LTB: Generalized Variational Integrators for Large-Scale Scientific Computation

LTB：用于大规模科学计算的广义变分积分器

• 批准号: 1001521
• 负责人: Melvin Leok
• 金额: $ 13.11万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-04 至 2011-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001521&HistoricalAwards=false
• 关键词: LTB; Generalized; Variational; Integrators; Large

Geometric integration is concerned with the construction of numerical methods that preserve the geometric structure of a continuous dynamical system. Many problems arising in science and engineering, such as solar system dynamics and molecular dynamics, are highly nonlinear, sensitive to small perturbations, and have underlying geometric structure that affects the qualitative behavior of solutions. The chaotic properties of these dynamical systems render prohibitively expensive the accurate computation of particular trajectories for long-time integration. As such, it is instead desirable to study numerical methods that preserve the geometry of a problem as they yield more qualitatively accurate simulations. The goal of this project is to generalize variational integrators based on a discrete Hamilton's principle to larger-scale problems arising from astrodynamics, molecular dynamics, and computational mechanics. This will involve incorporating methods from large-scale scientific computation, such as adaptivity, spectral methods, multi-resolution hierarchical techniques, and domain decomposition, while retaining the geometric preservation properties of variational integrators for Hamiltonian ODEs and PDEs.Computer simulations of complex physical systems have become an increasingly important complement to traditional experimental techniques as a tool for validating and guiding theoretical developments in science, as well as practical advances in technology and engineering. This research will improve our ability to accurately and efficiently compute the long-time behavior of complex systems, which is a fundamental aspect of the rational design of pharmaceuticals and high-performance composite materials. In addition, it has the potential to accelerate the pace of technological development by allowing the rapid prototyping of new and innovative industrial designs directly on the computer.

几何积分关注的是保持连续动力系统几何结构的数值方法的构造。在科学和工程中出现的许多问题，如太阳系动力学和分子动力学，是高度非线性的，对小扰动敏感，并具有潜在的几何结构，影响解决方案的定性行为。这些动力系统的混沌特性使得长时间积分的特定轨迹的精确计算过于昂贵。因此，人们希望研究保留问题几何形状的数值方法，因为它们可以产生更定性准确的模拟。这个项目的目标是推广基于离散汉密尔顿原理的变分积分器，以解决天体动力学、分子动力学和计算力学中的大规模问题。这将涉及纳入大规模科学计算的方法，如自适应性，光谱方法，多分辨率分层技术和区域分解，同时保留了Hamilton常微分方程和偏微分方程的变分积分器的几何保持性质。复杂物理系统的计算机模拟作为验证和指导理论的工具，已经成为传统实验技术的日益重要的补充。科学的发展，以及技术和工程的实际进展。这项研究将提高我们准确有效地计算复杂系统长期行为的能力，这是药物和高性能复合材料合理设计的一个基本方面。此外，它有可能通过允许直接在计算机上快速原型制作新的和创新的工业设计来加快技术发展的步伐。