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The present BESO method can be applied to the design of the cellular core in a sandwich structure. As illustrated in Figure 6.19, the sandwich structure is fixed at both ends of the skins. The designable core is a rectangle of 160 mm in length and 40 mm in width, which is divided into 320× 80 four node plane stress elements. The two skins, which are 1 mm in thickness, are divided into 320 beam elements for each skin. It is assumed that the skins and the core are tied together. A vertical point force P = 1 N is applied at the middle point of the top skin. The materials of the skins and the core are assumed to have Young’s modulus 100 GPa and Poisson’s ratio 0.3, and Young’s modulus 1GPa and Poisson’s ratio 0.3 respectively.

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Figure 6.20 Evolution histories of mean compliance, volume fraction and topology of the sandwich

structure with m= 2 × 1 when BESO starts from the initial full design.

The volume constraint is 30 % of the core space. The filter radius and evolutionary volume ratio are selected to be rmin= 2 mm and E R = 1 %. The hard-kill BESO method is used.

Figure 6.20 shows the evolution histories of the objective function (mean compliance), the volume fraction and the topology for m= 2 × 1 when the BESO starts from the initial full design. It is seen that all topologies satisfy the prescribed periodic constraint. The mean compliance of the final topology is 8.33 Nm. Table 6.1 presents the optimal topologies and their mean compliances for various numbers of unit cells. A typical unit cell is given inside dashed lines except for m= 1 × 1. Figure 6.21 plots the values of the final mean compliance against the number of unit cells. It is seen that the mean compliance increases with the number of unit cells. When the number of unit cells increases, the total number of independent design variables decreases. Therefore, the solution of the conventional BESO method corresponding to the special case of m= 1 × 1 has the lowest mean compliance among all optimal designs. On the other hand, the optimal topology depends on the aspect ratio of the unit cell. For example, the optimal topologies for m= 2 × 1 and m = 1 × 2 are totally different, even though their total numbers of unit cells are equal.

The BESO method may also start from an initial guess design. Such an initial design does not have to be an ‘intelligent’ guess or a near-optimum. Indeed, it could be a highly nonoptimal design. As an example, Figure 6.22 presents the evolution histories of the mean compliance, the volume fraction and the topology for m= 2 × 1 when the hard-kill BESO starts from the initial guess design shown at the top left corner. The initial guess design bears no resemblance to the final optimal topology. At the beginning of the evolution process, both the mean compliance and the volume decrease, which implies that the overall stiffness of the

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Table 6.1 Optimal designs and their mean compliances for the sandwich structure under various periodic constraints.

m2 = 1 m2 = 2 m1 = 1 C = 7.97 Nm C = 10.47 Nm m1 = 2 C = 8.33 Nm C = 10.66 Nm m1 = 4 C = 10.38 Nm C = 11.31 Nm m1 = 8 C = 12.20 Nm C = 12.93 Nm

structure is increased due to the optimization algorithm despite the fact that the volume of the structure is reduced. The volume fraction remains constant after the 7th iteration and the compliance continues to decrease until it converges to a constant value, 8.33 Nm, which is the same as that of the solution shown in Figure 6.20. One advantage of starting from an initial guess design with a volume close or equal to the target volume is that, while the volume is kept constant, the improvement in the objective function can be clearly seen in its evolution history. Another advantage is that this procedure is computationally more efficient as only a portion of all elements in the full design domain needs to be included in the finite element analyses. However, it should be pointed out that BESO starting from an initial guess design may sometimes converge to a local optimum (Huang and Xie 2007).

6.5.2.3 A Bridge-type Structure

Figure 6.23 shows an optimization problem of designing a bridge-type structure. The design domain is a rectangle of length L= 240 and depth H = 60 (thickness t = 1). The deck at the bottom with length L= 240, depth h = 5 (and thickness t = 1) is supported at two corners. A vertical load P = 100 is applied to the mid point of the deck. The design domain is divided into 240× 60 four node plane stress elements and the nondesignable deck is modelled with 240 beam elements. The nodes of the beam elements are connected to those of the plate elements at the bottom side of the design domain. It is assumed that the materials for the design domain and the deck are the same with Young’s modulus E = 2000 and Poisson’s ratio ν = 0.3. The volume fraction constraint is 30 % of the design domain. The hark-kill BESO method with

E R= 2 % is used in this example.

Figure 6.24(a) shows the optimal design for m= 1 × 1, which is the same as the conven- tional optimal design without any periodic constraint. The mean compliance of this design is 1.12. Figures 6.24(b) and (c) show optimal designs for m = 4 × 1 and m = 6 × 1 respectively.

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Figure 6.21 Effect of the number of unit cells on the mean compliance.

Figure 6.22 Evolution histories of mean compliance, volume fraction and topology of the sandwich

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Figure 6.23 Design domain, loading and boundary conditions of a 2D bridge-type structure.

Their mean compliances are 1.53 and 1.78, which are higher than that of the conventional de- sign. Similar to the example discussed in the previous section, the mean compliance increases with the total number of unit cells.

To study the influence of the deck stiffness, the above problem is reconsidered by increasing the deck depth h from 5 to 50. The optimal design for m= 4 × 1 is shown in Figure 6.25. The result is significantly different from the design shown in Figure 6.24(b). This demonstrates that the optimal topology may depend on the stiffness of nondesignable parts.