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

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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.