5.4.1
Introduction of Zhou and Rozvany (2001) Example
The structure shown in Figure 5.3(a) is used by Zhou and Rozvany (2001) to show the breakdown of hard-kill optimization methods, such as ESO/BESO. In the example, Young’s modulus is taken as unity and Poisson’s ratio as zero. If hard-kill ESO/BESO methods are applied to an FE model with 100 four node plane stress elements shown in Figure 5.3(b), the element in the vertical tie with the lowest strain energy density will be removed from the ground structure. The mean compliance of the resulting structure will be much higher than that of any institutive design obtained by removing one element in the horizontal beam.
After removing an element in the vertical tie, the resultant structure becomes a cantilever where the vertical load is transmitted by flexural action. The region with the highest strain energy density is at the left-bottom of the cantilever. According to the BESO algorithm, an ele- ment may be added in that region rather than recovering the removed element in the vertical tie. Therefore, Zhou and Rozvany (2001) conclude that hard-kill optimization methods such as ESO/BESO may produce a highly nonoptimal solution. In fact, soft-kill optimization algorithms such as the level set method using continuous design variables may also produce a similar result (Norato et al. 2007). To overcome this problem, the essence of such a solution needs to be examined first.
5.4.2
Is it a Nonoptimal or a Local Optimal Solution?
Obviously, the answer cannot be easily found by simply comparing the values of the objective function. Let us reconsider the above example with a volume fraction of 96 %. Hard-kill optimization methods such as ESO will remove the four elements from the vertical tie as shown in Figure 5.3(c). This design is certainly far less efficient than an intuitive design which removes four elements from the horizontal beam.
It is known that the SIMP method with continuous design variables guarantees that its solution should be at least a local optimum. Therefore, this topology optimization problem is tested by the SIMP method starting from an initial guess design (with xi= 1 for all elements
in the horizontal beam and xi = xmin= 0.001 for the four elements in the vertical tie). It is found that when p ≥ 3.1 the final solution converges to the structure shown in Figure 5.3(d), which is exactly the same as the initial guess design. Because xminis small, the SIMP solution in Figure 5.3(d) can be considered to be identical to the ESO/BESO solution in Figure 5.3(c).
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60 Evolutionary Topology Optimization of Continuum Structures
(a)
(b)
(c)
(d)
(e)
Figure 5.3 Zhou and Rozvany (2001) example: (a) design domain, load and support conditions; (b) a
coarse mesh; (c) a highly inefficient local optimum for Vf = 96 % from ESO; (d) a highly inefficient
local optimum for Vf = 0.96 from SIMP when p ≥ 3.1; (e) an optimal solution for Vf = 50 % from
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Comparing BESO with Other Topology Optimization Methods 61
These results demonstrate that the above solutions from ESO/BESO and SIMP are essentially a local optimum rather than a nonoptimum. Theoretically it may be more appropriate to call such a solution a highly inefficient local optimum than a nonoptimum.
The occurrence of the above 0/1 local optimal design is caused by the large penalty p in the optimization algorithms. Hard-kill ESO/BESO methods have an equivalent penalty of infinity and therefore fail to obtain a better solution once they reach the highly inefficient local optimum. Similarly, the soft-kill BESO method with a finite penalty may also fail because a large penalty ( p≥ 1.5) is normally required for topology optimization.
The exact value of the penalty p that is large enough to cause a local optimum is dependent upon the optimization problem. For the original Zhou and Rozvany (2001) example given in Figure 5.3(a), the SIMP method will produce a much more efficient solution than the one shown Figure 5.3(d) when p= 3 is used. However, if we modify the original problem slightly by reducing the vertical load from 1 to 0.5, the SIMP method with p= 3 will again result in the highly inefficient local optimum shown in Figure 5.3(d).
5.4.3
Avoidance of Highly Inefficient Local Optimum
It is well-known that most topology optimization problems are not convex and may have several different local optima. At the same time, most global optimization methods seem to be unable to handle problems of the size of a typical topology optimization problem (Bendsøe and Sigmund 2003). Therefore it is unwarranted to completely dismiss the merit of any optimization method just because it may sometimes produce a local optimum.
Once the essence of the problem is identified (i.e. the optimization algorithm falls into a local optimum), the problem can be avoided by suppressing the local optimum outside the optimization algorithm. For example, Huang and Xie (2008, 2009) suggest checking the boundary and the loading conditions after each design iteration. Once the predefined boundary or loading condition is changed due to element elimination, the optimization program should not proceed any further until some corrective measures are put in place. One effective measure is to use a much finer mesh to discretize the initial design.
For the Zhou and Rozvany example (2001) with a 50 % volume fraction, an optimal solution shown in Figure 5.3(e) has been obtained by the hard-kill BESO method using a very fine mesh (Huang and Xie 2008). It has a mean compliance of 378.4, which is close to that of a simple beam-tie solution proposed by Zhou and Rozvany (2001) of which the mean compliance is 387.5. Huang and Xie (2008) also obtain an optimal solution using the ESO method with the same fine mesh. The ESO solution has a mean compliance of 380.5, which is slightly worse than the BESO result. It is noted that the technique of mesh refinement has been used to tackle this type of problem by several other researchers too (Edwards et al. 2007; Norato et al. 2007; Zhu et al. 2007; Liu et al. 2008).
5.5
Conclusion
This chapter has compared various aspects of the current ESO/BESO methods with those of the SIMP method and the continuation method.
Without a mesh-independency filter, the ESO method with an infinite penalty may produce a good solution when the optimization parameters are properly chosen. However, the ESO
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solution is highly dependent on the used parameters and may sometimes become unstable. In contrast, the BESO method with a finite penalty usually yields a more stable and better solution with an objective function very close to that of the continuation method.
With a mesh-independency filter, all four topology optimization methods considered in this chapter produce almost identical topologies. The values of the objective function from both soft-kill and hard-kill BESO algorithms are much lower than that of the SIMP method and very close to that of the continuation method. The main advantage of the present BESO method is that less iterations are required to obtain high quality designs. The hard-kill BESO method produces an almost identical solution to that of the soft-kill BESO method. The effect of the penalty exponent p becomes negligible once it exceeds a certain value. Therefore, in view of its high computational efficiency, the use of the hard-kill BESO method is highly recommended for the compliance minimization problems.
Like many other optimization methods, BESO cannot guarantee to obtain a global optimum. To avoid a highly inefficient local optimum, it is suggested to check the boundary and the load- ing conditions in each design iteration. Generally, if a breakdown of boundary support is de- tected during the optimization process, it may well indicate that the used mesh is too coarse and a finer mesh should be adopted for the initial design in order to achieve a satisfactory solution.
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