## 3 Case Study
### Stability Evaluation Based on Genetic Algorithm and Discrete Element Analysis
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Type**} & **Identification** & **Deformation Modulus** & **Shear Strength** & **Parameters** \\ \cline{3-5}  & **Number** & **(GPa)** & **f** & **c (MPa)** \\ \hline \multirow{4}{*}{Shear Zone} & LS321 & 3.75 & 0.40 & 0.06 \\  & LS331 & 3.85 & 0.38 & 0.07 \\  & LS325 & 3.8 & 0.38 & 0.06 \\  & LS421 & 4.14 & 0.45 & 0.07 \\  & f101 & 2.45 & 0.43 & 0.08 \\  & f102 & 2.46 & 0.43 & 0.08 \\ Fault & f143 & 3.65 & 0.35 & 0.05 \\  & f141 & 3.64 & 0.35 & 0.05 \\  & f106 & 4.35 & 0.35 & 0.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Summary Table of Shear Zone and Fault Parameter Values. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Lithology**} & **Rock** & **Headingal Layer** & **Weathering and Unloading State** & **Deformation Modulus** & **Poisson’s Ratio** & **Density** & **Shear Strength** \\ \cline{3-5}  & **Category** & **Lithological Layer** & **Unloading State** & **(GPa)** & **Ratio** & **(kN/m3)** & **f** & **c** \\ \hline \multirow{4}{*}{Basalt} & \multirow{4}{*}{\ \begin{tabular}{c} \(\text{columnar-jointed}\) \\ \(\text{basalt}\), agglomerate \\ \(\text{lava}\) \\ \end{tabular}  } & \multirow{4}{*}{ \begin{tabular}{c} slightly fresh, \\ no unloading \\ \end{tabular}  } & \multirow{4}{*}{9} & \multirow{4}{*}{0.24} & \multirow{4}{*}{27.2} & \multirow{4}{*}{1.1} & \multirow{4}{*}{1.1} \\  & & & & & & & \\ \cline{1-1} \cline{4-1}  
where T\({}_{i}\) represents a selected monitoring indicator and \(\omega_{i}\) is its corresponding weight coefficient. Selection Operator  
A roulette wheel selection strategy was employed to probabilistically select individuals for reproduction based on their relative fitness. The higher the fitness value, the greater the probability of being selected. This approach increases the likelihood of high-performing solutions passing their genes to the next generation. The probability of selecting an individual is calculated according to Equation (2):  
\[p_{i}=\frac{f(x_{i})}{\sum_{j=1}^{N}f(x_{j})} \tag{2}\]  
where p\({}_{i}\) represents the probability of individual i being selected, f(x\({}_{i}\)) denotes the fitness function value of individual x\({}_{i}\), and N represents the population size. The selected individuals underwent crossover (e.g., simulated binary crossover) with a predefined probability to generate offspring. Mutation operations (e.g., Gaussian or uniform mutation) were applied to introduce genetic diversity and avoid premature convergence. The implementation is described in Equation (3):  
\[x_{k}^{\prime}=\alpha x_{i}+(1-\alpha)x_{j},\alpha\in[0,1] \tag{3}\]  
where x\({}_{k}\) denotes the new individual generated through crossover, x\({}_{i}\) and x\({}_{j}\) represent the two parent individuals, and \(\alpha\) is the crossover factor, which controls the genetic contribution of the parent individuals to the offspring. Mutation  
With a small probability, certain genes of the offspring were mutated to introduce diversity. Mutation can be represented as Equation (4):  
\[x_{k}^{\prime\prime}=x_{k}^{\prime}+\delta,\delta\sim N(0,\sigma^{2}) \tag{4}\]  
where \(x_{k}^{\prime\prime}\) denotes the mutated individual, \(x_{k}^{\prime}\) represents the state of the individual before mutation, \(\delta\) denotes the perturbation following a normal distribution, and \(\sigma^{2}\) is the variance of the normal distribution. Replacement  
Offspring were evaluated and inserted into the population, replacing less-fit individuals. Elitism was adopted to ensure that the best solution of each generation was preserved in the next generation without modification. Termination Criteria  
The algorithm terminated when either the maximum number of generations was reached or the improvement in fitness fell below a predefined threshold over successive generations. The best-performing chromosome at termination was selected as the optimal parameter set.