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.

拓扑优化是结构及其组件高效布局设计的有效方法。这种方法在固体力学领域中适用于线性弹性和小位移条件。然而，在大位移条件下弹性结构的拓扑优化已在文献中略有讨论。本研究提出了一种用于受几何非线性行为影响的平面结构拓扑优化分析的数值公式。该公式通过惩罚方法将位置有限元耦合到固体各向同性材料。使用了高阶位置有限元，可以实现对机械场评估的高精度。正如预期的那样，所提出的公式实现了文献中针对几何线性条件可用的基准响应。然而，考虑几何非线性条件的拓扑优化分析导致最终几何形状与线性条件下预测的几何形状大不相同。两个应用程序证明了所提出的数值方案的准确性，并强调了在实际工程设计中正确处理几何非线性效应的重要性。