## 2 Numerical Methodology
### Optimization Approach  
In this study, the modeFRONTIER 2024R1.1 software is utilized to implement the Non-Dominated Sorting Genetic Algorithm (NSGA) II in conjunction with CONVERGE 3.0 CFD to optimize the operational factors of the methane/diesel RCCI engine [53]. NSGA II is considered a robust optimization design of experiments algorithm, and robustness means the capability of the optimization algorithm to gain the maximum or minimum value of the maximized or minimized objective functions, respectively, even if the starting point is different from the final solution. The implemented optimizing algorithm in mode FRONTIER needs to be initialized by random or specified experimental designs (DOE). Equation (8) shows the minimum DOE size. \[\text{Number of designs}=2\times M\times B \tag{8}\]  
where \(M\) and \(B\) are the number of input variables and objectives, respectively. Figure 2 illustrates the flowchart of the NSGA II algorithm, and the basic steps to implement this algorithm are as follows [54]. Before executing the selection, the citizens are ranked based on the degree of nondomination of each member. First, the non-dominant members are identified as the current citizens. Then, all these members are deemed to compose the first nondominated front within the citizens and are given a high dummy fitness amount. To preserve variety among the citizens, these labeled citizens are then distributed along with their values of dummy fitness. Sharing is attained by employing a selection process depending on weakened fitness amounts, where each citizen's original fitness amount is divided by a factor relevant to the number of nearby citizens, leading to several optimal points for the citizens. After that, the nondominated individuals are \begin{table}
\begin{tabular}{c c} \hline
**Process** & **Model** \\ \hline Turbulence & RANS and RNG k-\(\varepsilon\) \\ Combustion & SAGE \\ Drop collision & NTC method \\ Break-up process of diesel spray & Kelvin–Helmholtz and Rayleigh–Taylor (KH-RT) \\ Evaporation of droplets & Frossling model \\ Droplet coalescence & Post-collision outcome model \\ Spray droplet interaction with solid surfaces & Rebound/slide model \\ Formation of NO\({}_{x}\) & Zeldovich mechanism \\ Soot oxidation rate & Hiroyasu–NSC empirical soot model \\ \hline \end{tabular}
\end{table} 
Table 3: Processes and their corresponding models. disregarded temporarily to process the rest of the citizens similarly and classify candidates for the next nondominated front. These nondominated results are granted an updated dummy fitness value, which is kept less than the lower value of the shared dummy fitness value of the prior front. This process persists until all citizens are sorted into several fronts and arranged in descending order, where the first front has a higher fitness value, the second one has a lower one, and so on [55]. NSGA-II performs two operations to create new designs: mutation and crossover. Crossover comprises several designs in which the parent generation subrogates its genetic information to construct a new design for the following generation. The objective is to produce individuals with better traits than their parents while preserving the variety of the citizens. Mutation introduces random alterations to the genetic information of a design, generating a new variant in the subsequent generation. Doing so presents a degree of randomness in the search. It enables exploring various configuration space regions to prevent the optimization algorithm's early convergence. Crossover also assists in improving the convergence. The maximum probability for the crossover from other studies is around 0.6 to 0.95. In contrast, the mutation probability is significantly low in the range of 0.001 to 0.05, which can result in a high level of mixing and exploitation with comparatively minimal exploration [54]. In the present study, the peak firing pressure (PFP) and the indicated specific fuel consumption (ISFC) are selected as objectives of the optimization study.