Mesh Adaptation

The need for adapted meshes comes from the drive for larger meshes. As the number of mesh points increases, the computational work required to solve the problem increases faster. If, as is the case in many other disciplines, suitable meshes can be found to capture the solution with high accuracy but with fewer mesh points on the finest mesh, then the work required would be dramatically reduced.

My work, started during my PhD and continued as part of my ROPA grant, was to introduce adaptive meshing to EHL. By observation of the known solution properties one would expect the mesh to be finest around the pressure spike/ridge. The contact area - especially the inflow start of it, would need to be captured accurately. The free surface of the cavitation region boundary would be expected to be important. However in the low pressure regions where the deformation is small and changes to the fluid properties minimal then a coarser mesh could be suitable.

Such an approach was implemented by the progressively finer meshes in the multigrid scheme being increasingly adapted. This was done under several different sets of criteria for assessment purposes, namely pre-defined boxes requiring more and less refinement; refinement based o some monitor function, eg the pressure; and refinement based on a multigrid convergence property. Taking care about where to adapt is especially important around the cavitation boundary, as this must be free to shift positions so points either side must be included.

The reduction of work has not included any reduction in the deformation calculation solve yet, as the multilevel multi-integration scheme seemed to couple the regularity of the points into its effectiveness. Reductions in the calculations required elsewhere, though, have made up for this, certainly in these initial stages of this work.

Below we can see computational results from an adapted rnu. The picture on the left shows a pressure based refinement criteria, with the individual points representing the mesh points used on that grid. On the right are computational timings for three mesh levels under each of the three chosen refinement techniques.

Grid Refinement type Time for 10 iterations (s) Saving on unadapted case (s) Percentage time saved 129x129 Geometry 19.5 11.8 37.7 129x129 Pressure value 20.5 10.8 34.6 129x129 Error test 30.4 1.0 3.1 257x257 Geometry 56.0 61.2 52.2 257x257 Pressure value 58.7 58.4 49.9 257x257 Error test 80.9 36.3 31.0 513x513 Geometry 237 245 51.3 513x513 Pressure value 218 264 55.6 513x513 Error test 276 206 44.3

This work has been published in the proceeding of ECCOMAS 2001 as well as other experiments given in my PhD thesis.