Design Space Exploration and Optimization of Structures under Consideration of Buckling Conditions

A look into the literature of the last five years shows a growing interest in the consideration of buckling in the optimization of structures. Buckling is a stability problem. The load factors are determined by solving a generalized eigenvalue problem with the underlying structural stiffness matrix and the geometric stiffness matrix. Similar to structural dynamics, multiple eigenvalues may already exist or arise during optimization due to symmetries. The inherently associated numerical difficulties in calculating the sensitivities increase the complexity of the task compared to standard weight or compliance optimization. In addition, the order of the buckling shapes may change during the optimization. Therefore, a mode tracking method is used to assess the correlation of the eigenmodes. The eigenmodes of the previous optimization step may be used as starting vectors for the iterative eigensolver. In this paper, first an example of a shape optimization is presented and contrasted with a sampling analysis. Afterwards, a topology optimization using buckling constraints is conducted. Load factors can be used in the objective function or as constraints. Additional result variables not used as design constraints in the optimization can also be exported. All simulations will be performed with the commercial FEA software PERMAS using examples from the literature.