# Shape Optimization of Structures by Biological Growth Method
## 2 Materials and Methods
### Shape Optimization--CAO Method 2  
Shape optimization (CAO method) allows one to reduce local stress concentrators (notch stresses) by adaptive growth until a uniform stress distribution over the entire surface of the component is achieved. After shape optimization using CAO, the notched stresses can be reduced by more than 50%, causing the resulting design to have a higher fatigue life than the original design. In the new design, the low loaded regions could be replaced by holes, which again reduced the stress concentrators using CAO and resulted in a design with homogeneous stresses along the internal contour lines but also already with a lower mass of the part [25].  
The procedure for simulating the growth mechanism of biological structures using the finite element method (FEM) can be described by the following steps:  
1. Create an FEA model with an initial design proposal.
2. Cover the model with a thin layer of finite elements that have a much smaller Young's modulus value than the original elements.
3. Apply a load to the model, thereby elastically modifying both the surface layer and the underlying structure.  
Figure 1: Pull rod (**a**) dimensions, (**b**) load and boundary conditions for simulation and optimization, (**c**) distribution of equivalent von Mises stresses before optimization, and (**d**) distribution of equivalent von Mises stresses after optimization.  
4. Multiply the incremental displacements by an appropriate factor to obtain visible shape corrections. The multiplied displacements shall be added to the coordinates of the nodal points on the grid surface.
5. Subsequently, the inner edge of the surface layer is displaced outward so that the thickness of the surface layer is constant throughout. It is also possible to move the nodal points further inward to avoid strong deformations of the mesh.
6. Repeat step 3 but with the new FEM model optimized according to step 5.
7. Perform elastic control cycles to determine the degree of optimization with a high value of Young's modulus (also on a surface layer that was previously soft).
8. Stop the optimization procedure if the stress concentrators are sufficiently reduced or if there are dimensional design constraints [6].  
The quality of design element relief by the CAO method mainly depends on the topology of the initial design. Using the CAO method always results in a design with a higher fatigue life, but sometimes, low-load areas remain in the component. Thus, shape optimization cannot create new holes in the structure but only shapes the already existing contour lines of the design. We use the SKO method to remove these redundant regions [25; 26; 27; 28; 29; 30; 31; 32; 33].  
Although the procedure for changing the [PERSON]'s modulus of the SKO method is analogous to that of the CAO method, there are several significant differences between the two. The first difference is that the reference stresses of the two methods are not the same. For the SKO method, the contour lines of the optimized structure (called boundaries) are represented by the temperature isoclines inside the region under study. The stresses calculated for the merged points are averaged over the nodes. This leads to a halving of the stresses in the boundary region, where the stress values gradually change from high to approximately zero. To maintain a good approximation between the reference stresses of the two methods, a relationship is given:  
\[\sigma_{ref}^{\text{SKO}}=\frac{\sigma_{ref}^{\text{CAO}}}{2}, \tag{11}\]  
where \(\sigma_{ref}^{\text{SKO}}\) is the reference stress for the SKO method, and \(\sigma_{ref}^{\text{CAO}}\) is the reference stress for the CAO method [25].