Topology optimization is an effective approach for the efficient layout design of structures and their components. This approach is well-established in the solid mechanics' domain for linear elastic and small-displacement conditions. However, the topology optimization of elastic structures under large-displacement conditions has been marginally addressed in the literature. This study proposes a numerical formulation for the topology optimization analysis of plane structures subjected to geometrically nonlinear behavior. This formulation couples positional finite elements to the solid isotropic material with penalization method. High order positional finite elements have been utilized, which enable high accuracy on the mechanical fields' assessment. The proposed formulation achieves the benchmark responses available in the literature for geometric linear conditions, as expected. Nevertheless, the topology optimization analysis accounting for geometric nonlinear conditions leads to final geometries largely different from those predicted in linear conditions. Two applications demonstrate the accuracy of the proposed numerical scheme and emphasize the importance of handling properly the geometric nonlinear effects into real engineering design.
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