Achievement of Global Second Order Mesh Convergence for Discontinuous Flows with Adapted Unstructured Meshes

Abstract

In the context of steady CFD computations, some numerical experiments point out that only a global mesh convergence order of one is numerically reached on a sequence of uniformly refined meshes although the considered numerical scheme is second order. This is due to the presence of genuine discontinuities or sharp gradients in the modelled flow. In order to address this issue, a continuous mesh adaptation framework is proposed based on the metric notion. It relies on a L p control of the interpolation error for twice differentiable functions. This theory provides an optimal bound of the interpolation error involving the Hessian of the solution. From this estimate, an optimal metric is exhibited to govern the adapted mesh generation. As regards steady flow computations with discontinuities, a global second order mesh convergence should be obtained. To this end, a higher order smooth approximation of the solution is reconstructed providing an accurate and reliable Hessian evaluation. Several numerical examples in two and three dimensions illustrate that the global convergence order is recovered using this mesh adaptation strategy.

On the use of anisotropic a posteriori error estimators for the adaptative solution of 3D inviscid compressible flows

“…We next look at the effect of varying these adaptation parameters on the quality of the solution, mesh characteristics, etc. Finally we compare the results obtained with our a posteriori estimator with those obtained with the anisotropic a priori estimator introduced in [37]. In this section, we have replaced u h by M h to make explicit that the error estimators and the error on the solution are all computed using the local Mach number M h instead of any of the flow variables ρ h , (ρu) h , etc.…”

“…After documenting this solver/mesher loop, the impact of the two main adaptation parameters associated with our a posteriori estimator is analyzed. We finish this section by briefly comparing our solutions with those obtained through the latest a priori error estimators developed in [37]. Section 4 presents the conclusions of our analysis.…”

“…We compare meshes and solutions obtained with our a posteriori H 1 (Ω)-error estimator with the ones obtained with the a priori L p (Ω)-error estimators introduced in [37]. The idea is to calculate a metric tensor M = M(P ) as in Eq.…”

“…shocks in compressible flows or solidification interfaces, are meshed with high-aspect ratio tetrahedra, allowing the computation of very accurate solutions with a minimal number of elements. Surprisingly the a priori error estimators carry over to inviscid flows with shocks in spite of the fact that the Hessian of a piecewise continuous functions is not per se defined as a function [37]. Practically, for these inviscid flows, an approximate Hessian can be calculated from the solution on any mesh and used to generate the next adapted mesh, hoping that the whole adaptation/solution process converges.…”

Output-Adaptive Tetrahedral Cut-Cell Validation for Sonic Boom Prediction

“…10 This alignment issue has also given rise to hybrid methods [11][12][13] where near-body unstructured grid solutions are interpolated to shock-aligned structured grid methods to increase accuracy. The hybrid methods are hindered by the interpolation process, so adaptive grid methods 14,15 are employed to improve the accuracy of unstructured grid methods for long propagation distances. These previous adaptive methods have used only primal solution information (Mach and density) to drive the adaptive process.…”