## 4 Numerical Examples
### Example 3: A Michell Truss  
The ground structure for the 662-bar Michell truss shown in Figure 15a was studied by [PERSON] and [PERSON] [31; 32]. The data are consistent with the literature. The width and height of the whole structure are 8 and 4, respectively. Three nodes on the left side, centered on the horizontal mid-axis, are fixed to a rigid wall of height 2, these are indicated as filled (\"\(\bullet\)\"), and other nodes that are not filled (\"\(\circ\)\") are free. Nodes are connected by a potential bar unless this creates an overlapping bar.  
In the referred literature, the modulus of elasticity is 1, and the cross-sectional area is chosen from the set \(\{0,1\}\) for all bars. The objective function is the minimum strain energy, and the constraint function is a certain volume. The design size variable is discretized into a \(0/1\) variable, and a series of nonconvex continuous relaxations based on convergent branch and bound algorithms is used to address discrete truss variables. The referred literature achieved topology optimization for this large-scale truss by discrete size variable optimization. The optimum configurations of the work of [PERSON] is showed in Figure 15b.  
We apply this example to our problem to optimize the size and topology for a large-scale truss with a nodal displacement constraint of the loading point and minimum weight objective. The structural parameters are set as follows: the modulus of elasticity is \(10^{3}\); the density is 1; the load \(P_{y}=1\); the nodal displacement of the loading point is limited from \(-0.0453\) to \(0.0453\); the initial cross-sectional area is 1, and it is permitted to vary between 0.0001 and 2. The parameters of GA are set as follows: the population size is 100, the maximum number of evolutions is 30, and the initial probabilities of crossover and mutation are 0.9 and 0.05, respectively. The maximum of the first-level iteration is 150, and the convergence accuracy is 0.005; the maximum of the second-level iteration for sizing optimization is 20, with a convergence accuracy of 0.05.  
Figure 16 illustrates the topology evolution. Gradually changing configurations are generated in the iterative procedure, and 11 typical configurations are chosen. Figure 17 shows the optimum configurations of this paper. The weight iteration history of this paper is presented in Figure 18. Tables 5 and 6 offer the optimal solutions of variables, structural weight, critical constraint, and structural analysis. The optimal weight found by this paper is 50.522. The critical constraint is zero, implying the validity of the presented result. The optimization process converged by 124 iterations, and a total of 314 topology variables were labeled. In particular, the method in this paper converged by 139 structural analyses, which only took 1.29% of that in the literature. Thus, the method in this paper achieves a valid optimum result and a concise topology configuration with fewer structural analyses. This advantage is significant because truss optimization problems with mixed variables of a similar scale usually require thousands of structural analyses in past literature [31].  
Figure 15: (**a**) Ground structure of the 662-bar truss; (**b**) The solution of the literature.  
Figure 16: Topology evolution of the 662-bar truss.  
Figure 17: Optimization results in this paper.  
Figure 18: Iteration history for the 662-bar truss.