The importance of mesh adaptation for higher-order solutions of PDEs

Eric Liu, Massachusetts Institute of Technology

Modern computational fluid dynamics (CFD) has the ability to explore problems that are more complex than ever before, through the availability of increasingly more powerful computing resources. However, many common CFD practices are not advancing to match. For example, many CFD practitioners use relatively low-order (second or even first order) solution approximations, but higher-order solutions offer substantial efficiency gains. Additionally, many engineers continue to generate meshes by hand, relying on some prior knowledge of qualitative flow features. Accuracy could be improved by uniformly refining these meshes. This process is manual and very slow. Furthermore, changes in the flow conditions (e.g., from sweeping through several angles of attack) often requires a new mesh at each parameter set. Using the wrong mesh produces results that often appear reasonable to the "eyeball-norm" despite being extremely inaccurate, as we also will show. Due to the prevalence of CFD-based design in the aerospace industry, the AIAA has hosted several drag prediction workshops in the past decade to assess the state-of-the-art in CFD when applied to representative problems of turbulent flow over aerodynamic bodies. From their efforts, we see that while conditions are improving, there is still significant scatter in the numerical predictions made by current CFD codes. A strong relation between solution quality and mesh topology also has been shown, further indicating that current mesh design practices are insufficient. In the present work, we will use a Discontinuous Galerkin Finite Element Method to demonstrate efficiency improvements obtainable through higher-order methods and the necessity of output-based mesh adaptation to realize these benefits. Our mesh adaptation strategy produces DOF (degree of freedom)-optimal meshes, i.e., giving the best answer for a given number of unknowns. Unfortunately, our adaptation scheme places great stress on the mesh generation tool, so we also will demonstrate a cut-cell approach which effectively completely alleviates the difficulty of mesh generation at the cost of greatly increased complexity in residual evaluation.

Abstract Author(s): Eric H. Liu