## 2 Materials and Methods
### Micro-Scale Spherical Surface Modeling via Genetic Algorithm
The flowchart of the genetic algorithm to construct the spherical surface modeling is shown in Figure 2b. From this structure, the genetic algorithm performs the first step to determine the initial population. To do so, Equations (4)-(6) are computed by employing \(m=10\), \(n=10\), \(\Delta=5\), and \(\delta=5\). Thus, the parent \(P_{1,1}\) is obtained by computing Equations (4)-(6) via surface coordinates whose sub-indexes are (\(i=0\), \(j=0\), \(\delta=5\), \(\Delta=5\)), the parent \(P_{2,1}\) is determined via sub-indexes (\(i=0\), \(j=5\), \(\delta=5\), \(\Delta=5\)), the parent \(P_{3,1}\) is deduced via sub-indexes (\(i=5\), \(j=0\), \(\delta=5\), \(\Delta=5\)), and the parent \(P_{4,1}\) is determined via sub-indexes (\(i=5\), \(j=5\), \(\delta=5\), \(\Delta=5\)). In this way, four origin coordinates (\(h\), \(k\), \(l\)) are obtained, and they are defined as the initial population. The initial population data are shown in Table 1, where the first column represents the origin coordinates, the second column represents the generation number, and the parents (\(P_{1,1}\), \(P_{2,1}\), \(P_{3,1}\)\(P_{4,1}\)) are pointed out in the third to sixth column. Then, the genetic algorithm performs the second step via crossover to compute the current children. To do so, the children (\(C_{1,l}\), \(C_{2,l}\), \(C_{3,l}\),\(C_{4,l}\)) are computed by substituting the parents (\(P_{1+2}\), \(P_{2+2}\),\(t\)) and \(q=0\) in Equations (7)-(10). Additionally, the children (\(C_{5,l}\), \(C_{6,l}\), \(C_{7,l}\),\(C_{8,l}\)) are computed by substituting the parents (\(P_{1+2}\), \(P_{2+2}\),\(t\)) and \(q=1\) in Equations (7)-(10). The data of the children are pointed out in the seventh to fourteenth column of Table 1.  
Then, the algorithm performs the third step to evaluate the fitness of the current generation. To do so, the coordinates (\(h\), \(k\), \(l\)) of each parent and son are replaced in Equation (11) to compute the objective function. The current generation fitness is pointed out in the fourth row of Table 1. Thus, the first generation is completed. In this case, the algorithm provides a low error from the initial population. Therefore, it is corroborated that the initial population provides the best candidates to speed out the convergence. Then, the algorithm performs the fourth step to select the next generation parents. To do so, the best current parents and children are selected via fitness. In this way, the parents \(P_{1,t+1}\) and \(P_{3,t+1}\) are selected from the pairs (\(P_{1,t}\), \(P_{2,t}\)) and (\(P_{3,t}\), \(P_{4,t}\)), respectively. In addition, the parents \(P_{2,t+1}\) and \(P_{4,t+1}\) are selected from the children (\(C_{1,t}\), \(C_{2,t}\), \(C_{3,t}\), \(C_{4,t}\)), (\(C_{5,t}\), \(C_{6,t}\), \(C_{7,t}\), \(C_{8,t}\)), respectively. In this case, \(P_{1,2}=P_{2,1}\), \(P_{3,3}=P_{4,1}\), \(P_{2,2}=C_{2,1}\), and \(P_{4,2}=C_{7,1}\). Figure 2: (**a**) Spherical surface to perform spherical surface modeling. (**b**) Flowchart of the genetic algorithm to determine spherical surface model. (**c**) Spherical surface provided by the surface model in Equation (1). Then, the algorithm performs the fifth step to mutate the parent \(P_{2,2}\), which is determined as the worst parent via fitness. To perform the mutation, a new parent is randomly generated from the search space and the fitness is computed via Equation (11). In this case, the new parent does not improve the fitness and the parent is not mutated. Next, the parent \(P_{2,3}\) is randomly selected to mutate the parameter \(h\), which is selected in random form. Thus, a new parameter \(h\) is randomly generated from the search space and the fitness of the parent \(P_{2,3}\) is computed via Equation (11). In this case, the fitness is improved by the new parameter. Therefore, the parameter \(h\) is mutated. Then, the algorithm performs the second step to create the children of the \((t+1)\) generation via Equations (7)-(10). Additionally, the fitness of these children is computed via Equation (11).