# Building Geometry Simplification for Improving Mesh Quality of Numerical Analysis Model
## 6 Results and Discussion  
### Simplification of Building Geometry  
A total of 40 geometries were simplified under six distance thresholds each. The average execution time of the implemented algorithm was 137.5 s (median 37.3 s) with Ryzen 7 3800X processor and 32 GB RAM. Figure 6 shows the simplification of a typical building model for which the majority of adjacent surfaces are orthogonal to each other. The gray surfaces and blue surfaces represent ordinary walls and openings respectively. Figure 6a shows Space 1E24 in the _Medical Clinic_ model. As the distance threshold (\(d_{threshold}\)) of the geometry simplification increased, the details of the geometry were gradually removed, and the overall geometry tended to be similar to a simple cube. The characteristic length of the original model (\(L_{\text{C0}}\)) was calculated as 0.848 m. As a result of applying the geometry simplification, an inner wall with a thickness of 0.124 m was removed under the condition of a distance threshold of 0.5\(L_{\text{C0}}\) (= 0.424 m). Wall faces with vertical distances of 0.353 m and 0.916 m were integrated with a threshold of 1.0\(L_{\text{C0}}\) (= 0.848 m) and 5.0\(L_{\text{C0}}\)(= 4.241 m), respectively. Figure 6b illustrates Space 04 from the _Academic Kingo_ model. For \(d_{threshold}=~{}0.5L_{\text{C0}}\), some parts of the ceiling surface parallel to the floor were united into the remaining non-horizontal ceiling. As the opening geometry was excluded from the simplification target, when the distance threshold is 1.0\(L_{\text{C0}}\) or greater, some of the wall face is shifted to a parallel plane to the face of the doors. When the model geometry contained several non-orthogonal and curved surfaces, the geometry mainly changed with a low threshold of below 1.0\(L_{\text{C0}}\) (Figure 7). This was considered to be because the length of the perpendicular from the center, which is a criterion for surface integration, was relatively short owing to the small angle difference between faces. Figure 7a is Space 123 of the _Academic Advanced Sample_ model. The circular column inside the original model was represented as the column of a regular 24-sided polygon as a generating surface mesh. Upon applying the simplification, it was transformed to an octagonal column (\(d_{threshold}=0.1L_{\text{C0}}\)) and square column (\(d_{threshold}=0.5L_{\text{C0}}\)). When the distance threshold was 1.0\(L_{\text{C0}}\)(= 0.756 m) or longer, the column was removed as the distance threshold exceeded the diameter (0.3 m) of the column. Figure 7b is Space B112 in the _OTC Conference Center_ model. Figure 6: Simplification of typical building geometry. (**a**) Space 1E24 of Medical Clinic, (**b**) Space 04 of Academic Kingo. As in the above examples, it can be observed that the number of surfaces decreases as the distance threshold increases. Figure 8 shows the averages of the geometric properties of the target model according to the simplification threshold. Based on the calculated results of the Bhattacharyya coefficient, the similarity with the original geometry tended to decrease as the degree of simplification increased. When the distance threshold was a maximum (\(d_{threshold}=10.0L_{C0}\)), the mean of the Bhattacharyya coefficient was 0.963, and the standard error was \(5.83\times 10^{-3}\) (Figure 8a). Based on the results of the statistical analysis, no significant change in the Bhattacharyya coefficient was observed according to the geometrical properties (ITC and characteristic length). In contrast, the distance threshold of the simplification had a linear relationship with the Bhattacharyya coefficient. From the result of regression analysis (Table 2), the similarity with the original geometry decreased by approximately 0.31% when the distance threshold increased by 1.0 \(L_{C0}\).