Structural Optimization Method 3.1 Introduction
3.9 Mesh-independence Studies
The filter scheme presented in Section 3.3.2 is based on an image-processing technique and works as a low-pass filter that eliminates structural components below a certain length-scale in the optimal design. One of the benefits of adopting such a filter scheme is that the optimal topology will no longer be dependent on the mesh sizes. This will be demonstrated below using two examples.
First the short cantilever example shown in Figure 3.5 is optimized using four different mesh sizes: 32× 20, 80 × 50, 160 × 100, 240 × 150, with the same filter radius rmin= 3 mm. Figure 3.21 gives the optimal topologies obtained from the different mesh sizes. It clearly shows that despite the significant differences in the mesh sizes the optimal topologies remain largely the same. The only difference in the topologies is that the boundary of the structure becomes smoother with mesh refinement.
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36 Evolutionary Topology Optimization of Continuum Structures
Figure 3.20 Computational time for each iteration.
(a) (b)
(c) (d)
Figure 3.21 Mesh-independent solutions for the cantilever from different mesh sizes: (a) 32× 20;
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Bi-directional Evolutionary Structural Optimization Method 37
(a) (b)
(c) (d)
Figure 3.22 Mesh-independent solutions for the beam from different mesh sizes: (a) 60 × 20;
(b) 90× 30; (c) 120 × 40; (d) 150 × 50.
Next, we solve the beam optimization problem with four different mesh sizes: 60 × 20, 90× 30, 120 × 40, 150 × 50. The optimal topologies are shown in Figure 3.22. Again, mesh-independent designs are obtained by adopting the proposed filter scheme for BESO.
3.10
Conclusion
We have presented in this chapter a new BESO method for stiffness optimization. The sensitiv- ity numbers of elements are based on the elemental strain energy density. The sensitivity num- bers used for material removal and addition are modified by introducing a mesh-independency filter which smoothes the sensitivity numbers throughout the design domain. Also, the con- vergence histories of the mean compliance and the structural topology are greatly improved by averaging the sensitivity numbers with their historical information.
Examples have shown the capability of the present BESO method to obtain convergent, checkerboard-free and mesh-independent solutions. The resulted topologies compare very well with those of the SIMP method. Furthermore, the BESO method may start from guess designs that are much smaller than the full design domain and may therefore save computational time for the finite element analysis.
The current BESO method is often called ‘hard-kill’ BESO method due to the complete removal of an element (as opposed to changing it into a very soft material). The main advantage of the hard-kill approach is that the computational time can be significantly reduced, especially for large 3D structures, because the eliminated elements are not involved in the finite element analysis. However, there have been some doubts among researchers about the theoretical correctness of the hard-kill ESO/BESO methods, especially after Zhou and Rozvany (2001) has showed that ESO fails on a certain problem. To gain a deeper understanding of the BESO method, we shall discuss an artificial material interpolation scheme for BESO in the next chapter and then present a comparison between BESO and various other topology optimization methods in Chapter 5.
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38 Evolutionary Topology Optimization of Continuum Structures
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