Mesh Adaptation for Computational Fluid Dynamics, Volume 2

4Goal-Oriented Adaptation for Inviscid Steady Flows

The purpose of this chapter is to introduce a mesh adaptation method, which will be closer to the objective of reducing the approximation error, taking into account the discretization of the PDE. In the goal-oriented approach, the user specifies a particular scalar output of the computation. The error in approximating this output through the discretization of the PDE is minimized. The goal-oriented approach relies on (1) a residual/local error estimate in terms of the metric, and (2) an adjoint state relating this estimate to the functional. The use of a continuous metric parameterization permits to build a smooth optimization problem solved via an optimality system. This method is an important method for this book. It is first explained for the case of steady Euler flows with application to sonic boom calculations.

4.1. Introduction

In the previous chapters, the adaptive specification of the mesh is mainly deduced from an interpolation error estimate for some fields (the sensors or the features) related to the PDE solution. Focusing on these interpolation errors is a limitation of this study. If for many applications, the simplicity of this standpoint is an advantage, there are also many applications where feature-based adaptation is far from being optimal regarding the way the degrees of freedom are distributed in the computational domain. Indeed, minimizing the interpolation error is not often so close to the actual objective ...