# Topology Optimization with Explicit Components Considering Stress Constraints
## 5 Numerical Solution Aspects
### L-Shaped Beam  
The design domain and boundary conditions of the L-shaped beam example are shown in Figure (a)a. A fixed constraint is applied to the top of the beam, and a downward force \(F=1\) is applied to the center of the right side. Obviously, stress concentration will \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**epsimin** & **raa0** & **albefa** & **asyinit** & **asyincr** & **asydecr** \\ \hline \(10^{-10}\) & 0.01 & 0.4 & 0.1 & 0.8 & 0.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: MMA optimizer parameter settings. occur in the right-angle area. The finite-element analysis uses a four-node rectangular bilinear element with an element size of 0.025 and a total of 4096 elements. Components are essential as the fundamental building blocks of the MMC method. With a smaller number of components, the limited descriptive ability of the geometry may fail to identify effective force transfer paths, causing the optimization process to oscillate significantly and leading to non-convergence. Conversely, a higher number of components introduces a large number of design variables, resulting in excessive computational demand. Considering both convergence and computational efficiency, this paper adopts 16 components. The initial layout of the components is shown in Figure 8, and the initial parameters of each component are set to \(ini\_val=[0.33\) 0.20 0.08 0.08 0.04]. In addition, each component also includes three parameters representing the position and angle, for a total of 8 parameters. Thus, the solution dimension of the problem is 128 design variables. The value of the upper limit of the global volume constraint is set to \(\overline{V}=0.4\), and the upper limit value of the global stress constraint is set to \(\delta_{lim}=70\), which is the same as the setting in some other literature [15; 36]. Figure 9 shows the final result of the optimization. After approximately 70 iterations, it converges stably, satisfying the volume constraint and the global stress constraint, with the minimum compliance being 178.288. Benefiting from the advantage that the MMC method can be directly integrated into the CAD/CAE system, the geometric boundary information is transmitted to ABAQUS. Based on the method mentioned in ABAQUS [37], the obtained geometric boundary, the created finite-element mesh (a hybrid mesh dominated by quadrilaterals with an element size of 0.025, consisting of 1622 elements in Figure 9 and 1703 elements in Figure 9), and the plotted von Mises stress distribution are shown in Figure 9. These results compare scenarios with and without considering the stress constraint. From the analysis results in Figure 9, it is evident that after adding the global stress constraint, the angle at the corner of the L-shaped beam becomes larger, and the stress concentration at the corner is significantly improved. While maintaining structural stiffness (the structural compliance only increases by 7.23%, from 166.261 to 178.288), the local maximum stress value of the structure is significantly reduced (the maximum von Mises stress value is reduced by 49.47%, from 137.640 to 69.548). Additionally, the overall stress distribution of the structure is more uniform. The iterative convergence curves of the objective function and the constraint function are shown in Figure 10.  
Figure 8: The design area and the initial layout of the components of the L-shaped beam. The iterative curve shows that in the early stages of optimization, the components search for the optimal structural form with large-scale movement and deformation, causing significant oscillations in the objective function and the constraint function. However, the compliance of the structure exhibits an overall downward trend. As optimization progresses, the objective function and the constraint function gradually stabilize and converge. Some key iterative steps in the optimization process are shown in Figure 11.  
Figure 10: The iterative convergence curves of the objective function and the constraint function of the L-shaped beam.