Mesh Adaptation for Computational Fluid Dynamics, Volume 2

7Goal-Oriented Adaptation for Unsteady Flows

Chapter 2 of this volume presents a transient fixed-point (TFP) multi-scale algorithm for the adaptation of mesh to an unsteady flow. Chapter 4 of this volume presents a fully anisotropic goal-oriented (GO) mesh adaptation technique for a steady flow, which uses an adjoint state and allows to define the best mesh for the most accurate evaluation of a given scalar output. The present chapter addresses the extension to unsteady flows of the GO method. Its rationale is to combine the GO of Chapter 4 of this volume with the TFP advances of Chapter 2 of this volume. However, the resulting GO-TFP has to be global in time, that is, the whole time interval needs to be computed before improved meshes are derived. Examples of applications concerns unsteady Euler flows.

7.1. Introduction

In this chapter, we define an extension of the feature-based TFP introduced in Chapter 2 of this volume to a GO formulation. To this end, several methodological issues need to be addressed. First, an error analysis based on the so-called state and adjoint systems is developed in order to set an unsteady mesh optimization problem. Second, as a necessary adaptation of the method introduced in Chapter 2, we define a global transient fixed-point (GTFP) algorithm for solving the coupled system formed, this time of three fields, the unsteady state, the unsteady adjoint state and the adapted meshes. By “global” we mean that mesh is not adapted successively time interval ...