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

该提案概述了以计算流体动力学（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根本弱点之一-耗散和/或色散误差，由于空间离散化的一阶项在控制守恒方程。在显着对比的现有技术，我们提出了一种方法，用于获得优化的数值制剂，基于最小化的目标函数，提供了一个估计的局部离散化误差。建议的优化策略将首次允许一个真正的自适应数值方法，产生一个“最佳情况”的解决方案，为一个特定的问题，使用一个特定的网格。这一努力的最终结果将是一个完整的，充分记录，并充分验证的数值框架，适用于不可压缩流体流动的控制方程，包括稳态和非稳态。一旦开发，可压缩流和其他守恒方程的方法的未来扩展将是直截了当的。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。利用由大学研究人员，工业专家，外展管理人员和高中教师组成的协调小组，我们将在密西西比的四到六所高中的物理课程中实施CFD项目模块。预计计算模拟的可视化，交互式的性质将对学生的学习产生积极的影响，更重要的是对学生的态度，科学和工程。该五年计划将进行评估，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。智力优势。在过去的四十年中，（一般）计算流体动力学的数值方法的主要研究重点一直是减轻离散误差所产生的近似的一阶（对流）条款。虽然取得了重大进展，但至今仍是主要焦点。一个真正的非增量的，跨越现有技术水平的阶跃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。到目前为止，所有的数值公式都是基于作为因变量场的函数的数值导数的明确规定。这些通常是复杂的，涉及高阶重建，限制器等，但是它们不允许数值近似响应于局部或全局数值误差的估计而被调整。所提出的策略采用一般的，自适应形式的数值近似，这是迭代优化的数值解本身的同时。实际上，使用反馈控制来规定数值离散化，以提供使数值误差最小化的优化解。初步结果表明，所提出的方法有可能减少数值误差的几个数量级与目前的方法，并在某些情况下，产生基本上为零的数值误差的解决方案。更广泛的影响。研究部分的影响将是巨大的，可能影响目前使用计算流体动力学的每一个科学和工程学科。人们还认为，新框架的发展将催生未来的研究工作，优化数值方法，将影响计算技术领域超越CFD。教育方面的影响也将是巨大的。既定目标是增加高中毕业生在科学和工程职业中的参与。参与者将从密西西比农村学区的高中中选出，这些学区教育的弱势群体和代表性不足的群体比例过高。该计划将使这些学生有机会与大学研究人员互动，并以他们目前无法使用的方式利用令人兴奋的科学工具。该计划还将促进对科学和技术的认识和欣赏，无论他们的职业选择如何，都将产生积极的长期影响。研究生和本科生的参与将产生额外的影响，其中至少包括一名女博士。学生谁已经与PI作为一个本科研究员，并已承诺在他的研究小组进行研究生学习。

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.