Article Contents
Article Contents

# Topological optimization and minimal compliance in linear elasticity

• * Corresponding author: Cornel Marius Murea
• We investigate a fixed domain approach in shape optimization, using a regularization of the Heaviside function both in the cost functional and in the state system. We consider the compliance minimization problem in linear elasticity, a well known application in this area of research. The optimal design problem is approached by an optimal control problem defined in a prescribed domain including all the admissible unknown domains. This approximating optimization problem has good differentiability properties and a gradient algorithm can be applied. Moreover, the paper also includes several numerical experiments that demonstrate the descent of the obtained cost values and show the topological and the boundary variations of the computed domains. The proposed approximation technique is new and can be applied to state systems given by various boundary value problems.

Mathematics Subject Classification: Primary: 49Q10, 65K10; Secondary: 74P05.

 Citation:

• Figure 1.  The geometrical configuration

Figure 2.  The fixed domain $D$ including the unknown domain

Figure 3.  Cantilever. Left: Geometrical configuration of $D$. Right: Convergence history of the objective functions for descent directions $i)$ given by (2.30) and $ii)$ given by (2.31)

Figure 4.  Cantilever. Initial (top, left), intermediate and optimal (bottom, right, after 50 iterations) domains using descent direction $i)$

Figure 5.  Cantilever. Optimal domains using descent directions $ii)$ given by (2.31) (left, after 38 iterations) and $iii)$ given by (2.32) (right, after 3 iterations) for the initial domain as in Figure 4

Figure 6.  Bridge. Left: Geometrical configuration of $D$. Right: Convergence history of the objective functions for descent directions $i)$ and $ii)$

Figure 7.  Bridge. Initial (top, left), intermediate and optimal (bottom, right, after 100 iterations) domains using descent direction $i)$

Figure 8.  Bridge. Optimal domain using descent directions $ii)$ after 80 iterations, for the initial domain as in Figure 7

Figure 9.  Bridge. Optimal domain using descent directions $i)$ after 100 iterations, for the initial domain $\Omega_0 = ]-1, 1[ \times ]0, 0.6[$, the bottom half of $D$

Figure 10.  Bridge. The meshes after the elastic deformations for the initial domain $\Omega_0 = ]-1, 1[ \times ]0, 0.6[$ and for the optimal domain $\Omega_{100}$ presented in Figure 9. The displacements were reduced by a factor $0.1$. The cost (1.7) decreases from $0.378632$ (left image) to $0.297857$ (right image)

Table 1.  The dependence of the cost given by (1.7) and of the displacement $\mathbf{y}^\epsilon(g_0)$ solution of (2.10) in the initial domain $\Omega_0 = ]-1, 1[ \times ]0, 0.6[$. The cost for $\mathbf{y}^*$ is $0.378727$

 $\epsilon$ $J$ $\left\| \mathbf{y}^\epsilon(g_0)-\mathbf{y}^*\right\|_{L^2(\Omega_0)}$ $\left\| \mathbf{y}^\epsilon(g_0)-\mathbf{y}^*\right\|_{H^1(\Omega_0)}$ 0.01 0.353644 0.097841 0.302369 0.005 0.369480 0.035030 0.112965 0.001 0.378150 0.002096 0.027376 0.0005 0.378506 0.000799 0.026536

Table 2.  The dependence of the cost given by (1.7) and of the displacement $\mathbf{y}^\epsilon(g_{100})$ solution of (2.10) in the final domain $\Omega_{100}$. The cost for $\mathbf{y}^*$ is $0.298536$

 $\epsilon$ $J$ $\left\| \mathbf{y}^\epsilon(g_{100})-\mathbf{y}^*\right\|_{L^2(\Omega_{100})}$ $\left\| \mathbf{y}^\epsilon(g_{100})-\mathbf{y}^*\right\|_{H^1(\Omega_{100})}$ 0.01 0.296596 0.018279 0.105998 0.005 0.297032 0.005071 0.084750 0.002 0.297813 0.002263 0.076559 0.001 0.298063 0.001925 0.073167
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