CAREER: Nonlinear Finite Element Manifolds for Improved Simulation of Shock-Dominated Turbulent Flows

对于具有边界层、激波和湍流的复杂流动，计算流体动力学(CFD)模拟对底层网格非常敏感。为了获得准确的预测，通常需要高质量的、高度精细化的、特征对齐的网格；这些都很难构建，而且需要大量的用户。该项目的主要目的是开发新的模拟技术，以消除现代CFD方法对激波为主的湍流底层网格的极端敏感性。该方法将在许多科学学科中有用，包括航空航天、天体物理、生物学和环境流动。这项研究工作将与一项教育计划相辅相成，该计划包括面向印第安纳州五年级学生的实践流体动力学模块，面向本科生的暑期研究计划，以及以在线内容、会议短期课程和开源软件为特色的非线性流形近似教育活动。该项目将为使用非线性流形作为高保真CFD模拟的基础发展理论和算法基础，这是一个尚未探索的研究前沿，以绕过现有方法在复杂流动现象方面的限制。具体的非线性流形将被构造来定制基本的近似空间以适应通用网格上的复杂流动特征：(1)表示边界层的三角流形，(2)表示粘性激波的成分映射，以及(3)表示更一般特征的自动编码器，包括湍流和流动不稳定性。与传统的基于线性近似空间的方法相比，通过将基本的近似空间定制为流动特征而不是网格，非线性近似可以显著提高每自由度的精度，并降低CFD预测对计算网格的敏感性。因此，该项目具有巨大的潜力，可以提供一种新的、变革性的CFD技术，提高激波主导的湍流高保真模拟的准确性、可靠性和自动化。新方法的可扩展实施将在开源的Julia Solver中传播，以促进该项目的发展向研究社区的过渡。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Computational Fluid Dynamics (CFD) simulations are notoriously sensitive to the underlying grid for complex flows with boundary layers, shock waves, and turbulence. High-quality, highly refined, feature-aligned grids are usually required to obtain accurate predictions; these are difficult and user-intensive to construct. The principal aim of this project is to develop new simulation technology that eliminates the extreme sensitivity of modern CFD methods to the underlying grid for shock-dominated turbulent flows. The method will be useful across many scientific disciplines, including aerospace, astrophysical, biological, and environmental flows. This research effort will be complemented with an education plan that includes a hands-on fluid dynamics module for Indiana fifth graders, a summer research program for undergraduates, and an education campaign on nonlinear manifold approximations that will feature online content, conference short courses, and open-source software.This project will develop the theoretical and algorithmic foundations for using nonlinear manifolds as the basis for high-fidelity CFD simulations, an unexplored research frontier, to circumvent the limitations of existing methods with regard to complex flow phenomena. Specific nonlinear manifolds will be constructed to tailor the underlying approximation space to complex flow features on generic grids: (1) trigonometric manifolds to represent boundary layers, (2) compositional mappings to represent viscous shocks, and (3) autoencoders to represent more general features, including turbulent eddies and flow instabilities. By tailoring the underlying approximation space to the flow features instead of the grid, nonlinear approximations can dramatically improve the accuracy per degree of freedom and reduce the sensitivity of CFD predictions to the computational grid relative to conventional methods based on linear approximation spaces. As such, this project has great potential to deliver a novel and transformational CFD technology with enhanced accuracy, reliability, and automation of high-fidelity simulations of shock-dominated turbulent flows. A scalable implementation of the new approach will be disseminated in an open-source Julia solver to facilitate transition of the developments in this project to the research community.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.