Modification of Adapted Grids on the Base of a Geometrical Element Quality

The background of another new option of the TAU adaptation is that the error es­timation, of advanced adjoint-based methods as well as of simple differences-based methods, determines a new point density for the initial grid. On the other hand, the local discretization error depends on the local resolution and probably on the element shape and alignment. So the ideal grid adaptation would adapt the point density without changing or at least without worsening the element shape. This is impossible for a hierarchical conform refinement as the TAU adaptation performs.

The preferable solution was to evaluate the influence of the element shape on the local deiscretization error and to consider the result when determining the refine­ment state. However, the attempt to estimate the numerical error by evaluating the numerical fluxes in the control volume or at least the first derivative of a variable for the methods used by TAU lead to very complex formulae and a large diversity of possible element configurations around a point. Because of the low prospect of success this trial was abandoned. In the mean time, a similar problem seems to be solved for the two-dimensional case, using symbolic computations [7].

Partial results of the analytical investigations suggest that the local discretization error is comparatively small for uniform grids. Therefore, it seems to be worthwhile considering a geometrical element quality [2,4, 12] for replacing the element shape related part of the local discretization error. The main problem of this approach seems to be that the converse argument is not true. Grids or areas with apparently low element quality may provide good dual grids in terms of rectangularity of dual edges and faces.