EU Regional School - Hartmann Seminar
Prof. Dr. Hartmann - Error Estimation and HP-Adaptive Mesh Refinement for Aerodynamic Flows
German Aerospace Center
C²A²S²E Center for Computer Applications in AeroSpace Science and Engineering
The requirements for numerical algorithms have changed. In addition to an accurate computation of an approximate solution it is of increasing importance to quantify uncertainties associated to the computational result. We present adjoint-based techniques to estimate the error of a numerical flow solution with respect to a given target quantity like an aerodynamic force coefficient. This estimate can be used to judge the overall accuracy of a computation, to enhance the computed value of the target quantity and to drive a solution-adaptive mesh refinement process. The error estimation procedure is extended to multiple target quantities. Considering arbitrary target quantities a residual-based error estimation technique can be devised which targets the resolution of all flow features in an adaptive algorithm.
The discontinuous Galerkin (DG) method offers the ability to achieve high orders of convergence on general unstructured meshes, but like any high order approach, it only yields the improved order globally if the underlying flow is smooth enough. If this is not the case, a globally high order approach will not yield enough improvement in accuracy to justify the additional cost. A suitable combination of local mesh refinement (h-refinement) and a local variation of the order of the scheme (p-refinement) can be employed to achieve an optimal efficiency of the overall scheme also for non-smooth flows including shocks. A smoothness estimation based on a truncated Legendre series expansion of the solution can be employed to locally select the more promising strategy. Numerical examples for inviscid, laminar viscous and turbulent viscous flows demonstrate the efficiency of the proposed algorithms.