Adjoint sensitivity analysis and optimization of transient problems using the mixed Lagrangian formalism as a time integration scheme

In optimization of transient problems, a robust, stable, and efficient numerical scheme for time integration is of much importance. Recently, the mixed Lagrangian formalism (MLF) has been proposed for the time integration of transient problems. MLF leads to an optimization problem for the computation of the state variables in each time step. It has shown a robust behavior, even in the presence of sharp gradients of the state variables in time. It has also been applied to a large variety of transient problems, including structural dynamics, multi-physics, and coupled problems, where it has shown its stability, robustness, and computational efficiency. Albeit the clear advantages of MLF, due to its nature, sensitivity analysis for responses of interest is challenging. However, adopting MLF within a first-order optimization framework while efficiently deriving its sensitivities will result in an efficient computational framework for the optimization of transient problems, while avoiding convergence issues in the time integration. This is done here by first reformulating the time integration scheme so as to enable working with more convenient functions. Then, for the sake of sensitivity analysis only, KKT conditions are formulated to replace the optimization problem of MLF in each time step. Finally, the sensitivity analysis is performed based on these KKT conditions. The sensitivity analysis is utilized here for the optimization of the dynamic response of a structure with tension-only yielding elements using viscous dampers.