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一个基本弱点--由于控制守恒方程中一阶项的空间离散而产生的耗散和/或色散误差。与当前技术状态形成鲜明对比的是，我们提出了一种基于最小化提供局部离散化误差估计的目标函数来获得优化的数值公式的方法。提出的优化策略将第一次允许一个真正自适应的数值方法，使用特定的网格网格为特定的问题产生“最佳情况”的解决方案。这项工作的最终结果将是一个完整的、充分记录的和充分验证的数值框架，用于不可压缩流体流动的控制方程，包括定常和非定常。一旦发展起来，未来将该方法扩展到可压缩流动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一个基本需求-吸引高中生从事科学和工程职业。利用一个由大学研究人员、工业专家、外展管理人员和高中教师组成的协调团队，我们将在密西西比州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、互动性将对学生的学习产生积极影响，更重要的是对学生对待科学和工程的态度产生积极影响。这项为期五年的计划将进行评估，以确定其在提高高中生对力学的概念理解方面的有效性，以及在鼓励学生高中毕业后追求科学和/或工程职业方面的有效性。智力上的功绩。在过去的40年里，(一般)计算流体力学数值方法的主要研究重点一直是减小一阶(对流)项近似引起的离散化误差。虽然取得了实质性进展，但很能说明问题的是，直到今天，这仍然是主要的重点。相对于当前技术水平，真正的非渐进式、阶段性的进步需要一个全新的框架，并形成了这一提议的动机。到目前为止，所有的数值公式都是基于数值导数作为因变量场的函数的显式规定。这些通常是复杂的，涉及高阶重建、限制器等，但它们不允许响应于对局部或全局数值误差的估计而调整数值近似。所提出的策略采用了通用的、自适应的数值逼近形式，这些数值逼近与数值解本身同时被迭代优化。实际上，使用反馈控制来规定数值离散化，以提供最小化数值误差的优化解。初步结果表明，与目前的方法相比，该方法有可能减少几个数量级的数值误差，并且在某些情况下可以得到数值误差基本为零的解。更广泛的影响。研究部分的影响将是巨大的，潜在地影响到目前利用计算流体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，这将对CFD以外的领域的计算技术产生影响。对教育的影响也将是巨大的。声明的目标是增加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出来，这些学区教育的弱势群体和代表不足的群体比例高得不成比例。该项目将允许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。该计划还将培养人们对科学技术的认识和欣赏，无论他们选择什么职业，科学技术都将产生积极的、长期的影响。其他影响将来自研究生和本科生的参与，包括至少一名女性博士生，她已经作为本科生研究员在PI工作，并承诺在他的研究小组继续研究生学习。

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.