Structure-preserving Numerical Methods for Engineering Applications

点击翻译按钮获取中文摘要

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.