# 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 initial structure was a spherical shell, the spherical radius being R = 10 m, and the spherical equation is \(z=\sqrt{R^{2}-\left(x-5\right)^{2}-\left(y-5\right)^{2}}-5\sqrt{2}\). The triangular mesh was divided on the 10 m \(\times\) 10 m square plane and projected onto the surface, and the pinned supports were set at the four corners. The thickness of the sphere was 0.05 m, the Young's modulus E = 3.0 \(\times\) 10\({}^{10}\) N/m\({}^{2}\), the density was 2500 kg/m\({}^{3}\), and the vertical uniformly distributed load was 10 kN/m\({}^{2}\). The initial size and supports are shown in Figure 1.  
First, three optimization schemes were designed to investigate the effect of the load term on the strain energy sensitivity. Scheme A: the structural weight was included in the load vector, and the load term was considered in the sensitivity, that is, the structural evolution was carried out according to Equation (7);  
Scheme B: the structural weight was included in the load vector, and the load term was not considered in the sensitivity, that is, structural evolution was carried out according to Equation (4), which means that the sensitivity was equal to the stiffness term value of the sensitivity in Scheme A;  
Scheme C: the structural weight was not considered in the load, and the load term was not considered in the sensitivity. However, the structural weight was included in the calculation of structural performance at each step, that is, structural evolution was carried out according to Equation (4), which is equivalent to obtaining the strain energy sensitivity according to the calculation principle of structural mechanics when structural weight is not considered. Figure 1: The initial structure and constraints. Figure 4 shows that the nodes on the middle and the diagonal line of the shell surface in Scheme A were slightly higher than that in Scheme B. The others in Scheme A were lower than that in Scheme B. The maximum height difference was up to \(-0.390\) m and \(0.580\) m. The strain energy at Step 120 was equal to \(12.2\%\) of that at Step 1 in Scheme A, and \(11.5\%\) in Scheme B. The shape difference at Step 248 is shown in Figure 5. Only the nodes on the edge of the shell surface in Scheme A were higher than that of Scheme B. The maximum height difference was up to \(-1.110\) m and \(0.710\) m. This was \(7.1\%\)\(\sim\)\(11.1\%\) of the span, which was obvious. Figure 6 shows the variation of strain energy in Schemes A and B. The strain energy of the structure at Step 248 was reduced to \(10.6\%\) of the initial structure in Scheme A, and \(10.8\%\) in Scheme B. The structural stiffness increased significantly, and the maximum vertical displacement of the structure at Step 248 was \(3.0\%\) of that at Step 1 in Scheme A, and \(2.2\%\) in Scheme B. The decline speed of the strain energy of the shell surface seemed to be similar whether or not the load term of the strain energy sensitivity was considered. Figure 4: The distribution of the surface height difference of Schemes A and B at Step 120. Figure 5: The distribution of the surface height differences of Schemes A and B at Step 248. However, we found some different points. Firstly, Scheme B finished the optimization at Step 248, and Scheme A could continue to decrease the strain energy to 8.65% of the initial structure at Step 1000. Secondly, t he difference of the strain energy and the structural weight increase rate between Schemes A and B for the same step denoted the cumulative impact on the optimization. In Figure 7, the difference in strain energy between Schemes A and B was largest at Step 112, which accounted for 6.6% of the strain energy in Scheme B. It was noted that the shape difference became greater as shown in Figures 4 and 5, although the difference in the strain energy between Schemes A and B was small at Step 248. Figure 8 shows the history of the structural weight increase rate of Schemes A and B.