Optimal Design of Energy Absorption Structures
8.5 Numerical Examples and Discussions
8.5.1
Example 1
The simply supported structure shown in Figure 8.3 is 100 mm long, 20 mm deep and 1 mm thick. The crushing displacement loading is applied at the centre of the top edge. The allowable maximum crushing force is set to be 20 kN and the maximum crushing distance umax is 20 mm. The material has Young’s modulus E = 200 GPa, Poisson’s ratio ν = 0.3, yield stress σy= 300 MPa and plastic hardening modulus Ep= 0.3 E. The design domain is discretized
using 200× 40 four node plane stress elements (which do not consider out-of-plane buckling). The mesh-independency filter radius is 3 mm.
In this example, BESO starts from the initial full design. Figures 8.4(a) and (b) show the evolution histories of the volume fraction, absorbed energy per unit volume and the maximum crushing force using criteria 1 and 2 respectively. In both cases, the absorbed energy per unit volume tends to increase as the volume decreases, and the maximum crushing force gradually decreases as the volume decreases until it satisfies the force constraint. It should be pointed out that the sudden jumps in the absorbed energy per unit volume in Figure 8.4(a) are caused
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(a)
(b)
Figure 8.4 Evolution histories of volume fraction Vf, absorbed energy per unit volume E/V and
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Figure 8.5 Optimal designs and their deformed shapes: (a) optimal design using criterion 1; (b) optimal design using criterion 2; (c) final deformation of optimal design (a); (d) final deformation of optimal design (b).
by significant changes of topology as a result of bar eliminations. Thereafter, the absorbed energy per unit volume recovers and the topology develops in the right direction. At the final stage of the evolution, the absorbed energy per unit volume converges to a maximum value while the volume oscillates within a narrow band of 1 % of the current volume as the force constraint is satisfied and violated alternately in two successive iterations.
The obtained optimal designs and their final deformation are given in Figure 8.5. Figure 8.6(a) shows the force-displacement curves of the optimal designs and the initial full design. It is seen that the initial full design does not satisfy the force constraint as the force exceeds the maximum allowable value of 20 kN long before the prescribed maximum crushing distance of 20 mm is reached. In contrast, the maximum crushing forces of both optimal designs obtained from the BESO method are within the allowable limit. Figure 8.6(b) shows the energy-displacement curves for the two optimal designs. It is seen that the design
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(b) (a)
Figure 8.6 Comparison of different designs: (a) force-displacement curves of optimal designs and initial full design; (b) energy-displacement curves of optimal designs.
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Table 8.1 Detailed comparison of various designs from four examples.
Design
Volume fraction,
Vf(%) Fmax F∗ W (J) e1(MN/m2) e2
Example 1 Initial full design 100 35.02 kN 20 kN –1 229.16 –
Design for criterion 1 45.7 19.98 kN 20 kN 289.70 316.96 0.724
Design for criterion 2 48.5 19.92 kN 20 kN 295.44 304.58 0.739
Example 2 Initial full design 100 35.64 kN 20 kN – 174.78 –
Design for criterion 1 41.9 19.96 kN 20 kN 222.27 265.24 0.741
Design for criterion 2 45.1 19.91 kN 20 kN 227.00 251.66 0.757
Example 3 Initial full design 100 102.82 N 70 N – 180.79 –
Design for criterion 1 51.8 69.58 N 70 N 0.7045 226.67 0.503
Design for criterion 2 56.9 69.99 N 70 N 0.7436 217.81 0.531
Example 4 Initial guess design 50 29.25 N 70 N 0.2568 85.59 0.183
Design for criterion 1 51.8 70.0 N 70 N 0.7018 225.89 0.501
Design for criterion 2 55.0 69.55 N 70 N 0.7310 221.50 0.522
1Value has no meaning as the design is unacceptable because F
max> F∗.
using criterion 2 absorbs slightly more energy than the design using criterion 1. This is mainly due to the fact that the former design is heavier than the latter, as shown in Table 8.1.
In Table 8.1 we compare the optimal designs and the initial design in various aspects. Firstly, it is noted that the absorbed energy per unit volume e1of each of the two optimal designs has significantly increased compared to the initial full design. The design using criterion 1 has the highest e1among the three designs. These results clearly demonstrate the effectiveness of the present BESO procedure for improving the energy absorption performance of structures. Secondly, the design using criterion 2 has higher e2(as well as higher volume) than the design using criterion 1. Note that e2is a measure of the total energy absorbed by the structure.
8.5.2
Example 2
In Example 2, the same design domain, supporting conditions and material as in Example 1 are used, but the structure is crushed at different locations as shown in Figure 8.7. The allowable maximum force is 20 kN and the maximum crushing distance is 15 mm.
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(a)
(b)
Figure 8.8 Evolution histories of volume fraction Vf, absorbed energy per unit volume E/V and
maximum crush force Fmaxfor example 2 using different criteria: (a) criterion 1; (b) criterion 2.
The evolution histories of the volume fraction, the absorbed energy per unit volume and the maximum crushing force are shown in Figure 8.8(a) for criterion 1 and in Figure 8.8(b) for criterion 2. These results indicate that optimal designs (at least local optimal designs) for both criteria have been obtained because the absorbed energy per unit volume converges to a maximum value at the final stage. Figure 8.9 shows the obtained topologies and their final deformation. Compared to the topologies in Figure 8.5, the two bars in the middle are
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Figure 8.9 Optimal designs and their deformed shapes: (a) optimal design using criterion 1; (b) optimal design using criterion 2; (c) final deformation of optimal design (a); (d) final deformation of optimal design (b).
now moved apart to support the two separate external loads. The force-displacement curves in Figure 8.10(a) illustrate that both optimal designs satisfy the force constraint whereas the initial full design does not. Figure 8.10(b) shows the energy-displacement curves of both optimal designs. More details about these designs are given in Table 8.1. Once again, the energy absorbed per unit volume of each of the optimal designs has been significantly improved compared to the initial full design. The design using criterion 1 is the best in terms of the energy absorbed per unit volume, e1. The design using criterion 2 has higher total absorbed energy, or e2, (as well as higher volume) than the design using criterion 1.
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(a)
(b)
Figure 8.10 Comparison of different designs: (a) force-displacement curves of optimal designs and initial full design; (b) energy-displacement curves of optimal designs.
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Figure 8.11 Design domain, displacement loading and support conditions for example.
8.5.3
Example 3
In the third example, a structure, 200 mm long, 30 mm high and 1 mm thick, is fixed along both ends. A rigid object collides with the structure at the centre of the top edge as shown in Figure 8.11. The material of the structure has Young’s modulus E = 1 GPa, Poisson’s ratio ν = 0.3, yield stressσy= 1 MPa and plastic hardening modulus Ep = 0.1 E. The allowable maximum
crushing force and crushing distance are set to be 70 N and 20 mm respectively. The design domain is discretized using 400× 60 four node plane stress elements. The mesh-independency filter radius is 3 mm.
Figure 8.12 shows the evolution histories of the volume fraction, the absorbed energy per unit volume and the maximum crushing force. The optimal designs obtained from using criteria 1 and 2 and their final deformation are presented in Figure 8.13. Figure 8.14(a) shows the force-displacement curves of the optimal designs and the initial full design. It is seen that both optimal designs satisfy the force constraint whereas the initial full design does not. The energy-displacement curves for both optimal designs are shown in Figure 8.14(b). Table 8.1 provides more details about these designs. Similar conclusions can be drawn as in previous examples, i.e. both optimal designs are significantly better than the initial full design in terms of energy absorbed per unit volume and the design using criterion 2 (which has more material) absorbs more energy than the design using criterion 1.
8.5.4
Example 4
To save the computation time, we may begin the BESO process from an initial guess design. In this example we solve the problem in example 3 by starting from the initial guess design shown in Figure 8.15, which occupies 50 % of the full design domain. Figures 8.16(a) and (b) show evolution histories of the volume fraction, the absorbed energy per unit volume and the maximum crushing force using criteria 1 and 2 respectively. It is seen that the absorbed energy per unit volume increases to a maximum value as the topology evolves. Meanwhile, the volume increases and then decreases as the maximum crushing force falls below and rises above the maximum allowable value. Similar to the previous examples, the absorbed energy per unit volume converges to a maximum value at the final stage, and the volume oscillates within a narrow band of 1 % of the current volume as the force constraint is satisfied and violated alternately in two successive iterations.
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(a)
(b)
Figure 8.12 Evolution histories of volume fraction Vf, absorbed energy per unit volume E/V and
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Figure 8.13 Optimal designs and their deformed shapes: (a) optimal design using criterion 1; (b) optimal design using criterion 2; (c) final deformation of optimal design (a); (d) final deformation of optimal design (b).
The optimal designs obtained from using criteria 1 and 2 are given in Figure 8.17 together with their final deformation. Compared to the topology results shown in Figure 8.13 from the initial full design, the optimal design using criterion 1 is almost identical whereas the optimal design using criterion 2 is slightly different. Figure 8.18(a) shows the force-displacement curves of the optimal designs and the initial guess design. It is seen that the two optimal designs satisfy the force constraint (Fmax= F∗) whereas the maximum crushing force of the initial guess design is far below the allowable limit. As shown in Figure 8.18(b) and Table 8.1, although the three designs use similar amount of material, each of the two optimal designs can absorb almost three times as much energy as the initial guess design. This highlights the substantial benefit that can be achieved from applying topology optimization techniques to the design of energy absorption structures.
A more detailed comparison of these designs is given in Table 8.1. The results are similar to those of Example 3. However, the computation time for example 4 is significantly less because the costly nonlinear finite element analysis is conducted on much smaller models.
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(a)
(b)
Figure 8.14 Comparison of different designs: (a) force-displacement curves of optimal designs and initial full design; (b) energy-displacement curves of optimal designs.
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(a)
(b)
Figure 8.16 Evolution histories of volume fraction Vf, absorbed energy per unit volume E/V and
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Figure 8.17 Optimal designs and their deformed shapes: (a) optimal design using criterion 1; (b) optimal design using criterion 2; (c) final deformation of optimal design (a); (d) final deformation of optimal design (b).
8.6
Conclusion
In this chapter, a topology optimization procedure for energy absorption structures has been developed using the hard-kill BESO method. Sensitivity numbers for two different criteria are derived in order to maximize the energy absorbed per unit volume at the end displacement or during the whole displacement history. Numerical examples show that an initial full design or a guess design which may not satisfy the force constraint can evolve to an optimal design which satisfies both force and displacement constraints. Compared to the initial full design or the guess design, the optimal design using criterion 1 or 2 leads to significantly improved energy absorption performance measure e1or e2. The results also show that the design using criterion 1 with less material has a higher e1than that using criterion 2; whereas, the design using criterion 2 with more material has a higher e2 than that using criterion 1. The BESO method presented in this chapter is applicable to the topology design of both 2D and 3D continuum structures for the purpose of substantially enhancing or optimizing their energy absorption capabilities.
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(a)
(b)
Figure 8.18 Comparison of different designs: (a) force-displacement curves of optimal designs and initial guess design; (b) energy-displacement curves of optimal designs and initial guess design.
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