Comparison of Reduction Methods for Finite Element Geometrically Nonlinear Beam Structures

Abstract

The aim of this contribution is to present numerical comparisons of model-order reduction methods for geometrically nonlinear structures in the general framework of finite element (FE) procedures. Three different methods are compared: the implicit condensation and expansion (ICE), the quadratic manifold computed from modal derivatives (MD), and the direct normal form (DNF) procedure, the latter expressing the reduced dynamics in an invariant-based span of the phase space. The methods are first presented in order to underline their common points and differences, highlighting in particular that ICE and MD use reduction subspaces that are not invariant. A simple analytical example is then used in order to analyze how the different treatments of quadratic nonlinearities by the three methods can affect the predictions. Finally, three beam examples are used to emphasize the ability of the methods to handle curvature (on a curved beam), 1:1 internal resonance (on a clamped-clamped beam with two polarizations), and inertia nonlinearity (on a cantilever beam).

Model order reduction based on direct normal form: application to large finite element MEMS structures featuring internal resonance

“…Furthermore, the method does not rely on the slow-fast assumption between master and slave modes, since the velocity is directly accounted by the parametrisation procedure. This is a major achievement since it offers a uniformly valid and simulation-free method that could be blindly applied to any structure without the need of extra assumptions [48][49][50][51], for the same computational cost of other methods. Normal form theory makes the distinction between resonant and nonresonant couplings.…”

“…ICE and MD assume the manifold to be velocity independent, assumption which is the more fulfilled, the larger the slow/fast separation between the slave and master coordinates [47][48][49]. In [48], it is estimated that a ratio between eigenfrequencies of the slave modes and those of the master modes of at least three ensures the correctness of MDs in the prediction of the hardening/softening behaviour [49][50][51]. The inclusion of velocity dependence in the MD approach have been proposed in [52] to overcome this limitation, and leads to similar formulations than those reported in [53].…”

“…A real formalism was used so that all along the calculation, the results are expressed in the form of oscillator-like equations. Damping and forcing were taken into account and illustrative examples on blades and beams were reported in [50,53]. At the same time, the computation based on SSM led by the group of Haller also proposed a direct computation that has been written in the open code SSM tool 2.0 [54].…”

A higher-order parametric nonlinear reduced-order model for imperfect structures using Neumann expansion

“…3, FOM and ROM backbones start departing for normalized amplitudes greater than 0.8, ultimately leading to a 1Hz difference at an amplitude of 1.2.) Interestingly, the aforementioned empiric rules used to select VMs find theoretical confirmation in [17,60], where it is argued that a slow-fast decomposition assumption has to be made for the MD-based quadratic manifold approach to work, indicating a threshold ratio of 4 between the linear eigenfrequencies (i.e., ω p /ω s ≥ 4, p = s). However, to the best of the authors' knowledge, there is no guarantee that this limit remains valid also in the MD-based linear manifold approach used in this work (i.e., where MDs are appended to the RB and additional independent reduced coordinates are introduced).…”

Low-Computational-Cost Technique for Modeling Macro Fiber Composite Piezoelectric Actuators Using Finite Element Method

“…Hence, the authors used the high-performance reduced FE square method to overcome the huge computational effort and evaluated their work in [ 19 ] for an industrial multiscale model. Many research works utilize the reduced-order model (ROM) technique to enhance the computational costs in different FE applications, some of which can be found in the literature [ 20 , 21 , 22 , 23 ].…”