## 5 Comparison of the Proposed Method with the Weighted Sum Methods
### Loads Are Borne by a Local Region of the Structure
#### 5.2.2 Problem 4  
Problem 4 considered two loading conditions with distributed loads (LC1 and LC2), as illustrated in Figure 25, where the load magnitudes are gradually changed by changing \(P_{1}\) and \(P_{2}\). In this problem,  
\[P_{1}=q_{1}\cdot l/4, \tag{21}\]  
and  
\[P_{2}=q_{2}\cdot l/4. \tag{22}\]  
The computational results for Problem 4 are listed in Table 9 with compliances \(c_{1}\) and \(c_{2}\) under LC1 and LC2, respectively. Stress Level 1 and Stress Level 2 are stress levels under LC1 and LC2, respectively. From the results given in Table 9, it can be seen that in Problem 4, the maximum compliance was always reduced from Stage 1 (TEWS method) to Stage 2 (BCM), because minimizing maximum compliance is the objective of the proposed method. For example, for \(P_{1}=2.0\) and \(P_{2}=8.0\), the maximum compliance reduced from 65.029 (under LC2) to 59.568 (under LC1) after applying the BCM method on the TEWS structure (see Table 9). In Problem 4, the stress level under more severe loading conditions was always reduced after Stage 2 of the proposed framework, along with a reduction in the maximum compliance. For example (see Table 9), when \(P_{1}=2.0\) and \(P_{2}=8.0\), the stress level under LC2 reduced from 0.06162 (TEWS) to 0.05960 (BCM); when \(P_{1}=3.0\) and \(P_{2}=7.0\), the stress level under LC2 reduced from 0.05567 (TEWS) to 0.05380 (BCM); and when \(P_{1}=9.0\) and \(P_{2}=1.0\), the stress level under LC1 reduced from 0.06717 (TEWS) to 0.06630 (BCM). This illustrates that controlling the compliance can help control the stress level under the same loading conditions. Figure 24: Structure performance using the weighted sum (WS) methods and the bisection constraint method (BCM) for Problem 3 when \(P_{1}=3.0\) and \(P_{2}=7.0\): (**a**) compliance; (**b**) stress level. Figure 25: Loading conditions for Problem 4: (**a**) Load case 1; (**b**) Load case 2. One important observation in Problem 4 is that not only are the loads borne by the local regions; the load types in both loading conditions are distributed loads. Under the effect of distributed loads, the structure is divided into two column-like separate parts, as shown in Table 9. This table also shows that the stress levels in structures obtained via the TEWS method under LC1 and LC2 appear to be not so different for most load combinations (e.g., 0.04648 compared to 0.04942 for \(P_{1}=4.0\) and \(P_{2}=6.0\); or 0.04845 compared to 0.04744 for \(P_{1}=5.5\) and \(P_{2}=4.5\)). At the same time, the compliances of the TEWS structures are approximately proportional to the load magnitudes applied to the structure (e.g., \(c_{2}/c_{1}=43.472/25.433\) is approximately proportional to \(P_{2}/P_{1}=6/4\) (see Table 9)). When pursuing the minimization of the maximum compliance (e.g., using the BCM method), the stress level under the less severe loading condition increases conversely. For example, for \(P_{1}=3.0\) and \(P_{2}=7.0\), the stress level under LC1 increased from 0.04483 (TEWS) to 0.13768 (BCM), even though the maximum compliance was reduced from 53.520 to 46.853. This observation implies that when the load/response is completely divided into two regions, the TEWS method generally provides better results, and the application of the BCM method becomes irrelevant from the perspective of the stress level. Additionally, in most load combinations, the BCM structures contain gray regions and sometimes even checkerboard patterns (see Table 9). Therefore, a post-optimization fine-tune operation is essential in order to obtain satisfactory results. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Load** & **Method** & **Topology** & \(c_{1}\) & \(c_{2}\) & \begin{tabular}{c} **Stress** \\ **Level 1** \\ \end{tabular}  & **Stress** & **Level 2** \\ \hline \multirow{3}{*}{\(P_{1}=2.0\), \(P_{2}=8.0\)} & TEWS & \multirow{3}{*}{\(\begin{array}{c}\ \  
It must be emphasized that the scenarios presented in Problem 3 and Problem 4 are highly uncommon because, in reality, loads are usually borne by the entire structure.