## 3 Materials and Methods
### Residual Stress Experimental Validation
#### 3.4.2 Model Validation  
The parameters were standardized to facilitate a direct comparison between the simulation outcomes and experimental findings. To validate the finite element model, the deep rolling parameters utilized were meticulously documented, as detailed in Table 5. The calculation of the rolling force followed a prescribed formula, denoted as Equation (3). \[\text{F}=\text{P}\pi\text{(D}_{\text{b}}/2)^{2} \tag{3}\]  
where F is the normal force exerted by the tool on the component surface, P is the hydrostatic pressure, and D\({}_{\text{b}}\) is the ball diameter. It was applied on the connector along the u direction of the local coordinate system. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Pressure (Bar)**} & **Rolling Speed** & **Rolling Offset** & **Ball Size** & **Number of** \\  & **(mm/min)** & **(mm)** & **Passes** \\ \hline
150 & 1800 & 0.1 & HG6 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Deep rolling parameters are used to verify the finite element model. The rotational speed of the ball and the corresponding step time for the rolling process were derived from the specimen, resulting in values of 5.2 rad/s and 0.3 s, respectively. Noteworthy is the imperative to nullify the rolling force at the simulation's conclusion to avert excessive stresses at the specimen's periphery. In implementing surface-to-surface contact, the penalty contact method and finite sliding were adopted for mechanical constraint formulation and sliding formulation, respectively. Within the finite element model, a penalty friction formulation was chosen, with a friction coefficient of 0.3 assumed due to its negligible impact on the simulation outcomes. The feed rate was simulated by advancing the ball longitudinally by an equivalent distance along the specimen's length. The deep rolling process simulation comprised a series of explicit dynamic simulations, each initiated from various starting positions of the ball. Notably, the stress and strain outcomes from each simulation were subsequently imported into the succeeding simulation as the initial state, ensuring continuity and coherence throughout the simulation sequence. Such meticulous attention to simulation parameters and methodology is paramount in ensuring the fidelity and reliability of the finite element analysis results. To ensure the accuracy of the FEM simulations, a detailed comparison is carried out between the simulation results and the experimental measurements. The primary goal of this comparison is to validate the FEM model by examining how closely the predicted results align with the actual experimental data. Any discrepancies or deviations identified during the comparison are carefully analyzed, and necessary adjustments are made to the FEM parameters or boundary conditions to enhance its predictive accuracy. For each condition examined in the study, various process parameters are systematically evaluated to identify optimal settings and minimize variability, thereby improving the robustness of the process. Each condition, as defined by the experimental design, is further evaluated in the FEM model to simulate real-world conditions. To quantitatively assess the differences between the FEM and experimental results, statistical analysis is conducted using the paired \(t\)-test in Minitab software (v.22.1.0). This statistical test helps determine whether the differences between paired samples (FEM and experimental data under the same conditions) are statistically significant. The paired \(t\)-test calculates a t-statistic using the Equation (4)  
\[\mathrm{t}=\mathrm{d}/(\mathrm{s}_{d}/n\ ^{1/2}) \tag{4}\]  
where \(d\) is the meaning of the differences between paired observations, \(S_{d}\) is the standard deviation of these differences, and \(n\) is the number of paired observations. By running the paired \(t\)-test for each condition, the accuracy of the FEM model is validated, and any inconsistencies can be addressed through iterative adjustments to improve the model's precision.