# Building Geometry Simplification for Improving Mesh Quality of Numerical Analysis Model
## 6 Results and Discussion
### Mesh Quality Improvement by Geometry Simplification  
A hexahedral mesh of each size was designed for the original and simplified geometries, and the respective mesh qualities were evaluated. The data of 10,661 cases were collected, except for some cases where simplification was not performed because the distance threshold value was too small, or where mesh generation failed owing to the excessively large mesh size when compared with the volume of the model. Among them, there were 10,549 datasets with approximately 1% of the outliers removed considering the distribution of the mesh quality. Among the mesh quality metrics, the maximum skewness formed a distribution with an extremely long tail on the right side (skewness of distribution 15.54); considering this distribution, the maximum skewness was log-transformed before the analysis. Figure 9 shows the proportion of hexahedron cells in the mesh and the relative number of cells according to the simplification threshold and mesh size. From the result of meshing the original geometry, the proportion of hexahedron cells among all cells tended to decrease as the mesh size increased (Figure 9a). Hexahedron cells have the advantage of having fewer numerical errors, and can be designed with a relatively small number of cells, but it is difficult to achieve a high-quality mesh design for complex geometries. The volume range of the target building models was 0.96-1062.6 m\({}^{3}\), and the characteristic length range was 0.015-1.025 m. As the mesh size was set to be larger relative to the volume and complexity of the target geometry, it was believed that the proportion of cells with geometries such as tetrahedron, pyramid, and prism increased to represent the detailed geometry of the model surface. The hexahedron cell ratio of the simplified model increased linearly according to the simplification threshold. For all mesh sizes, the hexahedron cell ratio of the model simplified with \(d_{threshold}=10.0L_{C0}\) increased by an average of 10.4% when compared with the original model. Figure 9b shows the number of cells standardized with the cell number of the original model. Overall, the number of cells decreased with simplification, and the effect of the computational cost reduction in the numerical analysis was expected. However, the number of cells exceeded the  
Figure 9: Statistical features of mesh according to simplification threshold and mesh size. original model in some conditions with \(d_{threshold}\leq 0.5L_{C0}\), particularly when the mesh size was 4.0 m. This was believed to result from the generation of cells other than the hexahedron, as described above, which could be prevented by setting an appropriate simplification threshold and mesh size. Figure 10 shows the changes in the maximum non-orthogonality and skewness, which affect the accuracy and stability of the FVM. Both metrics tended to decrease with an increasing simplification threshold, that is, the mesh quality improved. The total average maximum non-orthogonality of the original model was \(69.6^{\circ}\), and the log-transformed maximum skewness (\(\log{(\epsilon)}\))) was 0.82. With the geometry simplification, the metrics decreased to \(65.3^{\circ}\) and 0.47 with \(d_{threshold}=1.0L_{C0}\); under the condition of \(d_{threshold}=10.0L_{C0}\), the values decreased to \(51.0^{\circ}\) and \(-0.13\), respectively. In contrast, as a larger mesh was designed, the mesh quality improved on average; however, there was no consistent tendency. For example, the maximum skewness of the 4.0-m mesh was higher than that of the 2.0-m mesh under the \(d_{threshold}\leq 1.0L_{C0}\) condition. For a quantitative analysis of the effect of the geometry simplification on the mesh quality, a multiple linear regression analysis was performed (Tables 3-5). The independent variables included the ITC (\(C_{ITC0}\)) and characteristic length (\(L_{C0}\)) of the original geometry, the maximum size of the designed mesh (s), and the degree of simplification normalized with the characteristic length (\(d_{threshold}/L_{C0}\)).