A progressive method for shape, size, and topology optimization of trusses based on the bi-level approach
摘要
This paper presents a progressive method for shape, size, and topology optimization of truss structures based on the bi-level approach and plastic design method. In the truss simultaneous shape and size optimization problem, researchers commonly optimize two types of variables in a unified design space, which often results in structurally complex and statically indeterminate solutions due to the nonlinearity and nonconvexity of the optimization. However, practical design considerations and the compatibility condition-neglected in plastic design-require that the resulting structures be as simple and statically determinate as possible. The proposed progressive method iteratively performs bi-level optimization-where shape variables are handled at the upper level and member sizes at the lower level-followed by the topology optimization. At the lower level of the bi-level optimization, the size problem can be formulated as a linear programming problem, thereby promoting structural simplicity and static determinacy. In the bi-level framework, a linear penalty factor is introduced to restrict the buckling effects of compressive members. In the topology optimization, the structures are rationalized by member filtering, node merging, etc., and further simplified by the evolutionary member deletion process. The topology change may introduce equilibrium infeasibility in the lower-level problem, especially in three-dimensional problems. To address this, this study introduces the least squares methods and the infeasibility penalty to the bi-level framework, and also introduces a procedure of pre-check for equilibrium feasibility to the member deletion process. A comparative study using the Hemp cantilever example demonstrates the superiority of the bi-level approach over the simultaneous one in terms of computational efficiency, structural simplicity, and static determinacy. Several two- and three-dimensional examples further illustrate the effectiveness of the proposed progressive method.