An effective reduction technique is presented for flexible multibody systems, for which the elastic deflection could not be considered small. We consider here the planar beam systems undergoing large elastic rotations, in the floating frame description. The proposed method enriches the classical linear reduction basis with modal derivatives stemming from the derivative of the eigenvalue problem. Furthermore, the Craig–Bampton method is applied to couple the different reduced components. Based on the linear projection, the configuration-dependent internal force can be expressed as cubic polynomials in the reduced coordinates. Coefficients of these polynomials can be precomputed for efficient runtime evaluation. The numerical results show that the modal derivatives are essential for the correct approximation of the nonlinear elastic deflection with respect to the body reference. The proposed reduction method constitutes a natural and effective extension of the classical linear modal reduction in the floating frame.

This paper presents a novel model order reduction technique for 3D flexible multibody systems featuring nonlinear elastic behavior. We adopt the mean-axis floating frame approach in combination with an enhanced Rubin substructuring technique for the construction of the reduction basis. The standard Rubin reduction basis is augmented with the modal derivatives of both free-interface vibration modes and attachment modes to consider the bending–stretching coupling effects for each flexible body. The mean-axis frame generally yields relative displacements and rotations of smaller magnitude when compared to the one obtained by the nodal-fixed floating frame. This positively impacts the accuracy of the reduction basis. Also, when equipped with modal derivatives, the Rubin method better considers the geometric nonlinearities than the Craig–Bampton method, as it comprises vibration modes and modal derivatives featuring free motion of the interface. The nonlinear coupling between free-interface modes and attachment modes is also considered. Numerical tests confirm that the proposed method is more accurate than Craig–Bampton’s, a nodal fixed floating frame counterpart originally proposed in Wu and Tiso (Multibody Syst. Dyn. 36(4): 405–425, [2016]), and produces significant speed-ups. However, the offline cost is increased because the mean-axis formulation produces operators with decreased sparsity patterns.

We present an efficient method to significantly reduce the offline cost associated with the construction of training sets for hyper-reduction of geometrically nonlinear, finite element (FE)-discretized structural dynamics problems. The reduced-order model is obtained by projecting the governing equation onto a basis formed by vibration modes (VMs) and corresponding modal derivatives (MDs), thus avoiding cumbersome manual selection of high-frequency modes to represent nonlinear coupling effects. Cost-effective hyper-reduction is then achieved by lifting inexpensive linear modal transient analysis to a quadratic manifold (QM), constructed with dominant modes and related MDs. The training forces are then computed from the thus-obtained representative displacement sets. In this manner, the need of full simulations required by traditional, proper orthogonal decomposition (POD)-based projection and training is completely avoided. In addition to significantly reducing the offline cost, this technique selects a smaller hyper-reduced mesh as compared to POD-based training and therefore leads to larger online speedups, as well. The proposed method constitutes a solid alternative to direct methods for the construction of the reduced-order model, which suffer from either high intrusiveness into the FE code or expensive offline nonlinear evaluations for the determination of the nonlinear coefficients.

We present an enhanced version of the parametric nonlinear reduced-order model for shape imperfections in structural dynamics we studied in a previous work. In this model, the total displacement is split between the one due to the presence of a shape defect and the one due to the motion of the structure. This allows to expand the two fields independently using different bases. The defected geometry is described by some user-defined displacement fields which can be embedded in the strain formulation. This way, a polynomial function of both the defect field and actual displacement field provides the nonlinear internal elastic forces. The latter can be thus expressed using tensors, and owning the reduction in size of the model given by a Galerkin projection, high simulation speedups can be achieved. We show that the adopted deformation framework, exploiting Neumann expansion in the definition of the strains, leads to better accuracy as compared to the previous work. Two numerical examples of a clamped beam and a MEMS gyroscope finally demonstrate the benefits of the method in terms of speed and increased accuracy.

Common trends in model reduction of large nonlinear finite element (FE)-discretized systems involve Galerkin projection of the governing equations onto a low-dimensional linear subspace. Though this reduces the number of unknowns in the system, the computational cost for obtaining the reduced solution could still be high due to the prohibitive computational costs involved in the evaluation of nonlinear terms. Hyper-reduction methods are then used for fast approximation of these nonlinear terms. In the finite element context, the energy conserving sampling and weighing (ECSW) method has emerged as an effective tool for hyper-reduction of Galerkin-projection-based reduced-order models (ROMs). More recent trends in model reduction involve the use of nonlinear manifolds, which involves projection onto the tangent space of the manifold. While there are many methods to identify such nonlinear manifolds, hyper-reduction techniques to accelerate computation in such ROMs are rare. In this work, we propose an extension to ECSW to allow for hyper-reduction using nonlinear mappings, while retaining its desirable stability and structure-preserving properties. As a proof of concept, the proposed hyper-reduction technique is demonstrated over models of a flat plate and a realistic wing structure, whose dynamics have been shown to evolve over a nonlinear (quadratic) manifold. An online speed-up of over one thousand times relative to the full system has been obtained for the wing structure using the proposed method, which is higher than its linear counterpart using the ECSW.