Numerical analysis

For numerical calculations we used FreeFem++ [ffempp, Oh14]. FreeFem++ is developed mainly by O.Pironneau (Paris 6) and the current leader F.Hecht (Paris 6), in which the FEM can be computed with mathematical description and moved mesh $\mathcal{T}_h(\Omega ^{\varphi _{\epsilon }})$. FreeFem++ is powerful and has many features, for example, adaptive mesh refinement and interpolation of FE-functions on different meshes. There are applications of FreeFem++ to shape optimization (see e.g., [A-P06, Al07]). First, we show the necessity of constraints using the Poisson equation with Dirichlet boundary condition \begin{equation} -\Delta u=f~ \textrm{in }\Omega ;~ u=0~ \textrm{on }\partial \Omega \tag{4.1} \end{equation} From (2.46) , the energy decrease if $\mu _{\varphi }\cdot n \gt 0$. Then, taking $R$ such that $\Omega \subset B_{R}=\{(x,y):\, x^2+y^2 \lt R^2\}$, we have for $f=1$ $$ \mathcal{E}(u(\Omega );1,\Omega )\ge \mathcal{E}(u(B_R);1,B_R) \ge \mathcal{E}(u(B_{R+1});1,B_{R+1})\ge …$$ However the global optimal shape exists when constraint conditions of constant area are added. Indeed the global optimal shape $\Omega ^O$ must satisfy the following: \begin{equation} \int _{\partial \Omega ^O} (\partial u^O/\partial n)^2h\, ds=0,~ ∀h\in C(\partial \Omega ^O)~~\textrm{such that }\int _{\partial \Omega ^O}h\, ds=0 \tag{4.2} \end{equation} from which we can derive the condition that $\partial u^O/\partial n=\textrm{constant}$ on $\partial \Omega ^O$. For example $\Omega ^O=\{(x,y):\, x^2+y^2 \lt 1\}$, $u^O(x)=(1-r^2)/4,\, r^2=x^2+y^2$ if $f=1$ and the area is $\pi $. Even if the constraint condition of constant area is added, our numerical calculations do not have global optimal shape. In actual optimum designs, there are many restrictions such as shape, weight, balance in products, so it is wrong to provide constraint conditions to determine a global optimal shape mathematically. Together with the theoretical considerations in Section 2 and 3 and numerical calculations, we hope our results to become hints to be finding optimal shape in actual optimum designs.

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