# Topology and Response Surface Optimization of a Bicycle Crank Arm with Multiple Load Cases
## 3 Results and Discussion
### Response Surface Optimization  
From all the variation results by topology optimization, in general, there are two topological holes obtained for all the models. These holes create narrow shapes for the crank arm which consequently creates a high-stress concentration region (red circle) particularly from the bigger hole close to the crankshaft. Therefore, to improve this weakness, a response surface optimization method is implemented. This method can be considered as shape optimization. Two research variables have been chosen, i.e., the two topological holes from the simplified optimized crank arm, as seen more in Figure 15 as follows:  
The first hole radius near the crankshaft hole is assigned as P1 or the first variable while the hole radius near the pedal hole is P2, respectively. Each of these two variables is then varied in terms of radius increment from 0 to 5 mm and investigated following a design of experiment procedure in the ANSYS Response Surface Optimization Software (RSM) to obtain the optimum result. Figure 16 shows an example of how this radius evolves.  
Figure 14: Effect of material: (**a**) Aluminum alloy, (**b**) carbon fiber.  
Figure 15: Two topological radius P1 and P2 as the response surface optimization variables.  
The objective function for this process is stress minimization as discussed before. Then, the software makes sufficient iteration at near the optimum value to get the final minimum stress result. The result of the RSM process is presented in the surface-like graph which can be seen in Figure 17. In this case, the minimum point for P1 and P2 are obtained at 4.75 and 1.25 mm. This means that the best increment for both hole radiuses is 4.75 mm for the hole near crankshaft and 1.25 mm for the hole near the pedal. This method is implemented in all the samples and results in the data presented in Table 5. It shows the final maximum stress and the percentage of reduction after the response surface method is implemented.  
Figure 16: Example of how radius P1 evolves from P1a to P1c during the response surface optimization.  
Figure 17: Response surface optimization result.  
To improve the stress performance of the optimized crank arm, we applied the response surface optimization for all cases by topology optimization as summarized in Table 5. All the maximum stress of the crank arm has been successfully improved from their previous topology optimization result. The highest reduction is obtained by the 50% mass constraint crank arm with a 21.8% reduction. This once again, shows that increasing the hole radius by only around 0-2 mm will make significant differences to the obtained stress. Meanwhile, the lowest reduction is obtained by the 40% mass constraint with only 0.1%. However, an improvement still can be obtained using the RSM method.  
As a conclusion for this section, the newly optimized crank arm provides at least three benefits. First, the optimized geometry has a significantly lower mass than the original design which can be the basis of lightweight structures. The materials for the crank arm can be variously chosen depending on the cost consideration. In other words, this optimized crank arm design can reduce the manufacturing cost. Second, the stress distribution throughout the optimized crank arm is still considered safe. It is shown by the maximum stress value that it is much lower than the yield and tensile strength for all samples. Third, this optimized design provides a more aesthetic design compared to the solid-conventionally designed original crank arm. The aesthetic issue, as we know, can be a very sensitive matter to the customers.