Response sensitivity analysis of frame structures using finite elements based on three-field mixed formulation

Finite element response sensitivities represent an essential ingredient for gradient-based optimization methods used to solve problems in structural optimization, structural reliability analysis, structural identification and finite element model updating [1]. Finite element response sensitivities are also invaluable for gaining deeper insight into the effects and relative importance of the various geometric, material, and loading parameters defining the structure and its loading environment. Sensitivity analysis formulations have been developed for displacement-based finite element models [2] and, recently, for force-based frame elements [3]. The advantages gained in response analysis by using finite element formulations more advanced than the classical displacementbased formulation can be further extended to the realm of response sensitivity analysis. A large body of research has been devoted to mixed finite element formulations since they were first introduced in the pioneering work of Pian [4]. Several finite elements based on different variational principles have been developed and their accuracy and performance have been thoroughly analyzed and improved. Nowadays, mixed finite elements are well established and largely adopted tools in a wide range of structural mechanics applications. Multi-field mixed finite element formulations were proposed, among others, for finite elements widely used in the structural engineering community such as frame elements. Mixed frame elements are more accurate in nonlinear analysis than displacement-based elements and are a possible alternative to the recently established force-based elements. This paper focuses on the formulation of finite element response sensitivity analysis, using the Direct Differentiation Method (DDM) [2, 3], in the case of a nonlinear three-field mixed approach derived from the Hu-Washizu variational principle [5], considering both quasi-static and dynamic loadings. The general formulation for finite element response sensitivity analysis using the three-field mixed formulation is specialized and applied to frame finite element models. The results of the DDM are validated through the forward Finite Difference Method (FDM) using as application example a realistic steel-concrete composite frame structure subjected to quasistatic and dynamic loading, respectively. Both monolithic frame elements and composite frame elements with deformable shear connection [6] based on the three-field mixed formulation are included in this application example.