# Building Geometry Simplification for Improving Mesh Quality of Numerical Analysis Model
## 5 Evaluation of Simplification Effect
### Experiment Settings and Analysis  
The changes in geometric shape and mesh quality were analyzed according to the threshold of the proposed geometry simplification method. As the building model was mostly composed of perpendicular or parallel surfaces, in the simplification method, the distance threshold had a more dominant effect on the results than the angle threshold. Therefore, the angle threshold (\(\phi_{threshold}\)) was fixed at 90\({}^{\circ}\), and the degree of the simplification was controlled according to the distance threshold (\(d_{threshold}\)). Considering the differences in the scale and complexity of individual models, the distance threshold was standardized by a characteristic length, i.e., a metric in the length dimension representing the geometric properties. Each model was simplified with a distance threshold of 0.01, 0.05, 0.1, 0.5, 1.0, 5.0, or 10.0 times the characteristic length, and was compared with the original model. In view of a CFD simulation or fire safety analysis for the building, the geometry of the openings (windows and doors), which significantly affects the indoor aerodynamic environment, was excluded from the target of simplification. There is a trade-off relationship between the accuracy and the computational cost of the numerical analysis according to the number of meshes. The researcher controls the size and number of meshes considering the required accuracy of the model and available computational resources. Considering that, mesh sizes of 0.25, 0.5, 1.0, 2.0, and 4.0 m were designed for each model.  
The applicability of the geometry simplification was evaluated in terms of the degree of change in the geometry and mesh quality. The similarity between the original and the simplified model was quantitatively evaluated by the Bhattacharyya coefficient between D2 shape distributions. The D2 shape distribution was calculated from the distance between 1,000,000 pairs of points that were randomly extracted from the surface of the target geometry. The mesh quality was evaluated based on the maximum non-orthogonality, maximum skewness, and ratio of hexahedron meshes. In addition, the number of cells was evaluated as it was proportional to the computational cost of the numerical analysis model. The change in each index according to the simplification threshold was analyzed using  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Properties** & **Volume (m\({}^{3}\))** & **Surface Area (m\({}^{2}\))** & **Characteristic Length (m)** & **ITC** \\ \hline Average & 170.4 & 233.6 & 0.592 & 4.25e-2 \\ Standard deviation & 210.8 & 228.7 & 0.223 & 0.101 \\ Minimum & 0.960 & 37.6 & 1.47e-2 & 6.46e-4 \\ Maximum & 1062.6 & 1251.7 & 1.025 & 0.583 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Geometric properties of the building geometry dataset.  
a linear regression model based on ordinary least squares. Unlike other metrics, the skewness does not have an upper limit. Typically, high-quality meshes have a maximum skewness between 0 and 4; however, extremely skewed cells with skewness values of \(1\times 10^{10}\) or more can be generated near the surface of some complex geometries. To minimize the effects of these outliers in the statistical analysis, the elliptic envelope method [43] was applied to the data from the mesh design to remove outliers of 1% of the entire data.