# Preconditioning the Quad Dominant Mesh Generator for Ship Structural Analysis
## 4 Discussion and Conclusions  
Let us now consider a more involved geometry describing a superstructure of a large yacht. This geometry will be denoted by \(G_{3}\) and is depicted in Figures 10 and 11. The generated mesh will have more than hundred thousand elements and we report the quality measures for the preconditioned recombination and the full recombination algorithms. The geometry, together with details of the performance of two algorithms on the web surfaces, is presented in Figure 6.  
In order to emphasize the context of the mesh generation in ship structural analysis, let us emphasize that we are aiming for the coarsest mesh which satisfies the meshing rules posed by the classification societies. There are two reasons for this. One is the stability of the solution method which depends on the condition number of the finite element stiffness matrices which in turn deteriorate with aggressive mesh refinement. In other words, the element shape has much more adverse effects on the accuracy than the element size. The other is that there is a lower limit on the mesh size which is incurred by the material properties (thickness of the plates). Subsequently, the rules of the classification societies are to be interpreted as ingredients of the cost functions for the optimization and the quality measure of the mesh should further be penalized by the number of elements. This is precisely what we have achieved with our algorithm. We have chosen the combination of  
Figure 5: A comparison of the meshes for a web surface with openings from the geometry \(G_{2}\). We see that the full recombination algorithm produces an almost fully quadrilateral mesh, but the quality of the mesh, as measured by the SIGE indicator, for the full recombination algorithm is lower. In (**a**), we have the mesh generated by pyREMAKmsh and in (**b**) the fully recombined mesh. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Geometry \(G_{3}\)** & \(\overline{\eta}(\mathcal{T})\) & \(\xi(\mathcal{T})\) & \(\tau(\mathcal{T})\) & \(\delta(\mathcal{T})\) & \(|\mathcal{T}|\) & **tr. [s]** & **rec. [s]** \\ \hline pyREMAKEmsh & 94 & 78 & 94 & 1.95 & 128,053 & 21.73 & 115.35 \\ Full recomb. & 85 & 60 & 99.8 & 18.37 & 232,720 & 154.27 & 122.58 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The mesh quality of the superstructure of a large yacht (geometry \(G_{3}\)). Figure 6: A comparison of the meshing algorithm on an example of a superstructure of a large yacht (geometry \(G_{3}\)). The details of the web surfaces are compared using the SIGE indicator. In (**a**), we have the mesh generated by pyREMAKmsh and in (**b**) the fully recombined mesh. the algorithms (recombination and triangulation) together with the preconditioning steps (Boolean operations and virtual stiffeners) which optimize precisely such cost functions. A further argument for the preference of coarser meshes stems from the fact that we intend this algorithm to be used in the early stages of the design process when many design alternatives are tested, and this procedure allows for a cheap assessment if the proposed design adheres to the rules of the classification society. This design might be discarded later due to other reasons, so having to repeat only a cheap and reliable procedure is a big gain for a designers who might get to test more alternatives which are all conforming. We see that the pyREMAKEmsh algorithm typically achieves the quality restrictions as posed by the classification societies. It should be noted that on challenging geometries smaller in size (containing web surfaces with many openings), it is possible that the preconditioned Packing for Parallelograms algorithm (as implemented in pyREMAKEmsh can have as many as 10% triangles (geometries \(G_{2}\), \(W_{1}\) and \(W_{2}\)). However, those triangles do not affect, as measured by the SIGE indicator, the high stress area adversely. On the other hand, the full recombination algorithm can achieve and almost exclusively quadrilateral mesh at the expense of a lower element quality.