From Microscopic to Macroscopic: A Multiscale Numerical Approach

从微观到宏观：多尺度数值方法

• 批准号: 9805582
• 负责人: Shlomo Ta'asan
• 金额: $ 25.5万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-09-15 至 2001-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9805582&HistoricalAwards=false
• 关键词: Microscopic; Macroscopic; Multiscale; Numerical; Approach

Ta'asan9805582 The investigator and his colleagues develop multiscalenumerical techniques for dealing with phenomena on a wide rangeof scales. The main aspect of the research is the numericalconstruction of macroscopic equations in the form of partialdifferential equations from microscopic models such as moleculardynamics models. The numerical techniques focus on fluidsmodeling starting from molecular models. Special emphasize isgiven to models involving mixtures of several types ofatoms/molecules that may be governed by different interactions.Such molecular models give rise to complicated physical phenomenaincluding multiphase flows. A sequence of models is constructednumerically in an hierarchical manner starting from moleculardynamics. Each level in this hierarchy represent a larger scaledynamics (in time or space or both) of the previous level, and isconstructed using statistical tools (regression analysis) appliedto the data resulting from a simulation of the previous level.The finest level of modeling is given by Hamiltonian dynamics andis deterministic. Stochastic particle dynamics models areconstructed from the finest level model, and represent thephysical phenomenon on time scale of the order of the relaxationtime. Further coarsening (averaging) in space-time results indeterministic density evolution models, which can be identifiedas finite difference approximations to some partial differentialequation that is the macroscopic limit. Numerical techniques have been used extensively for severaldecades to obtain approximate solutions of partial differentialequations in different fields, and have improved significantlyscientific insights and engineering tasks. The derivation ofthose equations, which describe physical phenomena in differentareas, was done by analytical techniques whose use is limited torelatively simple cases. The treatment of more complex modelscannot, in general, be done analytically and it calls for anotherapproach. This is the subject of this work: the use of numericaltechniques for 'deriving' macroscopic dynamics equations frommore basic principles given at microscopic levels. This task iscarried out by large scale simulation of the detailed microscopicmodels and appropriate statistics. The problems treated in thiswork are limited to multiphase fluid flows, which are crucial inmany areas of applications, including medical inhalation devices,auto and aerospace industries (engines), computer industries(printers), and more. The techniques developed here, however, arequite general and apply to other areas in chemistry, physics andeconomics, from fundamental questions to real world applications.

研究人员塔阿桑9805582和他的同事开发了多标度计数技术来处理大范围内的现象。研究的主要方面是从微观模型(如分子动力学模型)数值构造以偏微分方程组形式表示的宏观方程。数值技术侧重于从分子模型开始的流体建模。特别强调了几种可能受不同相互作用支配的原子/分子混合物的模型，这种分子模型产生了包括多相流在内的复杂物理现象。从分子动力学出发，以层次化的方式构造了一系列模型。该层次结构中的每一级代表上一级的更大尺度动力学(在时间或空间上或两者)，并使用统计工具(回归分析)来构建，该统计工具应用于上一级的模拟产生的数据。最精细的建模由哈密顿动力学给出，并且是确定性的。随机粒子动力学模型是由最细粒子模型构造的，它在松弛时间量级的时间尺度上表示物理现象。时空中的进一步粗化(平均)导致了不确定的密度演化模型，它可以被识别为对一些偏微分方程的有限差分近似，即宏观极限。几十年来，数值技术被广泛地应用于不同领域的偏微分方程解的近似求解，并极大地提高了科学洞察力和工程任务。这些描述不同地区物理现象的方程的推导是通过分析技术完成的，其使用仅限于相对简单的情况。一般说来，对更复杂的模型的处理不能通过分析来完成，它需要另一种方法。这就是这项工作的主题：使用数值技术，从微观层面上给出的更基本的原理‘推导’宏观动力学方程。这项任务是通过对详细的微观模型进行大规模模拟和适当的统计来完成的。这项工作中处理的问题仅限于多相流体流动，这在许多应用领域都是至关重要的，包括医疗吸入器、汽车和航空航天工业(发动机)、计算机工业(打印机)等。然而，这里开发的技术是一般性的，适用于化学、物理和经济学的其他领域，从基本问题到现实世界的应用。