## 5 Numerical Validation of the Model
### Modal Analysis
#### 5.1.1 Square Wall  
The analyzed square wall had a width of 2.5 m and a thickness of 0.3 m. For this wall, the mesh was designated as \(n\times m\), where \(n\) represents the number of DFM elements along the wall's width, and \(m\) represents the number of DFM elements along its height. For instance, a wall modeled with a single DFM element was labeled as 1 \(\times\) 1. Examples of the meshes used for each model are illustrated in Figure 8.  
The mode shapes for the primary modes are depicted in Figure 9 for both the DFM and the reference 3D FE model. While minor discrepancies were observed, for instance, in the displacement at the midpoint of the wall for the second OOP bending mode, the overall deformed shapes produced by both models were, fundamentally, identical.  
The DFM's estimation of the natural frequencies demonstrated significant accuracy for all but the coarsest 3 \(\times\) 3 mesh, which exhibited an error of approximately 12%. The highest error observed with more refined meshes did not exceed 8%, as shown in Figure 10. This figure also highlights how the error varies with mesh refinement: while finer meshes enhance accuracy for certain modes (e.g., Mode 1 and Mode 5), they do not uniformly reduce the error across all modes. Notably, the error for the first vertical tension/compression mode (Mode 4) remains consistently around 8%. This persistent error in Mode 4 suggests an inherent limitation in the DFM's simplified approach to modeling complex structural behaviors. Nonetheless, the achieved accuracy is deemed acceptable for the seismic analysis of masonry buildings.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & Stiffness & \\ Young’s modulus & \(E_{m}\) & 44 MPa \\ Shear modulus & \(G_{m}\) & 22.2 MPa \\ \hline  & Shear strength & \\ Tensile strength & \(f_{t}\) & 18.4 kPa \\ Cohesion & \(c\) & 27 kPa \\ Friction coefficient & \(\mu\) & 0.6 \\ \hline  & Tri-linear behavior & \\ Cracks initiation & \(F_{cr}/F_{u}\) & 0.5 \\ Displacement at maximal & \(d_{u}/d_{cr}\) & 6. \\ strength & \\ \hline  & Lintel properties & \\ Lintel Young modulus & E\({}_{l}\) & 9 GPa \\ Lintel Poisson’s ratio & \(\
u_{l}\) & 0.2 \\ Lintel section & A\({}_{l}\) & \(5\times 5\) cm\({}^{2}\) \\ Lintel density & \(\rho_{l}\) & 700 kg\(\cdot\)m\({}^{-3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Material properties of the masonry.  
Figure 8: Examples of the mesh used for the square wall: (**a**) 8 \(\times\) 8 mesh with the DFM ; (**b**) 20 \(\times\) 30 \(\times\) 5 mesh with the 3D cubic FE model.  
Figure 9: Mode shapes and frequencies for the primary modes computed with the 10 \(\times\) 10 DFM and the reference 3D FE model.  
The evolution of the natural frequency of the 3 rd mode, as depicted in Figure 11, highlights the trend in error determination for this mode and explains why the error did not decrease with finer meshes. Coarser meshes tended to underestimate this frequency, while finer meshes tended to overestimate it. As a result, the 7 \(\times\) 7 mesh achieved a more accurate approximation of this frequency compared to even finer meshes.  
This modal analysis of a square wall demonstrated that the DFM accurately captures the primary mode shapes and their corresponding natural frequencies with good precision, even with a limited number of DOF.