Consider the problem of shape optimization of a cantilever which applies load to the other end which is calculated many times in the world of shape optimization. The H1-gradient method used $b_{E}$ (3.31) . The calculations are done for the example given by Allaire [6.5.2, Al07]. The initial shape and its boundary conditions are shown in Figure 4 (a); $u=0$ on $\Gamma _D$, the surface force $g=(0,-1)$ is given on $\Gamma _N$, stress free on $\Gamma _F$. Here the singular points $\{\gamma \} ^D=\{(0,4),(0,2),(0,-2),(0,-4)\}$ appear. In the process of optimization, $\Gamma _N$ is fixed, $\Gamma _D$ allows deformation only in $x_2$-axis direction and $\Gamma _F$ is deformable freely, that is, $$ M(\Omega )= \{\eta \in H^1(\Omega ,\mathbb{R}^2):\, \eta \cdot n=0 \textrm{on }\Gamma _D,\, \eta =0 \textrm{on }\Gamma _N\}$$ The optimal shape in [Fig.6.11, Al07] is obtained by setting the vector field $\mu _h$ is $0$ near $\{\gamma \} ^D$ using the cut-off funtion. However, our optimal shapes are obtained allowing points $\{\gamma \} ^D$ to move in $x_2$-direction.

Fig. 4 (b) No-hole: Vector fields obtaind by H1-gradient method, and optimized shape after 400 iterations

Unlike Figure [Fig.6.11, Al07], Figure 4 (b) shows that the upper and lower fixed parts are spreading and the compliance descend from $8.92223$ to $1.60046$. In Figure 4 (c), the body force $f=(0,-0.1)$ is added. Optimization tries to become thick at the fixed part and thin at the tip. The compliance descend from $8.92223$ to $5.00514$.

Fig. 4 (c) The body force $f=(0,-1)$ is given. Recalculated, so the shape is slightly different from the SoptS-AMS[Oh22].

Figure 4 (d) shows an optimal shape corresponding [Fig.6.12, A-P06] that four holes are made in 4 (a).

Fig. 4 (d) Cantilever with four holes, Initial shape, Vector field after H1-gradient method, Optimized shape

Figure 4 (e) shows an optimal shape under the body force $f=(0,-0.1)$.

Fig. 4 (e) Cantilever with four holes, where $f=(0,-0.1)$ is given. Recalculated, so the shape is slightly different from the SoptS-AMS[Oh22]

Figure 5 (a) shows an optimal shape corresponding Figure [Fig.6.13, A-P06] that seven holes are made in Figure 4 (a). The compliance decrease from $8.92223$ to $1.70947$ only by opening seven holes, and it decrease to 0.948022 in Figure 5 (a). There is no big difference between Figure 5 (a) and [Fig.6.13, A-P06].

Fig. 5 (a) Cantilever with seven holes Recalculated, so the shape is slightly different from the SoptS-AMS[Oh22]

Figure 5 (b) shows an optimal shape under the body force $f=(0,-0.1)$, and the compliance decrease from $9.28524$ to $6.96177$.

Fig. 5 (b) Cantilever with seven holes, where $f=(0,-0.1)$ is given. Recalculated, so the shape is slightly different from the SoptS-AMS[Oh22]

Figure 5 (c) shows an optimal shape of nine holes, where only the holes could be made deformable, and the compliance decrease from $4.56778$ to $3.9979$.

optimal shape with 9 holes, initial shape, Same deformation conditions as before, Boundaries other than holes are fixed. Recalculated, so the shape is slightly different from the SoptS-AMS[Oh22]

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