Topology optimization for initial post-buckling structural response: an application of Koiter asymptotic theory

Topology optimization enables the design of lightweight structures that achieve desired buckling performance. Linearized buckling analysis has been used to predict buckling for topology optimized structures, but it does not offer information about post-buckling performance. Koiter asymptotic theory predicts the initial post-buckling behavior without the computational cost or robustness challenges associated with computing the full nonlinear response. In this work, an asymptotic analysis is formulated and subsequently incorporated for continuum-based topology optimization problems. The derived asymptotic expansion coefficients are used to compute physically relevant load-generalized coordinate curve slopes. Results from three numerical examples demonstrate that accounting for the initial post-buckling regime within the topology optimization procedure yields designs that exhibit more gradual declines in their load-deflection curves. This is achieved by tailoring the designs to avoid localized buckling in individual internal support structures, which is associated with unfavorable structural collapse. The impact of incorporating pre-buckling nonlinearity is also studied, and it is found that these optimized results possess higher critical buckling load factors than conventional buckling load factor maximization. Ultimately, accounting for the initial post-buckling response with the asymptotic analysis provides the topology optimization procedure with additional information to identify designs that have more favorable responses after buckling.