Mesh Adaptation for Computational Fluid Dynamics, Volume 2

2Multi-scale Adaptation for Unsteady Flows

In this chapter, we describe an extension of the multiscale feature-based mesh adaptation method for the calculation of an unsteady flow. The mesh adaptation relies on a metric-based method controlling the L^p-norm of the spatial interpolation error. Mesh-adaptive time advancing is achieved because of a transient fixed-point adaptation algorithm to predict the solution evolution coupled with a metric intersection in time procedure. In time direction, we impose equidistribution of error, that is, minimization in L^∞. This adaptive approach is illustrated with an incompressible two-phase flow using a level set formulation.

2.1. Introduction

Multi-scale metric-based mesh adaptation methods need to be carefully extended before they efficiently apply to unsteady flows. We restrict our discussion to numerical unsteady schemes using time advancing. We consider adaptation methods changing the spatial mesh during the simulation time frame. Two options are usually considered. In the first option, mesh changes during the time advancing, for instance, between tⁿ and tⁿ⁺¹. Due to the difficulties in building a time derivative between meshes of different connectivities, the mesh is generally only deformed, for example, as in the moving finite element (Baines 1994). In the second option, the mesh does not change during the time advancing, but instead at a time level, for example, tⁿ. Once the mesh is changed, the solution is advanced from tⁿ to t