Gradient-based structural optimization with geometric nonlinearity and dynamics via differentiable analysis

Gradient-based structural optimization provides an efficient approach for improving the performance of engineering structures, but its application to problems involving geometric nonlinearity and dynamic responses remains challenging. The main difficulty lies in the accurate and efficient evaluation of design sensitivities, because structural responses are implicitly governed by nonlinear equilibrium equations, incremental-iterative solution procedures, and time-integration schemes. This study presents a gradient-based structural optimization framework for size and shape design of structures exhibiting geometrically nonlinear static and dynamic behavior. The structural analysis procedure, including nonlinear internal force evaluation, equilibrium iteration, and dynamic time integration, is formulated as a differentiable computational process, enabling the sensitivities of response-based objective and constraint functions with respect to cross-sectional and geometric design variables to be obtained consistently. Automatic differentiation is employed as a convenient tool for sensitivity evaluation, thereby avoiding cumbersome manual derivations and reducing the numerical errors associated with finite-difference approximations. Based on the obtained sensitivities, a gradient-based optimization algorithm is used to iteratively update the structural design. Several numerical examples, including a slender truss column subjected to eccentric loading, a spatial truss structure under dynamic loading, and a large-span bridge structure with acceleration control, are investigated to demonstrate the effectiveness of the proposed framework. The results show that the proposed approach can efficiently improve structural stiffness, suppress large nonlinear deformation, and reduce dynamic responses under prescribed material usage. The framework provides a general and extensible route for gradient-based structural optimization under complex nonlinear static and dynamic conditions.