Differentiable nonlinear analysis for structural optimization with geometric and material nonlinearities

Truss optimization under nonlinear conditions remains challenging due to the complex coupling between geometric and material nonlinearities, as well as the difficulty in obtaining accurate and efficient sensitivities. In this work, a differentiable nonlinear analysis framework is proposed to enable gradient-based truss optimization while consistently accounting for both geometric and material nonlinearities. The framework incorporates hyperelastic and elastoplastic material models, where the latter is regularized via a smoothing strategy to ensure differentiability. To efficiently compute design sensitivities, a hybrid approach combining automatic differentiation and the adjoint method is developed. The consistent tangent stiffness matrix obtained from the forward nonlinear analysis is directly utilized, while automatic differentiation is selectively applied to evaluate derivatives with respect to design variables, including both element cross-sectional areas and nodal coordinates. This strategy preserves the sparsity and computational efficiency of classical finite element formulations while avoiding cumbersome analytical derivations. The proposed framework is integrated into a gradient-based optimization scheme for structural weight minimization subject to displacement and stress constraints. Numerical examples, including cases dominated by material nonlinearity, geometric nonlinearity, and their coupling, are presented to validate the accuracy of sensitivities and demonstrate the necessity of considering both effects. The proposed framework establishes a unified approach for differentiable truss optimization, enabling the systematic treatment of complex nonlinear behaviors within gradient-based design.