# The Influence of the Load Term of the Strain Energy Sensitivity in the Shell Shape Optimization
## 6 Case Study
### Example 1: the Shape Optimization of A Spherical Shell
The rate of increase in the structural weight represents the increase percentage, which was equal to the value of the ratio between the structural weights at each evolutionary step, and that of the initial structure subtracts one. During the process of shape adjustment, the structure weight in Scheme A increased more slowly than that in Scheme B after Step 50. The structural weight increased by 24.8% of the initial structure in Scheme A, and 61.5% in Scheme B at Step 248. These cumulative impacts on the structural evolution were due to the load term of the strain energy sensitivity. Figure 6: The change ratio of the strain energy of Schemes A and B.  
Figure 7: The difference of the strain energy between Schemes A and B at the same step. To investigate the numerical relationship between the load term, the stiffness term, and the whole value for the sensitivity, the vector length of each sensitivity component and the whole value were calculated according to Equations (8) to (10) for each iteration step, as shown in Figure 9. It can be seen that the load term vector lengths of the strain energy sensitivity were relatively small at the initial stage of evolution, and the stiffness term vector lengths and whole vector length values decreased sharply. The load term vector length and stiffness term vector length were getting closer and closer after Step 40. The influence of the load term on the whole sensitivity value gradually increased. In order to further analyze the influence of the load term in the sensitivity, the sensitivity of the nodes located in the structural typical position, i.e., no. 17 and 51 shown in the Tables 1 and 2, was selected for observation. Table 1 shows the 17 th node's load term value \(\mathsf{S}_{\mathrm{NF}}\) of the sensitivity and  
Figure 8: The change ratio of the structural weight of Schemes A and B.  
Figure 9: The vector length of the load term, the stiffness term, and the whole value of the structural strain energy sensitivity in Scheme A.  
the whole sensitivity value \(\rm S_{NT}\) for partial evolution steps in Scheme A. It can be seen from the table that the signs of the load term value and the whole value were always contrary, and the ratio of their vector length increased significantly after the 30 th step. The whole value of sensitivity approached zero, and the load term value of sensitivity became small. The results show that the influence of the load term became obvious. Table 2 shows the ratio of the load term \(\rm S_{NF}\) of strain sensitivity for node no. 51 compared to its whole value \(\rm S_{NT}\), with positive and negative values. The ratio \(\rm S_{NF}/S_{NT}\) is obviously larger than that 17 th node. Combining the results of the analysis in Figure 9, during the convergence stage the load term of sensitivity would slightly offset the stiffness term, making the vector length of the strain energy sensitivity slightly smaller during the convergent phase. In order to discuss the role of each item in the load item, another two schemes were designed. Scheme D: the load term was considered in the sensitivity calculation, but the structure density was set as 0. In this scheme, the sensitivity load term was the first term of Equation (15) only, namely the external load term. Scheme E: the load term was considered in the sensitivity calculation, but the external load was set as 1N/m\({}^{2}\). In this scheme, the sensitivity load term can be approximately considered as containing only the weight, and the structural weight was the main load. We found it interesting that the results of Schemes A, D, and E were completely the same, they had the same change in the strain energy ratio \(\rm C_{i}/C_{1}\), the self-weight, and \(\rm S_{NF}/S_{NT}\). The results of Scheme D and E are not shown here. This means that the weight and vertical uniform load played the same role, and the sensitivity load term, the sensitivity stiffness term, and the strain energy sensitivity kept a constant ratio which was mainly determined by the thickness of the shell for the same initial structure.