Mixed Lagrangian Response and Sensitivity Analysis for Optimization of Elasto-Plastic Structures with Nonlinear Viscous Dampers

Nonlinear fluid viscous dampers are highly effective devices for passive control of structural systems. Their reliable analysis and design optimization require robust computational tools capable of accurately evaluating the nonlinear dynamic structural response. A robust time integration approach based on the mixed Lagrangian formalism (MLF) is presented for dynamic response analysis and optimization. MLF has demonstrated strong robustness in nonlinear transient analysis, but its use in gradient-based optimization of structures equipped with nonlinear fluid viscous dampers has been largely unexplored. Structures exhibiting elasto-plastic behavior with kinematic hardening and equipped with nonlinear fluid viscous dampers are considered. The dampers are modeled through a Maxwell representation combined with a fractional power-law constitutive relation. The proposed approach addresses the numerical challenges posed by nonlinear velocity-dependent damper forces, which are characterized by fractional exponents and rate-sensitive dissipation mechanisms. The proposed framework is assessed through numerical examples involving seismic retrofit with nonlinear fluid viscous dampers, including a three-dimensional irregular frame structure with a double setback. The results show that the proposed MLF-based approach provides accurate and stable response predictions and efficient optimized designs. As a direct sensitivity-analysis approach, the present formulation is computationally convenient when the number of design variables is less than or equal to the number of response functions to be differentiated. Moreover, unlike an adjoint sensitivity approach, it can be carried out concurrently with the forward response analysis, without requiring a subsequent backward-in-time solution.