## 4 The TABASCO Algorithm in Use
The effectiveness of the proposed sorted compliances optimization concept (see [44]) can be controlled not only by the \(\beta\) parameter; the adaptive technique for the width of the interval [N\({}_{1}\), N\({}_{2}\)] can be suggested. In the first exploration phase of the calculations, when the preliminary layout of the structure is being sought, the interval [N\({}_{1}\), N\({}_{2}\)] is proposed to be wide. Then, in the exploitation phase, in order to guide the calculations to a black/white (solid/void) structure, the interval is successively reduced. Based on the numerous tests performed [44], the following unified strategy can be proposed: if the iteration number is lower than 25, then N\({}_{1}\) = 0.02 N and N\({}_{2}\) = 0.98 N for the second phase; while, if the iteration number is greater than 25, N\({}_{1}\) = 0.6 N and N\({}_{2}\) = 0.98 N, where N is a number of elements (cells). As one can notice, the solution is free from the checkerboard effect, without any of the additional filtering or techniques reported in, e.g., [47]. The gray elements are absent in the solutions obtained with the proposed method, which is a great advantage compared with the OC method, for which the scale of occurrence of gray elements in the solution is dictated by additional parameters. To complete the discussion of generating the optimized topology conducted for the test structure, the comparison with the results found using the well-known optimality criteria method [40] was prepared. The final compliance for the mentioned code was 136.55 Nmm. The long cantilever structure shown in Figure 8 has been chosen as the next test example. Figure 7: Compliance history for the short cantilever. The figure below provides a closer look, referring to the narrowed interval of objective function values. In this example, the value of load equals P = 100 N, the parameter a is equal to a = 10 mm, while for the Young modulus and Poisson ratio, E = 10 GPa and \(\upnu\) = 0.3, respectively. The mesh of finite elements consists of 14,400 square elements, 240 \(\times\) 60, regular mesh, 4-node elements, an element edge length 1 mm. The calculations were performed for the volume fraction 0.5, and the resulting topology is presented in Figure 9, while the iteration history is presented in Figure 10. For the generated topology of the second test structure, the value of the compliance equals 365.69 Nmm, and for original, former FCA [44], the value of the compliance equals 371.97 Nmm. Figure 8: Long cantilever: computational model including supports and applied loads. Figure 10: Compliance history for the long cantilever. The figure below gives a closer look, referring to the narrowed interval of objective function values. Figure 4:0 Long cantilever results: the final topology. To sum up the investigation, the snail-like structure has been selected (see Figure 11) as the third test example. The values of parameters P and a equal P = 100 N and a = 10 mm, while the Young modulus and Poisson ratio equal E = 10 GPa and \(\
u\) = 0.3, respectively. The assumed volume fraction for this example is 0.5. The finite element mesh consists of 19,200 square elements (240 \(\times\) 80, regular mesh, 4-node elements, element edge length 1 mm). The final topology and the iteration history illustrating the fast convergence to the optimal solution are presented in Figures 12 and 13, respectively. The compliance value for the final topology of the third test structure equals 12.659 Nmm, and for original, former FCA [44], the value is 12.662 Nmm. This result has been obtained for \(\beta\) = 0.1. Figure 4:1 Snail-like structure: computational model including supports and applied loads. Figure 4:2 Snail-like structure results: the final topology.