Development of robust and efficient algorithms for solving aerodynamic flows

建议的研究重点是改进计算流体力学(CFD)算法，以用于模拟湍流空气动力学流动。基于CFD的数值分析工具已经成为航空航天和汽车工业设计的基本组成部分。在最短的时间内获得准确的建模解决方案，同时降低计算成本，是行业内CFD工具成功的关键。目前的CFD工具通常需要大量的专业知识，而不是手头的物理问题，这可能会阻碍工具的有效性。这项研究将寻求开发方法，以减少特定于工具的知识要求，同时产生尽可能快的CFD算法来模拟湍流。提出了以下方法。从基于多重网格的求解器开始，可以基于傅立叶分析来优化影响求解器性能的参数，以最大化误差模式的传播和衰减，同时最大化解的精度。因此，可以基于局部流动条件和网格几何形状自动选择求解器参数。为了缩小最优参数组合的范围，有必要研究预处理和粗化技术的最佳组合，以减小离散的数值刚度和传播速度的差异。虽然傅立叶分析提供了对算法性能和稳定性的估计，但它在数值稳定性极限方面也可能具有误导性。用代数稳定性理论(或能量方法)研究数值稳定性，可能会得到性能更好、更一致的最优参数。总而言之，这些研究应该会提高CFD社区对数值性能的理解，并导致更好地实现解算器，其中用户只需专注于所建模问题的物理。

The proposed research focuses upon improving computational fluid dynamics (CFD) algorithms for use in modelling turbulent aerodynamic flows. Numerical analysis tools based upon CFD have become a fundamental part of design in aerospace and automotive industries. Obtaining accurate modelling solutions in minimal time while reducing computational cost is critical to CFD tool success within industry. Current CFD tools typically require a substantial amount of specialized knowledge beyond the physical problem at hand which may hinder the tool's effectiveness. This research will seek to develop methods for reducing the tool-specific knowledge requirement while producing CFD algorithms which are as fast as possible for modelling turbulent flow. The following approach is proposed. Beginning with a multigrid-based solver, parameters which affect solver performance may be optimized based upon Fourier analysis to maximize propagation and damping of error modes while maximizing the accuracy of the solution. Solver parameter selection may thus be automated based upon local flow conditions and mesh geometry. To reduce the range of optimal parameter combinations, it is necessary to investigate the best combination of preconditioning and coarsening techniques to reduce discrete numerical stiffness and disparity in propagative speeds. While Fourier analysis provides an estimate of an algorithm's performance and stability it can also be misleading with respect to numerical stability limits. An investigation of numerical stability using algebraic stability theory (or energy methods) may yield optimal parameters with better and more consistent performance. Together these investigations should improve the CFD community's understanding of numerical performance, and lead to better implemented solvers where the user need only focus on the physics of the problem being modelled.