Hybrid proper orthogonal decomposition formulation for linear structural dynamics

Hybrid proper orthogonal decomposition (PODh) formulation is a POD-based reduced-order modeling method where the continuous equation of the physical system is projected on the POD modes obtained from a discrete model of the system. The aim of this paper is to evaluate the hybrid POD formulation and to compare it with other POD formulations on the simple case of a linear elastic rod subject to prescribed displacements in the perspective of building reduced-order models for coupled fluid–structure systems in the future. In the first part of the paper, the hybrid POD is compared to two other formulations for the response to an initial condition: an approach based on the discrete finite elements equation of the rod called the discrete POD (PODd), and an analytical approach using the exact solution of the problem and consequently called the analytical POD (PODa). This first step is useful to ensure that the PODh performs well with respect to the other formulations. The PODh is therefore used afterwards for the forced motion response where a displacement is imposed at the free end of the rod. The main contribution of this paper lies in the comparison of three techniques used to take into account the non-homogeneous Dirichlet boundary condition with the hybrid POD: the first method relies on control functions, the second on the penalty method and the third on Lagrange multipliers. Finally, the robustness of the hybrid POD is investigated on two examples involving firstly the introduction of structural damping and secondly a nonlinear force applied at the free end of the rod.