# Shape Optimization of Structures by Biological Growth Method
## 2 Materials and Methods
### Application of the SKO and CAO Method for Optimization of 2D Structural Element
Before starting the topological optimization of the selected component, it was necessary to set two main criteria: the goal and the constraints of the topological optimization, which will guide the Abaqus program in the computation. Only one optimization constraint was set: a reduction in the design volume by 70%. The result of the topological optimization of the design element at computation cycle \(n=30\) is shown in Figure 5a. The finite element mesh of the model after topological optimization is shown in Figure 5b. It consisted of 3354 elements and 10,851 nodes (Figure 5b). In some regions the mesh was not ideal, but we assumed that these were low-load regions in the center of the geometry, which would not negatively affect the computation of the structural element under study. The static analysis of the 2D structural element after topological optimization is shown in Figure 6. In the extracted geometry, it can be seen that the shape of the component at the stress concentrator location (point \(A\), Figure 2a) did not change even after topological optimization, since the SKO method cannot generate geometry outside the design contour lines. Its main goal is to lighten the given component in low-load regions. For this reason, it can be expected that the value of the maximum stress at a given (critical) location will not decrease. Figure 3: Static analysis of the initial design model of the structural element: (**a**) von Mises field of equivalent stresses and (**b**) field of resultant displacements. Figure 4: Topological optimization at \(n=20\) computation cycles. Figure 5: (**a**) Structural element after topological optimization. (**b**) Finite element mesh of the structural element model after topological optimization. Figure 6: Static analysis of the 2D structural element model after topological optimization: (**a**) von Mises field of equivalent stresses and (**b**) field of resultant displacements. Topological optimization negatively affected the resulting displacements and equivalent stresses in the contour lines of the initial design. Despite the high percentage increase (+52.94%) in the maximum value of the resulting displacement, its overall value of \(u_{max}=0.156\) mm was acceptable. It was already worse with the distribution of the equivalent stress in the investigated element. The case of the CAO method on a 2D structural element was performed on the same structural element as for the SKO method (Figure 1). The optimization strategy of the biological growth method consisted of the successive application of topological optimization (SKO), which guaranteed us a minimum weight of the component, and then by applying shape optimization (CAO) we could obtain a design whose elements around the contour lines were modified (grey/shrank) so that the structure exhibited a constant stress on its surface. In other words, by using shape optimization, we could remove stress concentrators but also less stressed areas around the contour lines of the design [27; 31]. The CAO method could not create new holes in the design but only modified the already existing contour lines to the final optimal shape. Also in this case, the optimization result was affected by the number of computational cycles. A larger number of cycles increased the computation time but provided better quality results. The results of the shape optimization for \(n=20\) cycles are shown in Figure 7.  
Again, it is necessary to set the input parameters that will guide the Abaqus program in the computation. The chosen shape optimization objective for the structural element under study was to minimize the maximum stress variations in the given design elements. In other words, the Abaqus program tried to adjust (shift) the nodes of the mesh so that the elements with different stresses, after optimization, were as close as possible to the reference (most occurring) value of the equivalent stress in the design. This resulted in a more homogeneous distribution of the equivalent stress in the component.