulerian setting, where they could impact both the understanding and the reliability of time integrators for computational fluid dynamics. The goal of this research project is thus to develop novel, geometrically-based Eulerian time integrators for the class of problems whose dynamics is described by an action principle, possibly including dissipation and forcing---which encompasses the canonical Euler and Navier-Stokes equations, as well as many other models. Eulerian discretizations of the Hamilton-Pontryagin principle will be explored, and combined with mathematical and numerical tools such as Discrete Exterior Calculus, the semigroup of positive doubly-stochastic matrices, and implicit functions. Resulting integrators are expected, just like in the Lagrangian setting, to respect the structure of the physics, i.e., to introduce no artificial numerical loss of crucial physical quantities such as energy or circulation.The proposed research activities aim at developing an infrastructure for predictive and high-order accurate simulations of fluid-mechanical systems that combine the power of modern applied geometry with modern computational mechanics. In particular, it promises the introduction of novel variational fluid simulation algorithms: this innovative computational approach relies on a multidisciplinary effort drawing upon techniques from geometric mechanics, discrete geometry, numerical analysis, and graphics, thus promising a broad theoretical and practical impact. The development of such variational integrators from a unified geometric standpoint represents a stepping stone for our long-term goal of solving complex physical phenomena such as a flowing dress, a swimming fish or splashing water, the simulation of which requires considerable improvement of the current state of the art to become commonplace. The research experience acquired during this project is to be disseminated to a wide range of audiences through publishing in mathematics, engineering and computer science journals, books, and conferences, as well as on our web sites, in summer schools, workshops, and other educational activities. Outreach efforts at our three institutions include the recruitment of students from underrepresented groups to help with this research project, leveraging existing efforts for enhancing the participation of women and minorities in scientific research.