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，其中许多现在可以在现成的求解器中使用。然而，能量漂移和数值耗散的问题，即使在高精度的算法仍然经常困扰工程应用。几何时间积分器最近已被证明是非常有用的阐明和解决这些问题的固体力学。然而，这些贡献并没有结转到欧拉设置，在那里他们可能会影响的理解和计算流体动力学的时间积分的可靠性。因此，本研究项目的目标是开发新的，几何为基础的欧拉时间积分的一类问题，其动态描述的作用原理，可能包括耗散和强迫-其中包括规范欧拉和Navier-Stokes方程，以及许多其他模型。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，如离散外部微积分，正双随机矩阵半群和隐函数。就像在拉格朗日设置中一样，期望所得积分器尊重物理结构，即，拟议的研究活动旨在开发一个基础设施，用于流体机械系统的预测和高阶精确模拟，将现代应用几何学的力量与现代计算力学联合收割机相结合。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，从几何力学，离散几何，数值分析和图形技术，从而有希望广泛的理论和实践的影响。这种变分积分器从统一的几何观点的发展代表了一个垫脚石，我们的长期目标，解决复杂的物理现象，如流动的衣服，游泳的鱼或飞溅的水，其模拟需要相当大的改进，目前的最先进的成为司空见惯。在这个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，以及在我们的网站上，在暑期学校，研讨会和其他教育活动中传播给广泛的受众。我们三个机构的外联工作包括从代表性不足的群体中招募学生来帮助这一研究项目，利用现有的努力来加强妇女和少数民族对科学研究的参与。

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.