## 4 Numerical Examples
### Example 2: A Cantilever Truss  
A large-scale cantilever truss is adapted from the numerical example in the research of [PERSON] and [PERSON] [31]. As shown in Figure 11, this cantilever truss has 63 bars. The width and height of the rectangular truss are 3 and 2, respectively. The nodes on the left side are pinned, and a vertical external load is applied at the bottom of the right-hand side that equals 1. Nodes are connected by a potential bar unless this creates an overlapping bar. The density is 1, and the modulus of elasticity is 1000. The original cross-sectional area for each bar is 0.1 and is permitted to vary from 0.01 to 10. The nodal displacement of the loading point is limited from \(-\)0.05 to 0.01. The parameters of GA are set as follows: the population size is 150, the maximum number of evolutions is 100, and the initial probabilities of crossover and mutation are 0.9 and 0.1, respectively. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Design Variables** & **Data from [29]** & **Data from [30]** & **Present** \\ \hline
**A1** & 1.081 & 0.954 & 1.60 \\
**A2** & 0.539 & 0.539 & 1.59 \\
**A3** & 0 & 0.141 & 0 \\
**A4** & 1.081 & 0.954 & 2.39 \\
**A5** & 0.954 & 0.539 & 0.80 \\
**A6** & 0.440 & 0.287 & 0.79 \\
**A7** & 0 & 0.141 & 0 \\
**A8** & 0.141 & 0 & 0 \\
**A9** & 0 & 3.813 & 0 \\
**A10** & 0.270 & 0.440 & 1.13 \\
**A11** & 0.270 & 0.440 & 0.20 \\
**A12** & 0.539 & 0.220 & 0.20 \\
**A13** & 0.141 & 0.220 & 1.13 \\
**A14** & 0.440 & 0.347 & 1.13 \\
**A15** & 0 & 0.141 & 0 \\
**Iteration number** & & & 7 \\
**Labeled variables** & & & 5 \\
**Critical constraint** & \(-\)0.00088 & \(-\)0.0649 & 0.000 \\
**Weight (lb.)** & 77.84 & 74.33 & 75.376 \\
**Percentage difference (\%)** & 5.57 & 0.23 & — \\
**Structural analysis** & 8000 & 8000 & 15 \\
**Percentage difference (\%)** & 0.19 & 0.19 & — \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Optimization results of the 15-bar truss. Figure 11: Ground structure of the 63-bar truss. Figure 12 illustrates the topology evolution. Gradually changing configurations are generated in the iterative procedure, and eight typical configurations are chosen. Figure 13 shows the optimized configurations of this paper compared with the same method without the LCS strategy, and the weight iteration history for the two methods is presented in Figure 14. The method without the LCS strategy yields a result of nonconvergence and a middle topology configuration, and the method with the LCS strategy obtains a more concise, effective, and valid configuration. The optimal solutions of variables, structural weight, critical constraint, and structural analyses are compared in Tables 3 and 4. The critical constraint of the method without the LCS strategy is 5.492 \(\times\) 10\({}^{-2}\), which indicates the iteration's nonconvergence. The method with the LCS strategy converges after 57 iterations and 66 analyses with 51 topology variables labeled, showing the efficiency of the proposed method. The contrast of  
Figure 14: Iteration history for the 63-bar truss. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**NO.**} & \multicolumn{2}{c}{**Area**} & \multicolumn{2}{c}{**Area**} & \multirow{2}{*}{**No.**} & \multicolumn{2}{c}{**Area**} \\  & \multicolumn{2}{c}{**LCS/No LCS**} & & & \multicolumn{2}{c}{**LCS/No LCS**} \\ \hline
**1** & 0.1830 & 0.4293 & 15 & 0 & 0.1 & 43 & 0 & 1.0140 \\
**2** & 0 & 0.1 & 16–18 & 0 & 0 & 44–47 & 0 & 0 \\
**3** & 0.4113 & 0.2727 & 19 & 0.1 & 0 & 48 & 0.1 & 1.3853 \\
**4** & 0.5575 & 0.3942 & 20–26 & 0 & 0 & 49 & 1.4154 & 0.1386 \\
**5** & 0.4056 & 0.1349 & 27–28 & 0 & 0.1 & 50–51 & 0 & 0 \\
**6** & 0 & 0.1 & 29–35 & 0 & 0 & 52 & 0 & 0.1 \\
**7** & 0 & 0 & 36 & 0 & 0.1 & 53 & 0 & 0 \\
**8** & 0 & 0.1 & 37 & 0 & 0.4385 & 54 & 1.6756 & 0 \\
**9-11** & 0 & 0 & 38 & 0 & 0.1 & 55–56 & 0 & 0.1 \\
**12** & 0 & 0.1470 & 39 & 0 & 0 & 57–61 & 0 & 0 \\
**13** & 0 & 0 & 40 & 0.4103 & 0 & 62 & 0 & 0.1026 \\
**14** & 0.1826 & 0.1860 & 41–42 & 0 & 0 & 63 & 0 & 0.1414 \\ \hline \hline \end{tabular}
\end{table}