## 5 Comparison of the Proposed Method with the Weighted Sum Methods
### Loads Are Borne by the Entire Structure
#### 5.1.2 Problem 2
As shown in Figure 22, the VM stress distributions in both the TEWS and BCM structures under LC2 are nearly uniform. However, the VM stress distributions in the two structures under LC1 are uneven. At the same time, the maximum VM stress under LC1 in the BCM structure occurs near the left upper corner (circled regions in Figure 22b). These are local regions where it is still possible to strengthen the structure to improve the withstanding of LC1; i.e., there is still space to add material around the stress concentration areas. Note that LC1 is an ill-loading condition in this case, and it is straightforward to fine-tune the structures to better withstand ill-loading conditions. Based on the above discussion, the BCM method can be utilized in this case; however, the obtained structure must be fine-tuned. Figure 21: Structure performance using the weighted sum (WS) methods and the bisection constraint method (BCM) for Problem 2 when \(P_{1}=1.0\)\(P_{2}=9.0\): (**a**) compliance; (**b**) stress level. The calculation results for other load combinations are listed in Table 7. The compliances and the stress levels under LC1 and LC2 in the structures obtained via the TEWS and BCM methods are shown in this table. The cases marked in red are scenarios in which the stress level appears to be higher in the structures obtained via the BCM method (Stage 2) than in the structures obtained via the TEWS method (Stage 1). However, the key point is that the stress level under the more severe load case in the BCM structure is lower than that in the TEWS structure. The maximum stress level occurs when applying the less severe load case and can be easily addressed by fine-tuning the structures, which is essential due to the appearance of unreasonable regions in the structure, represented by elements with intermediate densities. \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}  & \begin{tabular}{c} **Stress** \\ **Level 2** \\ \end{tabular}  \\ \hline \multirow{6}{*}{\(P_{1}=0.5\), \(P_{2}=9.5\)} & \multirow{6}{*}{ \begin{tabular}{c} TEWS \\ (Stage 1) \\ \end{tabular}  } & \multirow{6}{*}{ \begin{tabular}{c} **328.70** \\ **328.70** \\ \end{tabular}  } & \multirow{6}{*}{\ \begin{tabular}{c} **328.70** \\ **328.70** \\ \end{tabular}  } & \multirow{6}{*}{ \begin{tabular}{c} 0.116 \\ **0.164** \\ \end{tabular}  } \\ \cline{1-1}  
It is also noted that in Problem 2, the stress level may be higher under the loading condition possessing lower compliance compared with the loading condition possessing higher compliance. For example, when \(P_{1}=3.0\) and \(P_{2}=7.0\), the compliance in the structure obtained via the TEWS method under LC1 is much lower than that under LC2 (129.94 compared with 235.84). However, the stress level under LC1 is higher than that under LC2 (0.223 compared with 0.190) according to Table 7. Stage 2 helped reduce the compliance under LC2. At the same time, the compliance under LC1 increased, worsening the stress level under this loading condition; i.e., the stress level became much higher in