# 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  
The selected 2D steel structural element is fixed at the top side, and a static load is applied to the area \(b\times c\); the resulting force is \(F=1000\) N. The dimensions of the element and the defined load parameters are \(a=150\) mm, \(b=50\) mm, \(c=10\) mm (Figure 2a), volume \(V=125\),\(000\) mm\({}^{3}\), and mass \(m=0.981\) kg. The basic material properties of the steel used are density \(\rho=7850\) kg\(\cdot\)m\({}^{-3}\), [PERSON]'s modulus \(E=210\) GPa, [PERSON]'s ratio \(\mu=0.3\), and yield strength \(R_{e}=235\) Mpa. The point \(A\) in Figure 2a represents the region in which we assume the stress concentrator to occur under the given loading conditions. First, a meshed FEM model with assigned material properties and properly defined boundary conditions must be created. Figure 2b shows the defined boundary conditions of the initial design model of the investigated structural element, which has been modeled as a 2D model with an assigned required thickness of 10 mm. The finite element mesh consists of square elements with a side length of 2.50 mm, whose sides are loosely arranged along the contour lines of the design. The element is formed by eight nodes (quadratic element), with each node taking six degrees of freedom. A finer mesh (a mesh with smaller elements) will provide us better quality results but will negatively affect the computation time, so it is important to choose a mesh with a suitable number of elements and nodes. The finite element mesh of the initial structural design model consists of 2000 elements and 6241 nodes. The static analysis of the initial structural element design model yielded a field of equivalent stresses according to [PERSON] theory and a field of resultant displacements (Figure 3). Already from the boundary conditions for a given structural element, it can be assumed that the critical region (the maximum stressed region with the occurrence of a stress concentrator) will occur at point \(A\), and the maximum displacements will be located at the end of the structure. Abaqus was used for FEA computations of static loads and beam optimization. Figure 2: (**a**) The 2D structural element and (**b**) structural element model with boundary conditions. A maximum equivalent stress of \(\sigma_{max}=78\) MPa was generated on the model under investigation (Figure 3a). The yield strength of the selected material \(R_{e}=235\) Mpa was located at a sufficient distance from the maximum stress occurring on the loaded structure, even when taking into account computational errors or undesirable effects that were not considered in the model design. Thus, it could be concluded that the investigated structural element was satisfactory for the given loading conditions, which provided us the possibility to optimize its topology in order to lighten the structure. The maximum displacement of the investigated element was \(u_{max}=0.102\) mm (Figure 3b). The SKO method is used to find the optimum structural topology of a given structural element and can be used to remove low-load areas in the structure, resulting in a significant reduction in the weight of the product. The density of the finite element mesh affects the computation of the topological optimization. While a finer mesh will provide us a better design topology with respect to the given loading conditions, such a model may be characterized by a very complicated design, which may contain, for example, very thin partitions, which could make it technologically impossible to produce such a component. Therefore, it is very important to choose an appropriate element size before starting the topological optimization. Another important input for the implementation of topological optimization is the number of cycles of the computation. A greater number of cycles increases the computation time but provides better results. Figure 4 shows the topological optimization with \(n=20\) computation cycles.