A Combined Entropy and Output-Based Approach for Mesh Refinement and Error Estimation Using Finite-Element Formulations

Abstract

As aerospace design programs rely more on computational fluid dynamics (CFD) in all phases of the engineering design process, there is a growing interest in obtaining high-fidelity solutions using high-order finite-element methods. Such aspirations require considerable research on error estimation and mesh adaptation techniques to obtain high-quality results with lower computation costs. Adaptive mesh refinement approaches generally rely on an error indicator to drive the adaptation. Feature-based error indicators are based on directly computable solution characteristics and provide a relatively inexpensive way to target areas of the mesh for refinement. However, these indicators are not robust as they assume local error depends solely on the local resolution of the mesh. Particularly for hyperbolic problems, such as convection-dominated flows, these methods may inadequately refine areas essential for the accurate prediction of integrated forces and moments.

Alternatively, goal-oriented error indicators specifically target areas of the computational domain that are critical to the prediction of a specific output. These methods are particularly effective at accounting for propagation effects through the use of adjoint solutions that provide the sensitivity of the output to local residuals. This dissertation presents a strategy for mesh adaptation driven by a new error indicator that combines two previously-investigated adjoint-based indicators: one based on a user-specified engineering output such as drag or lift coefficient, and the other based on entropy variables. The novelty of this approach lies not only in its construction but also in its implementation with multiple types of high-order, finite-element discretizations.

Using the entropy-variable indicator to adapt a mesh is computationally advantageous since it does not require the solution of an auxiliary adjoint equation, which is costly for unsteady problems. However, the entropy-variable indicator targets any region of the domain where spurious entropy is generated, regardless of whether or not this region affects an engineering output of interest. Conversely, an indicator computed from an engineering output generally targets only those regions important for the chosen output, though it is more computationally taxing because of the required adjoint solution. Approximations in the adjoint calculation reduce this cost, at the expense of indicator accuracy. In combining these indicators, the objective is to maintain the low cost of approximate adjoint solutions while achieving improved indicator accuracy from the entropy variables. To further reduce the computational cost, various modifications to the combined approach are also analyzed. In this dissertation, we demonstrate the benefits of the combined approach for not only steady-state simulations, where only spatial error is considered, but also unsteady simulations where both spatial and temporal errors must be considered. Various applications of fluid governing equations with multiple types of mesh adaptation strategies are considered to show how robust this novel combined approach currently is.

Much of the work presented in this dissertation is in the context of the Discontinuous Galkerin (DG) finite-element discretization. To not only further demonstrate the robustness of the novel combined approach, but to also investigate a different finite-element discretization, a code was written as part of this dissertation that incorporates the Streamline-Upwind Petrov-Galerkin (SUPG) method. Specifically for moderate orders of accuracy, SUPG requires fewer degrees of freedom than DG for comparable accuracy. Comparisons of output error estimates using the combined approach for both SUPG and DG methods are included in this work to further demonstrate the potential of this new error estimation approach.