# Specific Modeling Issues on an Adaptive Winglet Skeleton
## 2. Criticalities and Approach  
The just mentioned stress concentration has occurred on the zone highlighted in Figure 4, depicting the stress map for the aforementioned load case 1. This stress distribution has been assumed as the standard reference load (RL). The type of solicitation is in line with the bending action of the load case, that produces a compression of the skin in span direction and its bending around the spar cap along the chord. This, jointly to the dramatic variation of the local thickness (passing from the overlap of the skin laminate and the cap to the skin alone), can justify the stress concentration. However, it is questionable whether the discretization could have a role on the estimate of the stress level. To answer this question a parametric study has been organized, considering progressive refinement levels of the mesh on the zone of interest. This skin  
Figure 4. Stress concentration on the upper zone of the skin, close to the rear spar. region, as shown in Figure 5, is constituted by two consecutive lines of seven 19 \(\times\) 19 mm plate elements, towards the span direction. It covers the skin part upstream the spar cap and corresponds to the critical zone highlighted in Figure 4.  
At first, the elements have been split along the chord, since the highest stress gradient occurs along this direction. Then a further split has been performed along the spar, obtaining again square elements. This process -split along the chord and then along the span- has been repeated for two times, coming to the end to square elements 1/4 sized with respect to the original ones. To guarantee an adequate continuity with respect to the surrounding mesh, the boundary elements have been suitably split into triangles, fitting the nodes on the edges. The stress maps illustrated in Figure 6 show the effect of the level of refinement in terms of stress and distribution. The maximum normalized stress obtained has been summarized in Table 1 and plotted in Figure 7. \begin{table}
\begin{tabular}{c c c} \hline
**Case ID** & **Element Size (mm)** & **Normalised Stress (\%)** \\ \hline reference & 19 \(\times\) 19 & 100.0 \\ case a & 19 \(\times\) 9.5 & 107.3 \\ case b & 9.5 \(\times\) 9.5 & 97.3 \\ case c & 9.5 \(\times\) 4.75 & 130.9 \\ case d & 4.75 \(\times\) 4.75 & 154.5 \\ \hline \end{tabular} (*) The second dimension refers to the chord direction.  
\end{table} 
Table 1: Maximum normalized stress vs. mesh refinement level; normalization reference: maximum stress computed on the standard reference load (RL) distribution reported in Figure 4.  
Figure 5: Zone of the model considered for the parametric refinement. Figure 6: Normalized stress maps vs. mesh refinement level: 1 st chordwise split (**a**), 1 st spanwise split (**b**), 2 nd chordwise split (**c**), 2 nd spanwise split (**d**); normalization reference: maximum stress computed on the stress distribution reported in Figure 4 (RL). Figure 7: Normalized maximum failure index vs. mesh refinement level: 1 st chordwise split (**a**), 1 st spanwise split (**b**), 2 nd chordwise split (**c**), 2 nd spanwise split (**d**); normalization reference: maximum stress computed on the distribution reported in Figure 4 (RL). The refinement process highlighted, in general, an increasing trend in terms of stress level. This can be explained considering two numerical aspects. The first one is represented by the strain definition as ratio (gradient) of the local displacement over the element size: as this latter term goes to zero over a certain threshold, the division badly handled by the solver. The second aspect is strictly related to the inversion of the structural matrix, whose conditioning level is dramatically affected by its size. This aspect, for the specific case under investigation, however, does not play a fundamental role, since the increase of DOFs due to the local refinement is really modest with respect to the original size of the structural matrix.