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.

几何整合与保留连续动力系统几何结构的数值方法的构建有关。在科学和工程中产生的许多问题，例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，并且具有影响解决方案定性行为的潜在几何结构。这些动力学系统的混沌性能使特定轨迹的精确计算以长期整合的准确计算变得过高。因此，需要研究保留问题几何形状的数值方法，因为它们产生了更定性的准确模拟。该项目的目的是根据离散的汉密尔顿的原理概括变异积分器，以归因于天体动力学，分子动力学和计算机械师引起的大规模问题。这将涉及合并大规模科学计算的方法，例如适应性，光谱方法，多分辨率层次结构技术和域分解，同时保留了汉密尔顿odes和PDES的变异集成剂的几何几何保护特性，PDES.com.com.com.com.com.com.com的PUTENTING对复杂的物理系统进行了越来越多的辅助技术，并且是越来越重要的工具，作为传统的辅助技术，并且是一种辅助技术，是一种辅助技术，是一种辅助技术，是一种辅助技术，是一种辅助技术，是一种辅助技术。以及技术和工程学的实际进步。这项研究将提高我们准确有效地计算复杂系统的长期行为的能力，这是药物合理设计和高性能复合材料的基本方面。此外，它有可能直接在计算机上直接允许新的和创新的工业设计的快速原型制作来加速技术发展的速度。