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An appropriate value ofλ can be determined when both the volume and displacement con- straints are satisfied. For the ease of implementing the calculation,λ is defined as

λ = 1− w

w (6.18)

wherew is a constant ranging from wmin, e.g. 10−10, to 1.

To find the appropriate value ofw, the initial bound values of w are set to be wlower = wmin andwupper = 1. The program starts from an initial guess of w = 1 and the sensitivity number

is determined according to Equations (6.17) and (6.18). Thus, the threshold of sensitivity numbers can be determined when the volume constraint is satisfied by assuming that the density of element is xminor 1 if the elemental sensitivity number is smaller or larger than the threshold accordingly.

Then the displacement in the next iteration, ukj+1, can be estimated using Equation (6.10).

Thereafter, if uk_j+1> u∗_jwe updatew with a smaller value as follows ˆ

w = w + wlower

2 (6.19)

At the same time, we move the upper bound ofw so that wupper = w. On the other hand, if uij+1< u∗j, we updatew with a larger value as follows

ˆ

w = w + wupper

2 (6.20)

and the lower bound ofw is updated so that wlower = w.

With the updated w = ˆw, the above procedure is repeated until wupper− wlower is less

than 10−5. Therefore, a total of 17 iterations are needed to obtain an accurate Lagrangian multiplier. This procedure is typically used for an additional constraint except for the volume constraint. The cost for computing the Lagrangian multiplier is negligible in the following examples because the calculation only needs to update the relative ranking of sensitivity numbers according to Equation (6.17) in each iteration. For problems with more than one additional constraint, multiple Lagrangian multipliers will need to be introduced and a more sophisticated algorithm should be developed to improve the computational efficiency.

6.3.3

Examples

6.3.3.1 Local Displacement Constraint at a Roller Support

In this example, stiffness optimization is carried out for the structure shown in Figure 6.5. In addition to a common volume constraint, a displacement constraint is imposed on the horizontal movement at the roller support. The vertical load P= 100 N is applied to the

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74 Evolutionary Topology Optimization of Continuum Structures

(a) (b)

(c) (d)

Figure 6.6 Various optimal designs: (a) without additional local displacement constraint; (b) u∗_A= 1.4 mm ; (c) u∗_A= 1.2 mm (d) u∗_A = 1.0 mm.

middle of the upper edge. Due to the symmetry, the computation is performed on the right half of the design domain with 100× 100 four node plane stress elements. The material is assumed to have Young’s modulus E = 1 GPa and Poisson’s ratio ν = 0.3. The volume constraint V∗is 30 % of the design domain. The soft-kill BESO method with xmin= 0.001, p = 3, E R = 2 % and rmin = 1.5 mm is used.

The optimal topology without any displacement constraint is given in Figure 6.6(a) for the purpose of comparison. Its mean compliance is 191 Nmm and the horizontal movement of the roller support at point A is 1.43 mm. When the horizontal movement of the roller is constrained to be less than or equal to 1.4 mm, 1.2 mm or 1.0 mm, we obtain the topologies shown in Figures 6.6(b)–(d). Their mean compliances are 191 Nmm, 195 Nmm and 203 Nmm respectively. It is noted that there is an increase in the final mean compliance when a stricter displacement constraint is imposed.

Figure 6.7(a) presents evolution histories of the mean compliances for various displacement constraints. It shows that the objective function (mean compliance) is convergent at the final stage of the optimization process for all cases. Figure 6.7(b) gives evolution histories of the horizontal movement of the roller support. It is seen that the displacement converges to its constraint value at the final stage.

6.3.3.2 A Bridge-type Structure

In this example an optimization problem of designing a bridge is considered as shown in Figure 6.8. The design domain is a rectangle of the size L× H = 200 mm × 40 mm. The bottom deck of L× h = 200 mm × 1 mm is supported at two corners. The thickness of both the design domain and the deck is assumed to be 1 mm. A uniformly distributed load p= 1 N/mm is applied on the top surface of the deck. The design domain is divided

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Figure 6.7 Evolution histories: (a) mean compliance; (b) constrained displacement.

into 200× 40 four node plane stress elements and the nondesignable deck is divided into 200 beam elements. The nodes of the beam elements are connected to the corresponding nodes of the plate elements at the bottom side of the design domain. The material properties for the design domain are Young’s modulus E= 1 GPa and Poisson’s ratio ν = 0.3, and the material properties for the deck are assumed to be Young’s modulus E= 100 GPa and

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Figure 6.8 Design domain, loading and boundary conditions for a bridge-type structure.

Poisson’s ratioν = 0.3. The volume constraint V∗is 50 % of the design domain. The soft-kill BESO method is used with xmin = 0.001, p = 3, E R = 2 % and rmin= 3.0 mm.

Figure 6.9(a) presents the optimal design using the traditional BESO method without a local displacement constraint. It shows a tie-arch bridge with the deck suspended by a series of cables inclined at different angles. The mean compliance of this design is 93.5 Nmm and the vertical deflection at the middle of the upper edge (point A in Figure 6.8) is 2.7 mm. When the vertical deflection at A is constrained to be less than or equal to 2.4 mm and 2.0 mm, we obtain the topologies shown in Figures 6.9(b) and (c). With the additional displacement constraint, the deck is suspended by a series of cables that are almost parallel to each other.

(a)

(b)

(c)

Figure 6.9 Various optimal designs: (a) without additional local displacement constraint; (b)ν∗_A= 2.4 mm ; (c) ν∗_A= 2.0 mm.

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BESO for Extended Topology Optimization Problems 77

Figure 6.10 Evolution histories: (a) mean compliance; (b) constrained displacement.

The mean compliances are 95.8 Nmm and 99.0 Nmm forν∗A = 2.4 mm and ν∗A= 2.0 mm

respectively. Figure 6.10 shows evolution histories of the mean compliance and the constrained displacement. It is noted that both the mean compliance and the displacement are convergent at the final stage even though a few large jumps have occurred forν∗A= 2.0 during early

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