## 3 Experimental Tests and Results
### ZERO(tm) Machine Tests
#### 3.2.2 FEM-Based Validation of ZERO(tm) Machine Tests  
A second approach was undertaken to validate the correlation between the measured thermal error evolution and environmental temperature variations using FEM simulations. The ZERO(tm) machine was modelled using Altair Optistruct(tm) 2024 Software (Altair Engineering Inc., Troy, MI, USA), as shown in Figure 5b. A transient thermal analysis was conducted using the recorded environmental temperature variations from 23 to 26 February as input data. The analysis output, representing the TCP displacement at each of the 100 calculated steps, was then compared with the errors measured at the corresponding steps during the MIIM process. Several points within the point cloud were analysed to confirm the validity of the simulation results, which yielded good correlations across all tested positions. For clarity, the results for two significant and easily interpretable positions are presented below, where they are compared in detail with the experimentally measured errors from the MIIM tests. Figure 21 shows the calculated positions corresponding to Point P_11 and Point P_31 (refer to Figure 14). These positions are critical for determining the Y-axis error and are particularly sensitive to environmental temperature variations, as previously explained. Position 1 is expected to exhibit a greater degree of influence from temperature changes, with a proportional relationship to the intermediate position. This behaviour aligns with a convective heat transfer scenario and the linear expansion characteristics of the structural material. Figure 22 illustrates the measured data and the FEM model output obtained using the ambient temperature variations recorded during the test conducted from 23 to 26 February. The graph represents the measured E\({}_{\text{Y}}\) error evolution at Point P_11 alongside the FEM  
Figure 21: ZERO(tm) FEM calculated positions: (**a**) Position 1—maximum Y travel, and (**b**) Position 2—Intermediate Y travel. \begin{table}
\begin{tabular}{c c c c} \hline  & **Date** & **MAE [\(\upmu\)m]** & **RMSE [\(\upmu\)m]** \\ \hline Adjustment & 22 to 26 February & 6.9 & 9.6 \\ \hline Validation & 12 to 15 February & 10.1 & 14.1 \\ \hline \end{tabular}
\end{table} 
Table 9: Comparison metrics between measured values and first order model predictions. simulated value. This specific point corresponds to the maximum extension of the ram (Point 11 in Figure 14 and Position 1 in Figure 21a). Figure 23 presents the FEM results in the Y direction in the specific moment indicated with a circle in Figure 22, which is near the conclusion of the test, when the machine responds to the activation of the heating system and the error begins to reduce. The undeformed and deformed structures are displayed together in the image, illustrating how the contraction of the ram predominantly contributes to the error in the Y direction. The corresponding error values are also plotted. Figure 24 illustrates the measured data and the FEM model output obtained using the ambient temperature variations recorded during the test conducted from 23 to 26 February. The graph represents the measured \(\mathrm{E_{Y}}\) error evolution at point 31 alongside the FEM  
Figure 23: FEM-simulated Y error (mm). Second test, from 23 to 26 February. Position 1 (maximum ram extension). Figure 22: FEM simulation vs measured data: second test, from 23 to 26 February. Position 1 (maximum ram extension). simulated value. This specific point corresponds to the intermediate extension of the ram (Point 31 in Figure 14, Position 2 in Figure 21b). The FEM-based validation of the ZERO(tm) machine tests has demonstrated a strong correlation between the simulated and measured thermal error evolutions, with the FEM model accurately capturing the influence of environmental temperature variations on the Y-axis position errors.