## 3 Results and Discussion
### Optimization Process Results  
The initial step of the analysis of the results was the verification of the algorithm convergence. Figure 7 presents the evolution of the global objective function for all the cases simulated with CFD. It can be seen how the PSO-NS algorithm decreases the value of the objective function as it progresses, converging toward a minimum value. The definition  
Figure 5: Mesh generator: control points and block definition. Figure 6: Virtual injector model: a comparison between several values of injection pressure. of best particle refers to the one with the minimum value of the objective function until that current iteration. The capability of the PSO-NS for rapid convergence is observed, because the objective function decreases quickly from the beginning of the procedure. In this figure, the black dots represent the cases related to the standard particles, and the blue dots represent the particles that explore the domain. The best particle, the one with the lowest objective function, was obtained around the simulated case 720. Subsequently, an analysis of the optimization targets and constraints was performed, to verify the evolution of these parameters during the optimization process. Figure 8 shows the progression of these parameters for each simulated case. The results obtained for soot and efficiency are shown in the bottom graphs and show a great improvement in both variables. Although the optimization process was capable of reducing the soot emissions close to a virtual zero, the efficiency exhibits an improvement of 8.8% in relation to the baseline oxy-fuel combustion case (marked with the horizontal dashed line). Likewise, all the graphs evidence a common information, the high variability of the outputs at the beginning of the optimization process with results that do not meet the constraint values. As the simulations evolve, the majority of the points try to reach the optimized region. However, some points seem to be scattered with relatively bad results in terms of soot or efficiency. These dispersed points are explorer particles that seek new positions throughout the search space, away from the optimal region. Moreover, the top graphs represents the evolution of the constraints of the objective function: CO and maximum pressure respectively. In this figure, the blue dots represent each simulated case, and the black dashed line is the constraint, obtained from the baseline engine configuration operating under oxy-fuel conditions. Along the progress of the optimization, the blue dots tend to move far from their respective constraint value, especially on the reduction of the CO values, due to the high correlation between the reduction of CO emissions and the increase of the efficiency, also reported by [PERSON] et al. [40]. Finally, the optimized case is represented by the red dot in the four plots. Figure 7: Objective function convergence evolution. The dependency of the objective function value on the geometrical parameters is shown in Figure 9, where again the red dot represents the optimized case and the blue dots are the value of each parameter for each case. The geometrical parameter 1 (GP1) adjusts the depth of the piston and is presented in the left-top graph. The optimization process tends to move the value of GP1 toward regions near the maximum limit of this parameter, decreasing the height of the piston bowl. For the geometrical parameter 2, most points are concentrated near to the maximum limit of this parameter, suggesting that the bowl diameter increment would enhance the performance of the oxy-fuel combustion concept. The evolution of the geometrical parameter 3 during the optimization process starts with some cases in the minimum limit moving to an intermediate region of the total range, but nearer to the maximum limit. This evolution modifies the point GP3 in Figure 4, changing the initial geometry of the deep re-entrant piston toward an almost no re-entrant geometry.