# Multi-Scale Concurrent Topology Optimization Based on BESO, Implemented in MATLAB
## 6 Numerical Results  
In this section, different test cases regarding the presented formulation are illustrated in both 2D and 3D space. All test cases are taken from the literature and illustrate different applications of the presented formulation. In all test cases, the final geometry of both the macro and micro-scales is presented. The starting material densities for all test cases regarding the macro-scale were set equal to the volume fraction target value. In the micro-scale, the starting geometry was set as a solid domain with a hole at the center of the unit cell and a radius of \(1/3\) the dimensions of the periodic unit cell. A graphical representation of the unit cell geometry for both 2D and 3D is presented in Figure 1.  
To display the hole at the center of the 3D unit cell, the geometry presented in Figure 1b illustrates only half of the periodic unit cell geometry. Regarding the 2D test cases, three different examples were selected to be optimized. Each of these examples was chosen to illustrate a different application of the methodology. In Figure 2, the design domains along with the loading and boundary conditions for the three test examples are presented. Figure 1: Initial geometries of the periodic unit cell. (**a**) Initial geometry of 2D periodic unit cell. (**b**) Initial geometry of 3D periodic unit cell. Figure 2: The 2D test cases. (**a**) Cantilever beam with fixed hole. (**b**) L-Shape. (**c**) Long cantilever beam. The first test case presented in Figure 2a refers to a cantilever beam with a hole applied at one-third of its horizontal length and the middle of its vertical length. The second test case presented in Figure 2b refers to the \"L shape\" geometry with a load applied at its right edge, one-fifth from the bottom. The third test case presented in Figure 2c refers to a long cantilever beam with the load applied at the middle of its left side. In all test cases, the proportions of the horizontal and vertical dimensions are presented in their Figures. For the optimization procedure, the \(L\) value was considered equal to 1. There are two sets of parameters considered for the concurrent multi-scale TO formulation. The first set consists of the geometry parameters, whereas the second set consists of the optimization parameters. In all cases, the periodic unit cell was considered to have square geometry, with a difference in scale of \(10^{-5}\). The periodic unit cell geometry was discretized with a grid of \(50\times 50\) finite elements in the directions of the abscissa and ordinate. Additionally, the loading applied in all test cases denoted as \(P\) was equal to 1 and the material was set as an isotropic material with a Young's modulus \(E\) equal to 1 and Poisson's ratio equal to 0.25. The applied grid discretizing of each test case was different between them. Hence, the cantilever with a fixed hole test case was discretized with a grid of \(150\times 100\) finite elements in the directions of the abscissa and ordinate whereas, the L shape test case was discretized with a grid of \(80\times 80\) finite elements in the directions of the abscissa and ordinate. Lastly, the long cantilever test case was discretized with a grid of \(160\times 40\) finite elements in the directions of the abscissa and ordinate. Moving to the second set of parameters, i.e., the optimization parameters, in all test cases a maximum number of optimization iterations was set equal to 200 and an optimization tolerance of 1%. Additionally, the penalization factor for the SIMP interpolation scheme was set equal to 3 and the reduction rate for the target volume for the BESO approach was set equal to 5%. Regarding the periodic unit cell, a target volume fraction of 50% was considered for all test cases and a sensitivity filtering technique for the BESO-based approach. The filtering radius for the sensitivity filtering was set equal to 1.5 elements.