Bidirectional evolutionary structural optimization algorithm for multi-grade materials

The classical Bidirectional Evolutionary Structural Optimization (BESO) algorithm always obtains local optimal solution only, due to boundary problems occurring in the optimization process since it uses an optimization policy of life or death for a single material. In order to overcome this problem, a BESO algorithm for multi-grade materials was achieved by introducing linearly interpolated materials with multiple elastic modulus grades and using the evaluation index of material utilization constructed based on variation coefficient to determine the upgrade and downgrade of the elements ’ properties between different material grades. Then, through numerical examples of a deep beam, the optimization ability between the new algorithm and the classical BESO algorithm was compared, and the influence of quantities of material grades on optimization process and results was investigated as well. The results show that the solutions of the classical BESO algorithm for single material, and the BESO algorithm for bi-grade materials, tri-grade materials and tetragrade materials separately have compliance index values of 0.818×104 N∙mm, 0.785×104 N∙mm, 0.775×104 N∙mm and 0.768×104 N∙mm, as well as the time consumption is 4 min, 8 min, 12 min and 18 min, respectively, when the examples were all optimized to a converted volume rate of 0.3. Moreover, the variation coefficients of material 2 and material 3, which accounts for high proportions in the solution of the BESO algorithm for tri-grade materials, are 0.11 and 0.26 respectively, while that of single material in the solution of the classical BESO algorithm is 4.88, when the examples were all optimized to an absolute volume rate of 0.3. Consequently, compared with the classical BESO algorithm, the BESO algorithm for multi-grade materials can obtain solutions with lower compliance and more in line with the optimization goal, that are, better solutions; meanwhile, its evolved topology of bar-system structures are clearer and have a higher material utilization. Besides, the more grades of materials that the optimization is based on, the more the stress mechanism of the components described by the material distribution of different grades, and the more stress details of the topologies could the new algorithm obtain as well, but it will also increase the time consumption for optimization. Therefore, the optimization accuracy and efficiency should be well balanced to select the appropriate number of material grades in practical application. © 2022, Central South University Press. All rights reserved.