# Shape Optimization of Structures by Biological Growth Method
## 2 Materials and Methods
### Growth Optimization Method--CAO Method 1
This nature of nodal point displacements follows from Equations (2) and (10). If the equivalent stress in the element is greater than the average equivalent stress in the structure, then it follows from Equation (2) that the components of \(\mathbf{\varepsilon}^{\mathrm{R}}_{\mathrm{mm}}\) are positive and from Equation (10) that the forces are of such a nature that they expand the element. Conversely, when the equivalent stress in the element is less than the average equivalent stress in the structure, the element is compressed. \(\mathbf{\varepsilon}^{\mathrm{R}}_{\mathrm{mm}}\) are the normal components of the strain tensor, i.e., elements that lie on the main diagonal of the matrix representing strain tensor. From relation (2) it can be seen that the tensor \(\mathbf{\varepsilon}^{\mathrm{R}}\) does not contain shear components. Hence, a pure tension or pressure is assumed. Thus, a computation according to this relation would ensure a uniform increase or decrease in one element in all considered directions. For multiple elements, the direction of growth will result from the interaction of the elements with each other [26; 27; 28; 29]. Figure 1 shows the solution of the example of optimizing the stress distribution in the rod. The tie rod was formed by a flat piece of steel with a thickness of 10 mm. The material of the tie rod had [PERSON]'s modulus \(E=210\) GPa and [PERSON]'s ratio \(\mu=0.3\). A half-model of the structure with its dimensions is shown in Figure 1a. Due to the symmetry, it was not necessary to model the whole tie rod; its half-model with dimensions is shown in Figure 1a. The model contained approximately 140,000 triangular finite elements with 90,000 nodes. The average size of the finite elements was about 4 mm. In Figure 1b, the load and the applied boundary conditions are shown (marked by green arrows in Figure 1b). A force of \(F=10\),000 N was applied on the left side of the rod (marked with purple arrows in Figure 1b). The resulting field of equivalent stresses according to [PERSON] theory for the structure before optimization is shown in Figure 1c. The biggest equivalent stress according to [PERSON] theory was at point A and had the value 87 MPa. This stress was at the point of application of the boundary conditions (Figure 1b) and represented the stress at the point of contact between the pin and the pull rod. Obviously, this contact stress at the cylindrical surface could not be significantly affected by the shape optimization at the other edges of the component. Moreover, the allowable values of the contact stresses were higher than those of the other loading methods. Of interest from the point of view of shape optimization was the equivalent stress around point B, where the sharp edge, and hence the stress concentrator, was located. The magnitude of the equivalent stress according to von [PERSON] theory was approximately 72 MPa in this region before optimization. The optimization of the distribution of equivalent stresses was carried out by the growth method. During the optimization, only the changes of the upper edges marked in red in Figure 1b were allowed to be realized. The shape of the rod after 15 optimization steps is shown in Figure 1d. The field of equivalent stresses according to [PERSON] theory can also be seen in this figure. It can be seen from Figure 1d that the stress field was distributed more homogeneously, and the stress peaks at the transition point from the arc to the longitudinal part of the rod were removed. The equivalent stress according to [PERSON] theory in the vicinity of point C was reduced to approximately 26 MPa after optimization.