# The Influence of the Load Term of the Strain Energy Sensitivity in the Shell Shape Optimization
## 6 Case Study
### Example 2: The Shape Optimization of the Shell Roof In the Rolex Learning Center  
The Rolex learning center free-form roof in Switzerland was simulated. The plane size was 120 m \(\times\) 120 m, and a total of 42 key points (6 \(\times\) 7 and 20 m in grid size) were selected to form the initial surface by B-spline surface technique. The supports were three-way hinge supports, and the key points of the grid and support positions are shown in Figure 11. The concrete elastic modulus was \(E\) = 3 \(\times\) 10\({}^{4}\) MPa, the Poisson's ratio \(v\) = 0.2, the thickness was 0.2 m, and the uniform load \(q\) = 10 kN/m\({}^{2}\). This example was consistent with the example in literature [7], but Equation (15) was used for the optimization. All the nodes on the mesh except the supporting points were design variables.  
In order to investigate the effect of the load term on the strain energy sensitivity, the structural weight changes were compared for Schemes A and B. The displacement and the strain energy were  
Figure 11: The size of the initial structure and constraints.  
Figure 10: The difference of the strain energy between Schemes A and B at the same step for the different thickness t.  
close. Thus, only the results of the strain energy and the displacement in Scheme A are given here. Figure 12 shows the change of the structural strain energy. The strain energy of the final structure was 14.5% of the initial structure. In Figure 13, the difference in the strain energy between Schemes A and B was largest at Step 111, which accounts for 1.17% of the strain energy in Scheme B. Similar to the discussion in the first example, the thickness of the shell was thick, namely 0.2 m. The contribution of the plate element on the stiffness term was big, and the difference of the strain energy and structural weight between Schemes A and B at the same step was small. The role of the load term can be ignored.  
In Figure 14, the vertical displacement of the structure gradually decreased, and the displacement tended to be homogeneous. The corresponding maximum vertical displacement was 4.3% of the initial structure. In Figure 15, the weight of the structures increased all the time, but the increment of the structural weight was slightly lower when the sensitivity load term was taken into account. It can be  
Figure 12: The change ratio of the strain energy.  
Figure 13: The difference of the strain energy between Schemes A and B at the same step.  
seen from Figure 16 that the vector length values of the stiffness term and that of the whole sensitivity were close to each other, gradually decreasing with evolution and approaching to the value of the load term, while the vector length value of the load term of the structural strain energy sensitivity was at a much smaller proportion of the whole strain energy sensitivity than that in the first example. This also denotes that the role of the load term is small.  
Figure 14: The z-displacement distribution of several structures in the evolution process. (**a**) Step1, 100%; (**b**) Step 50, 70.75%; (**c**) Step 111, 38.94%; (**d**) Step 296, 14.57%.  
Figure 15: The change ratio of the structural weight.