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Conclusions for the Application
This contribution introduced three development directions of the TAU adaptation tool followed within the framework of the project MUNA, all aiming for improved adapted grids enabling for higher accuracy.
The first one, the investigation and use of the element decomposability, see Sect. 2, improves the edge refinement algorithm of the TAU adaptation. In effect as much as possible of the grid area is considered for re – or de-refinement, instead of having larger regions which are unintendedly excluded from adaptation. Under the assumption that the refinement indicator provides the correct measure for the necessity to refine edges, this step obviously improves the adapted grids. The better accuracy of the resulting solution was demonstrated for an example, using the adjoint-based error indication.
The second topic, see Sect. 3, provides an option to avoid some of the elements with a low geometrical quality introduced by standard adaptation. Because this feature is not as effective as previously thought and slightly increases the computational effort, it is switched off as a default. However, the grid modification seems to be a useful option in some special situations. The idea of avoiding the worst shaped bridging elements in an adapted grid suggests to try this method in configurations with some poorly shaped elements in grid regions affecting the global solution. The stabilizing effect of the grid modification could possibly be used in examples in which the computation converges very slowly after a grid adaptation or a restart is comletely impossible. At least the option generates slight grid variations better than inserting random points. So it is an instrument for further investigation of uncertainties caused by grid variations.
The use of the adjoint solution for an adjoint-based adaptation, see Fig. 5, significantly improves the accuracy of the result for the target functional obtained on the adapted grids for the investigated test examples. Because of the large effort which is needed for an adjoint-based adapted computation, compared to the conventional differences-based adaptation, a careful cost-benefit analysis has to be done. Some more time and additional tests are needed to find out classes of problems and configurations for which one or the other method is preferable.
