## 3 Results of Micro-Scale Spherical and Cylindrical Surface Modeling
To do so, the spherical surface model is sampled at the surface coordinates \((x_{ij},y_{i,j},z_{ij})\), \((x_{i+\delta j},y_{i+\delta j},\)\(z_{i+\delta j})\), \((x_{i+\delta j},y_{i+\delta j+\Delta},z_{i+\Delta})\), and \((x_{i+\delta j+\Delta},y_{i+\delta j+\Delta},z_{i+\delta j+\Delta})\) to obtain a good initial population \((h,k,l)\). In this way, Equations (4)-(6) are computed by employing \(m=180\), \(n=194\), \(\Delta=90\), and \(\delta=97\). Thus, the parent \(P_{1,1}\) is determined via surface data whose sub-indices are (\(i=0\), \(j=0\), \(\delta=97\), \(\Delta=90\)), the parent \(P_{2,1}\) is determined via (\(i=0,j=90\), \(\delta=97\), \(\Delta=90\)), the parent \(P_{3,1}\) is determined via sub-indices (\(i=97\), \(j=0\), \(\delta=97\), sub-indices \(\Delta=90\)), and the parent \(P_{4,1}\) is determined via sub-indices (\(i=97\), \(j=90\), \(\delta=97\), \(\Delta=90\)). By employing these surface data, four origin coordinates \((h,k,l)\) are computed via Equations (4)-(6). From this procedure, the parent \(P_{1,1}\) is represented by \((h=1218.8\ \upmu\)m, \(k=1173.2\ \upmu\)m, \(l=-3135.7\ \upmu\)m), the parent \(P_{2,1}\) is represented by \((h=1672.1\ \upmu\)m, \(k=1389.1\ \upmu\)m, \(l=-3813.6\ \upmu\)m), the parent \(P_{3,1}\) is represented by \((h=1302.1\ \upmu\)m, \(k=1276.6\ \upmu\)m, \(l=-3941.3\ \upmu\)m), and the parent \(P_{4,1}\) is determined by \((h=1762.91\ \upmu\)m, \(k=1516.8\ \upmu\)m, \(l=-3584.3\ \upmu\)m). These parameters represent the initial population, which provides good fitness to speed out the convergence. In addition, the ratio \(\tilde{R}\) is computed via Equation (11) by employing the origin coordinates \((h,k,l)\) and the surface points \((x_{i},\ j,y_{i,j},\ z_{i,j})\). Then, the genetic algorithm performs the second step via crossover to compute the children by substituting \(q=0\) and \(q=1\) in Equations (7)-(10). The probability of crossover is determined via fitness [52]. When the average fitness of the parents is improved, the crossover is carried out. This procedure leads to preventing the loss of candidates to achieve the convergence. Thus, the probability of crossover is in the interval from 0.0 to 0.5. This result is obtained by computing the same experiment several times. Then, the algorithm performs the third step to evaluate the fitness by substituting the coordinates \((h,k,l)\) of parents and children in Equation (11). Then, the algorithm performs the fourth step to select the parents of the next generation 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. Furthermore, the parents \(P_{2,t+1}\) and \(P_{4,t+1}\) are selected from the children\((C_{1,t},C_{2,t},C_{3,t},C_{4,t})\) and \((C_{5,t},C_{6,t},C_{7,t},C_{8,t})\), respectively. Then, the algorithm performs the fifth step to mutate the worst parent, which is determined via fitness. In addition, a parent is randomly selected to mutate one coordinate, which is selected in random form. The mutation probability is determined via fitness. For the mutation of the parent, if the new parent improves the fitness, the worst parent is mutated, if not, the worst parent is not mutated. For the parameter mutation, if the new parameter improves the parent fitness, the parameter is mutated. Thus, the mutation probability crossover is in the interval from 0.0 to 0.62. This result is obtained by computing the same experiment several times. 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). Thus, the population of the \((t+1)\) generation is completed. Then, the procedure to compute the \((t+1)\) generation is repeated until the optimal parameters (\(h,k,l\)), which minimize the objective function Equation (11) are obtained. The number of generations is deduced from the iterations number to obtain the optimized parameters (\(h,k,l\)).