# Topology Optimization Considering Porosity Defects in Metal Additive Manufacturing
## 6 Numerical Examples
### C-Shaped Bracket  
The design domain of the second example is a C-shaped area with a ratio 3:2 of length \(L\) to height \(H\), meshed with 3600 elements, as shown in Figure 8a. Its lower side is fixed, and a vertical load \(P=1\) is applied at the bottom of its right-hand side. The upper bound of volume ratio is 0.4. The evolutionary volume ratio \(c_{\sigma}\) is set to 0.04, the filter radius \(R_{min}\) is taken as 3, and the interval \(j\) of hole nucleation is 5. The initial shape is displayed in Figure 8b. At first, the level set topology optimization of hole nucleation combining BESO and topological sensitivity is employed. The optimized shape is shown in Figure 9. The optimization converges after 75 iterations, and the evolution of the volume fraction and the compliance are shown in Figure 10.  
Secondly, the preliminary optimization results of C-shaped bracket are imported to perform topology optimization considering the porosity constraint. The pores radius r is set to 0.5, and the tolerance \(\eta\) is taken as 0.15. The optimization converges after 15 iterations, and the evolution of the volume fraction, the compliance, and porosity constraint are shown in Figure 11. The preliminary optimization results and the optimization results considering porosity constraint are shown in Figure 12a,b, respectively, and the corresponding porosity constraint distribution clouds are shown in Figure 12c,d.  
Figure 7: Optimization results and porosity constraint distribution clouds of L-shaped beam, (**a**) optimized shape without considering porosity constraint; (**b**) optimized shape considering porosity constraint; (**c**) porosity constraint distribution clouds without considering porosity constraint; (**d**) porosity constraint distribution clouds considering porosity constraint.  
As shown in Figure 11, the optimization problem with porosity constraints converges after 15 iterations, the porosity constraint value decreases 15% from 187.1, but the objective compliance increases from 293.7 to 323.3. Without considering the porosity constraint, the maximum porosity constraint value appears near the reentrant corner of the optimization result shown in Figure 12c, while the porosity constraint value in other areas is almost zero. Therefore, under the effect of porosity constraints, the reentrant corner will be key optimization region of the structure. According to the optimization results of Figure 12d, under the effect of porosity constraint, the reentrant corner becomes smooth and presents a characteristic of fillet. The results show that the porosity constraint value decreases obviously, and the maximum value decreases from 18.9 to 10.4. Comparing the changes in compliance with or without considering porosity constraint, there is a deviation that is  
Figure 8: Optimum design problem of C-shaped bracket, (**a**) design domain and boundary conditions; (**b**) initial shape.  
Figure 10: Convergence history of C-shaped bracket by hole nucleation method.  
Figure 9: Preliminary optimized shape of C-shaped bracket.  
within the admissible range. The optimization results of the C-shaped bracket are similar to the L-shaped beam, which further proves the effectiveness of the proposed method.