## 3 Results and Discussion
### Numerical Simulations  
The numerical simulation results were recorded when the convergence plot was fully generated with the final minimum estimation error of the solution process for the selected nonlinear solver, as shown in Figure 11. It includes both physical interfaces in Table 4 that identify the operational states of the proposed actuator. Although the starting and ending iteration numbers for the operational states differ, they both end with the same estimation error of \(10^{-3}\). The opening state ends at the 186 th iteration number and the closing state starts at the previous state's last iteration number and ends at the 489 th iteration number. Figure 12 shows the numerical results of both the actuator's opening and closing operation states. They use the internal structure of the actuator that defines its operational mechanism in Figure 3. The external structure is hidden to prevent the use of results with multi-sliced graphs of results to present the internal structure, which can cause some confusion in analysis. Figure 11: Solver’s convergence plot of the estimation errors. The extension of the SMA spring activates the closing position of the actuator. It allows the rod and pintle to move in a downward position. However, the stretching of the SMA spring experiences a maximum stress of 461 MPa, while the other parts of the actuator are less stressed, as shown in Figure 12a. It is noticed that there is a difference of 9 MPa between the numerical and analytical maximum stresses. The difference between the numerical output stress and allowable maximum stress of the SMA spring is 229 MPa. It defines the operability of the actuator for a long time without overstressing. Figure 12b shows the strain of the actuator during the closing state with a maximum strain of the actuator that is recorded on the SMA spring. This strain was below the transformation train prescribed in the SMA properties, with a strain difference of 0.3% between them. Figure 12c shows the displacement that is moved by the pintle to close the IAC actuator. The red colour that indicates the maximum displacement appears on the parts which play a role in the pintle's movements. The red colour on the bias spring comes from the settings chosen to keep the axial pitch of this spring and this spring is identified as the force \(\mathrm{F_{b}}\) during simulation. The maximum closing displacement of the actuator is 6.48 mm, which indicates a difference of 0.02 mm from the analytical simulation. Figure 12d shows the stress of the actuator during the opening process. The maximum stress is 359 MPa, which appears in the springs. The SMA spring compresses the bias spring to regain its initial position and pulls the rod with a pintle to move upward, which leads to high stress like a bias spring. The color bar of stress clearly shows the bias spring as a part of the actuator that experiences stress. This stress is the lowest compared to the closing stress and the permissible working stress of the SMA spring. The strain from the actuator's opening state is shown in Figure 12e. Based on a predefined colourmap of the actuator's strain for the opening state, the bias spring shows the highest strain, and the SMA spring follows it. However, the recorded maximum strain of 2% is lower than the allowable working strain of the spring made from steel, which is between 15% and 20% [54,55]. This strain is also below  
Figure 12: Numerical analysis results of the SMA-based IAC actuator’s closing and opening operating mechanisms. (**a**) Induced stress of the actuator during closing state; (**b**) induced strain of the actuator during closing state; (**c**) closing displacement of the actuator; (**d**) stress induced by the actuator during opening state; (**e**) induced strain of the actuator during opening state; and (**f**) the actuator opening displacement. the maximum strain transformation of the SMA spring. Figure 12f shows the displacement of the actuator to the opening state.