High-order accurate adaptive numerical methods for fluid mechanics using unstructured meshes

为了使复杂流动的计算流体动力学（CFD）模拟结果在工程设计环境中有用，仿真软件（求解器）必须高效、鲁棒地生成准确的解，同时最大限度地减少人类设置模拟的时间。为了解决这些问题，我的研究小组开发了非结构化网格的高阶精确解决技术。我们的结果和其他人的结果一致表明，使用高阶方法可以更快地获得相同精度的流解。我的团队的长期目标是将高阶方法的好处从研究层面带到工业可用性。在短期内，我们将强调高阶方法在构型空气动力学、空气动力学优化和流固耦合中的应用。为了在高阶方法的背景下实现这些应用，我们将扩展现有的二阶技术或开发用于粘性网格生成、网格自适应和伴随问题解决的新技术。高阶离散化方案的粘性网格生成的关键问题是曲面的精确表示，包括创建具有弯曲边的网格单元，以避免在壁面上倒置的网格单元。我们已经在曲面边界网格划分方面取得了重大进展，并期望在此基础上进一步开展粘滞网格划分工作。高阶网格自适应程序将需要开发新的误差指标，因为二阶自适应方法常用的二阶导数已经被高阶求解器捕获。我们将研究使用高阶导数作为方向误差指标，并执行网格自适应来有效地控制误差。我的团队在高阶方法优化方面的早期努力，因难以计算足够精确的梯度而受阻。伴随问题的高阶解既要提供所需的梯度，又要在解的误差界计算中有应用。

For the results of computational fluid dynamics (CFD) simulations of complex flows to be useful in an engineering design context, the simulation software (solver) must produce accurate solutions efficiently and robustly while minimizing the human time to set up a simulation.      To address these issues, my research group has developed high-order accurate solution techniques for unstructured meshes.  Our results and those of others consistently show that a flow solution of the same accuracy can be obtained more quickly using high-order methods.  My group's long-term goal is to bring the benefits of high-order methods from the research level to industrial usability.      In the near term, we will emphasize application of high-order methods to configuration aerodynamics, aerodynamic optimization, and fluids-structures interaction.  To enable these applications in the context of high-order methods, we will extend existing second-order techniques or develop new techniques for viscous mesh generation, mesh adaptation, and solution of adjoint problems.      The key issue in viscous mesh generation for high-order discretization schemes is the accurate representation of curved surfaces, including creation of mesh cells with curved sides to avoid inverted cells at the wall.  We have already made significant progress on curved-boundary meshing, and expect to build on that in the viscous meshing work. High-order mesh adaptation procedures will require the development of new error indicators, as the second derivatives commonly used by second-order adaptive methods are already captured by a high-order solver.  We will investigate the use of higher-order derivatives as directional error indicators, and perform mesh adaptation to control error effectively.  My group's early efforts in optimization for high-order methods were stymied by difficulties in computing sufficiently accurate gradients.  High-order solution of adjoint problems should provide the required gradients, and also have application in computing solution error bounds.