Interpolation Parameters of the GSB Method

In the top left image of Fig. 12, the RMS deviations in the flap-wise bending deflec­tion are plotted over the minimum number of supporting edge midpoints NM, min. The radius of the additional support points was fixed at rFB = 0.15 m. In the GSB method the actual number of support points differs all over the wetted surface, as was explained on page 193. The support points of a given surface point are all the edge midpoints and additional support points within the support radius 5. The de­formed wetted surface used here as a reference was obtained with the FIE method and its default parameter settings. The deviations do not seem to be influenced by the number of support points. The RMS values hover at 1.7% for linear global inter­polation functions and at 0.12% for quadratic ones. Yet the wetted surfaces obtained with MLS are not completely identical, as is documented in the bottom left diagram. It shows the RMS deviation relative to the deformed surface resulting from the GSB method with NM, min = 8. These deviations, however, are at least one order of mag­nitude smaller than those relative to the FIE reference case.

The top right image of Fig. 12 shows the influence of the radius of the additional support points. Eight fixed radii are investigated as well as the alignment of the additional support points with the wetted surface. The number of supporting edge midpoints is NM, min = 8; the reference wetted surface as before resulted from the FIE method. The better the spatial arrangement of the additional support points ap­proximates the wetted surface, the lower the average deviations come to be. For both linear and quadratic global contributions to the interpolation function the smallest RMS values are achieved with an alignment of the additional support points. Next best is rFB = 0.15 m, which is approximately half the mean chord length. Between rFB = 0.276 m and rFB = 0.474 m a marked increase in deviations occurs. This is be­cause rFB becomes larger than the fixed support radius 5, and the additional support points cease to come into play. Case 0 in the bottom right image shows the distri­bution of the normalised deviations. These increase over the length of the span and values of Auy/uytip| > 2 are reached at the tip. Cases 0 and 0 highlight a general

problem of the GSB method: The deformation distribution is generally not approx­imated well by the global polynomial contribution to the interpolation function, not even by a quadratic one. This is to be compensated by the local RBF contributions of the edge midpoints and of the additional support points. The deviations become large in regions of the wetted surface far-off the support points, for instance at the leading and trailing edges. Consequently, case © with alignment of the additional support points with the wetted surface reveals significant improvement over cases with fixed values of rFB.

Ahrem et al. [1] propose breaking down the configuration into sections and ap­plying the GSB method on each one by itself. The projection results are smoothly interpolated by a partition of unity algorithm. In each section a different fit for the global polynomial contribution is obtained which results in a better approximation of the wetted surface. Simultaneously the memory requirements and the numerical effort are reduced. The downside is the introduction of yet another interpolation scheme, and this approach has not been included in the ACM.