Optimization of Nonlinear Structures Using Evolutionary Structural Optimization (ESO)

Evolutionary structural optimization (ESO) is based on the simple concept of systematically removing inefficient material from the structure after each finite element analysis, so that the resulting design is gradually evolved to an optimum. The bidirectional evolutionary structural optimization (BESO) method is a new version of the ESO method in which simultaneously removing and adding elements is allowed. Due to the importance of nonlinear structural analysis, in this study the BESO approach is used for nonlinear analysis of structures. The problems nonlinearity is assumed for the geometry, for the material, and for both geometry and material. In the first example, the BESO is applied to maximize the stiffness of a cantilever beam with a time dependent loading. Next, the BESO is applied to optimize the stiffness of a plate with the material nonlinearity. The results show that the nonlinear analysis leads to a much stiffer design. In the third example, a cantilever beam with both material and geometry nonlinearity is considered. The beam is also to be optimized for stiffness. The optimized shapes are compared for linear and nonlinear analysis against the SIMP.
Furthermore, effectiveness of the ESO is proved by applying them to some shape optimization problems. The aim is to find the best fillet and notch shape so that it possesses a lower stress concentration factor. Design boundary has been set with some control points and optimization process is only applied to these points. First a square plate with a circular hole at its center is optimized for minimizing the stress concentration. The obtained results for linear and nonlinear analysis using ESO are compared with the results obtained using the biological growth method. Then, a square plate with a rhombus hole is optimized for stress concentration. It is concluded that using ESO, the maximum stress concentration around the boundary of the hole can be significantly decreased with linear analysis and the ESO is a powerful alternative for the biological growth method. The ESO method is finally used for shape optimization of geometrically different fillet for minimization the stress concentration. The material is assumed nonlinear while there is geometrical nonlinearity for loading. The results are compared with that of Wu who has used the fully stressed design criterion. The results show that using the ESO, the stress concentration factor is significantly redused and in this case it is reduced by 22%. In this way, the optimum shapes have completely uniform stress in the boundary of the fillet. The results show that the ESO has a superior capability for shape optimization of fillets of nonlinear structures and in this case the maximum stress is reduced by 7.7%.
Furthermore, effectiveness of the ESO is proved by applying them to some shape optimization problems. The aim is to find the best fillet and notch shape so that it possesses a lower stress concentration factor. Design boundary has been set with some control points and optimization process is only applied to these points. First a square plate with a circular hole at its center is optimized for minimizing the stress concentration. The obtained results for linear and nonlinear analysis using ESO are compared with the results obtained using the biological growth method. Then, a square plate with a rhombus hole is optimized for stress concentration. It is concluded that using ESO, the maximum stress concentration around the boundary of the hole can be significantly decreased with linear analysis and the ESO is a powerful alternative for the biological growth method. The ESO method is finally used for shape optimization of geometrically different fillet for minimization the stress concentration. The material is assumed nonlinear while there is geometrical nonlinearity for loading. The results are compared with that of Wu who has used the fully stressed design criterion. The results show that using the ESO, the stress concentration factor is significantly redused and in this case it is reduced by 22%. In this way, the optimum shapes have completely uniform stress in the boundary of the fillet. The results show that the ESO has a superior capability for shape optimization of fillets of nonlinear structures and in this case the maximum stress is reduced by 7.7%.