Differentiating the structural response within a mixed Lagrangian time integration framework

• Novel sensitivity analysis approach for Mixed Lagrangian Formalism • Response gradients derived via differentiation of KKT conditions • Efficient first-order optimization of nonlinear transient dynamic problems • Validated through optimization of a shear frame with dampers and elastoplastic braces Response optimization of transient problems requires time-integration schemes that guarantee stability and computational efficiency across large and highly nonlinear design spaces. The Mixed Lagrangian Formalism (MLF) meets this need by reformulating the computation of the state variables in each time step as an optimization problem. MLF has proven to be robust and stable for a diverse range of transient nonlinear response problems, including structural dynamics and coupled systems, particularly when sharp temporal gradients are present. With MLF, the state variables at each time step are obtained by solving an optimization problem, which makes the computation of design sensitivities for optimization a particularly challenging task. In this work, we propose a novel sensitivity analysis approach for MLF. The sensitivity analysis is carried out through direct differentiation of the Karush–Kuhn–Tucker (KKT) conditions at each time step, relying on the implicit function theorem applied to the necessary optimality conditions. The resulting framework enables efficient first-order optimization of complex transient problems while avoiding convergence issues in the response time-history analysis. As an application, we demonstrate the approach in the optimization of the dynamic response of a structure equipped with tension-only elasto-plastic elements and viscous dampers.