CAREER: Optimized Computational Fluid Dynamics -- Towards Exact Numerical Methods for Conservation Equations

职业：优化计算流体动力学——迈向守恒方程的精确数值方法

• 批准号: 0645138
• 负责人: Dibbon Walters
• 金额: $ 41.2万
• 依托单位: Mississippi State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-05-01 至 2013-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0645138&HistoricalAwards=false
• 关键词: CAREER; Optimized; Computational; Fluid; Dynamics

This proposal outlines a research and education plan focused on computational fluid dynamics (CFD). The research component addresses one of the fundamental weaknesses of current numerical methods - dissipation and/or dispersion errors due to the spatial discretization of the first-order terms in the governing conservation equations. In marked contrast to the current state of the art, we propose a methodology for obtaining optimized numerical formulations, based on minimization of an objective function that provides an estimate of the local discretization error. The proposed optimization strategy will for the first time allow a truly adaptive numerical methodology that yields a "best case" solution for a particular problem using a particular grid mesh. The end result of this effort will be a complete, fully documented, and fully validated numerical framework for application to the governing equations of incompressible fluid flow, both steady and unsteady. Once developed, future extensions of the methodology to compressible flows and to other conservation equations will be straightforward. The educational component addresses a fundamental need for the nation and for the state of Mississippi - the attraction of high-school students to science and engineering careers. Using a coordinated team comprised of university researchers, industrial experts, outreach administrators and high-school teachers, we will implement a CFD project module into the Physics curriculum of four to six high schools in Mississippi. It is expected that the visual, interactive nature of computational simulation will have a positive impact on the students' learning, and more importantly on the student's attitude toward science and engineering. The five-year program will be assessed to determine its effectiveness in improving conceptual understanding of mechanics among high-school seniors, and in encouraging students to pursue science and/or engineering careers after high school. Intellectual Merit. The primary research focus in numerical methods for (general) computational fluid dynamics over the past four decades has been mitigation of discretization errors arising from the approximation of the first-order (convective) terms. While progress has been substantial, it is telling that it remains the primary focus to this day. A truly non-incremental, step-change advancement over the current state of the art requires an entirely new framework, and forms the motivation for this proposal. To date, all numerical formulations have been based on explicit prescriptions of the numerical derivatives as functions of the dependent variable field. These are often complex, involving higher-order reconstructions, limiters, etc., but they do not allow the numerical approximations to be adapted in response to estimates of the local or global numerical error. The proposed strategy employs general, adaptive forms of the numerical approximations, which are iteratively optimized concurrent with the numerical solution itself. In effect, the numerical discretization is prescribed using feedback control to provide an optimized solution that minimizes the numerical error. Preliminary results indicate that the proposed methodology has the potential to reduce numerical error by several orders of magnitude versus current approaches, and in some cases to yield solutions with essentially zero numerical error. Broader Impacts. The impact of the research component will be substantial, potentially influencing every scientific and engineering discipline that currently makes use of computational fluid dynamics. It is also believed that development of the new framework will spawn future research efforts into optimized numerical methods that will impact computational techniques in fields beyond CFD. The educational impact will also be substantial. The stated goal is the increased participation of graduating high-school students in science and engineering careers. Participants will be selected from high schools in rural Mississippi school districts, which educate disproportionately high percentages of disadvantaged and under-represented groups. This program will allow these students the opportunity to interact with university researchers and to utilize exciting scientific tools in ways that they currently cannot. The program will also foster an awareness of and an appreciation for science and technology that will have a positive, long-term impact regardless of their career choices. Additional impacts will arise from the participation of graduate and undergraduate students, including at least one female Ph.D. student who is already working with the PI as an undergraduate researcher and has committed to pursue graduate study in his research group.

本提案概述了一项以计算流体动力学（CFD）为重点的研究和教育计划。研究部分解决了当前数值方法的一个基本弱点-由于控制守恒方程中一阶项的空间离散而导致的耗散和/或色散误差。与目前的技术状况形成鲜明对比的是，我们提出了一种基于目标函数的最小化来获得优化数值公式的方法，该目标函数提供了局部离散化误差的估计。提出的优化策略将首次允许真正自适应的数值方法，为使用特定网格的特定问题产生“最佳情况”解决方案。这项工作的最终结果将是一个完整的、充分记录的、充分验证的数值框架，用于不可压缩流体流动的控制方程，包括稳态和非稳态。一旦发展起来，将来将该方法扩展到可压缩流和其他守恒方程将是直截了当的。教育部分解决了国家和密西西比州的基本需求——吸引高中生从事科学和工程职业。通过一个由大学研究人员、工业专家、外联管理人员和高中教师组成的协调团队，我们将在密西西比州的四到六所高中的物理课程中实施一个CFD项目模块。计算模拟的可视化、互动性将对学生的学习产生积极的影响，更重要的是对学生对科学和工程的态度产生积极的影响。这项为期五年的计划将被评估，以确定其在提高高中高年级学生对力学的概念理解方面的有效性，并鼓励学生在高中毕业后从事科学和/或工程职业。知识价值。在过去的四十年中，（一般）计算流体动力学数值方法的主要研究重点是减轻由一阶（对流）项近似引起的离散化误差。虽然取得了实质性的进展，但它仍然是今天的主要焦点。一个真正的非增量的、逐步变化的进步需要一个全新的框架，并形成了这个建议的动机。迄今为止，所有的数值公式都是基于作为因变量场函数的数值导数的显式公式。这些通常是复杂的，涉及高阶重建，限制器等，但它们不允许数值近似适应于局部或全局数值误差的估计。所提出的策略采用一般的、自适应的数值近似形式，并与数值解本身同时迭代优化。实际上，数值离散化是规定使用反馈控制，以提供一个优化的解决方案，使数值误差最小化。初步结果表明，与目前的方法相比，所提出的方法有可能将数值误差减少几个数量级，并且在某些情况下产生的解基本上具有零数值误差。更广泛的影响。研究部分的影响将是巨大的，潜在地影响到目前使用计算流体动力学的每一个科学和工程学科。人们还相信，新框架的发展将催生未来对优化数值方法的研究，这将影响CFD以外领域的计算技术。教育方面的影响也将是巨大的。该计划的目标是提高高中毕业生在科学和工程领域的参与度。参与者将从密西西比州农村学区的高中中挑选，这些学区教育的弱势群体和代表性不足的群体比例过高。这个项目将使这些学生有机会与大学研究人员互动，并以他们目前无法做到的方式利用令人兴奋的科学工具。该项目还将培养对科学技术的认识和欣赏，无论他们的职业选择如何，这将产生积极的、长期的影响。研究生和本科生的参与将产生额外的影响，包括至少一名女博士生，她已经作为本科生研究人员与PI一起工作，并承诺在他的研究小组中进行研究生学习。