Structure-preserving Numerical Methods for Engineering Applications

这个项目涉及的基础研究将为计算分析工具奠定基础，这些工具可以提高我们对复杂动态系统行为的理解和预测，范围广泛，包括天体动力学、粒子物理、地球物理流体动力学、等离子体物理和分子动力学。这项研究涉及几何数值积分(GNI)，它涉及到通过保留控制数学方程的几何性质来尊重问题的基本物理的数值方法。除了研究成果的直接影响外，该项目还通过面向各级学生的各种教育和推广活动产生了重大的广泛影响，包括为中学生开设夏令营，向他们传授机械系统和精确数值模拟的重要性。过去在几何数值积分领域的努力主要集中在为受保守强迫的低维、时不变的机械系统开发数值算法。这项研究工作将建立一个理论框架，以理解和扩展现有几何数值方法在显式时间相关、非保守强迫和约束方面的能力，并研究它们对具有实际意义的多自由度动力系统的数值性能。具体地说，研究计划将(I)发展具有外力的依赖时间的力学系统的结构保持方法，并将这些方法扩展到受约束或在非欧几里德流形上发展的系统，(Ii)提供结构保持积分方案与传统方法的全面、定量的比较，以选择特定的工程问题，包括无限维问题的离散化，以及(Iii)创建和传播一个文件齐全的工具箱，其中包含使用研究界熟悉的软件开发的各种结构保持数值方法。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project involves fundamental research that will lay the groundwork for computational analysis tools that can improve our understanding and prediction of the behavior of complex dynamic systems, as wide-ranging as astrodynamics, particle physics, geophysical fluid dynamics, plasma physics and molecular dynamics. The research involves geometric numerical integration (GNI), which involves numerical methods that respect the fundamental physics of a problem by preserving the geometric properties of the governing mathematical equations. Beyond the direct impact of the research outcomes, the project also has significant broader impact through various educational and outreach activities for students at all levels including a camp for middle school students to teach them about mechanical systems and the importance of accurate numerical simulation.Past efforts in the field of geometric numerical integration have mainly focused on developing numerical algorithms for low-dimensional, time-invariant mechanical systems subject to conservative forcing. The esearch effort will develop a theoretical framework to understand and extend the capabilities of existing geometric numerical methods with regard to explicit time-dependence, non-conservative forcing and constraints and investigate their numerical performance for dynamical systems of practical interest with many degrees of freedom. Specifically, the research program will (i) develop structure-preserving methods for time-dependent mechanical systems with external forcing and extend these methods to systems that are constrained or that evolve on non-Euclidean manifolds, (ii) provide a thorough, quantitative comparison of structure-preserving integration schemes with traditional methods for a selection of specific engineering problems including discretization of infinite-dimensional problems, and (iii) create and disseminate a well-documented toolbox that features a variety of structure-preserving numerical methods developed using software familiar to the research community.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.