Efficient nonlinear reduced order modeling for dynamic analysis of flat structures

In the present paper, strategies for reduced order modeling of geometrically nonlinear finite element models are investigated. Simulation-free, non-intrusive approaches are considered, which do not require access to the source code of a finite element program (e.g., proprietary knowledge). Our study focus on but is not restricted to flat structures. Reduction bases are generated using bending modes and the associated modal derivatives, which span the additional subspace needed for an adequate approximation of the geometrically nonlinear response. Moreover, the reduced nonlinear restoring forces are expressed as third order polynomials in modal coordinates. Consequently, the reduced systems can be effectively solved using time -integration schemes involving only the reduced coordinates. A bottleneck in the non-intrusive methods is typically the computational effort for precomputing the polynomial coefficients and generating the reduction basis. In this regard, we demonstrate that modal derivatives have several useful properties. In particular, the modal derivatives essentially provide all the information needed for generating the polynomial coefficients for the in-plane coordinates. For condensed systems, which ignores the inertia of the in-plane modes, we show that the modal derivatives can be used effectively for recovering the in-plane displacements. Based on these findings, we propose a methodology for generating reduced order models of geometrically nonlinear flat structures in a computationally efficient manner. Moreover, we demonstrate that the concepts extend also to curved structures. The modeling techniques are validated by means of numerical examples of solid beam models and continuously supported shell models. The computational efficiency of the proposed methodology is evaluated based on the number of static evaluations needed for identifying the polynomial coefficients, as compared to the state-of-the-art methods. Furthermore, strategies for efficient time integration are discussed and evaluated.