# Smooth Design of 3D Self-Supporting Topologies Using Additive Manufacturing Filter and SEMDOT
## 3 Numerical Experiments
### Different Target Volume Fractions  
A simply supported deep beam, as shown in Figure 4, is used to investigate the effects of different target volume fractions on compliance, the number of iterations and topologies. As illustrated in Figure 4, four bottom corners are prevented, and a unit vertical load (\(F_{y}=1\) N--Figure 4) is applied in the middle of the top surface. Six different target volume fractions (i.e., 0.15, 0.2, 0.3, 0.4, 0.5 and 0.6) are considered in this study. The design domain is discretized by a 60 \(\times\) 40 \(\times\) 20 finite element mesh.  
Figure 5a shows that with an increase in the target volume fraction, the performance of optimized topologies gradually improves, and there are some difficulties in convergence when small target volume fractions (\(V^{*}=0.15\) and \(V^{*}=0.2\)) are considered. Optimized topologies with different target volume fractions are shown in Figure 5b-g. There is a thin feature in the middle of the optimized topology, as shown in Figure 5b. The value of the element at the bottom of this thin feature is small, whereas elements above this bottom element have greater values, which violates the basic strategy of [PERSON]'s AM filter. The formation of this thin feature is because of the absolute error caused by the softmax function (mentioned in Section 2.1).  
Figure 4: Design domain of a simply supported deep beam.  
The first solution to solve this specific thin feature issue is to introduce the weighting factor of \(w^{*}\) to the location of element \(X_{3}\) in the supporting region (see Figure 1). Equation (7) is therefore rewritten as  
\[\Xi_{(i,j,k)}=\max(\xi_{(i-1,j,k-1)},\xi_{(i,j-1,k-1)},w^{*}\xi_{(i,j,k-1)},\xi_ {(i,j+1,k-1)},\xi_{(i+1,j,k-1)}) \tag{18}\]  
The related equations are accordingly changed.  
The reference design (Figure 6a) converges at 3.621 J after 86 iterations. For the rest of the self-supporting designs (Figure 6b-d), the maximum number of iterations (500) is reached. Among self-supporting designs, the use of a higher weighting factor \(w^{*}\) will generally contribute to relatively improved performance. More importantly, the thin feature  
Figure 5: Compliance, number of iterations and optimized topologies with different target volume fractions: (**a**) compliance and number of iterations, (**b**) \(V^{*}=0.15\), (**c**) \(V^{*}=0.2\), (**d**) \(V^{*}=0.3\), (**e**) \(V^{*}=0.4\), (**f**) \(V^{*}=0.5\), (**g**) \(V^{*}=0.6\).  
issue in Figure 5 has been solved by introducing the weighting factor \(w^{*}\), as illustrated in Figure 6b-d.  
Another solution is to use a fine mesh, as it can provide more design freedom. The mesh size of 90 \(\times\) 60 \(\times\) 30 is used as an example. As the element size is scaled with a certain ratio, the filter radius is scaled with the same ratio to make sure its absolute value remains constant. Figure 7a shows that the self-supporting design case with the fine mesh for \(V^{*}=0.15\) converges at 3.484 J after reaching the maximum number of iterations (500), and the topological boundary error gradually decreases to almost 0% when the converged topology is obtained. The performance of the fine mesh case is 4.32% better than that of the optimized topology in Figure 5b (3.641 J). Figure 7b shows a much better self-supporting design than that in Figure 5b in terms of manufacturability. There is almost no structural resemblance between coarse mesh (Figure 5b) and fine mesh cases (Figure 7b) as [PERSON]'s AM filter is highly mesh-dependent. Compared to the strategy of introducing the weighting factor \(w^{*}\), using the fine mesh is a more general way to solve the thin feature issue.  
Figure 6: Optimized topologies with different weighting factors \(w^{*}\): (**a**) reference, \(C=3.621\) J. (**b**) \(w^{*}=1.01\), \(C=3.665\) J. (**c**) \(w^{*}=1.015\), \(C=3.662\) J. (**d**) \(w^{*}=1.02\), \(C=3.659\) J.