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Research Papers

Simulation-Free Hyper-Reduction for Geometrically Nonlinear Structural Dynamics: A Quadratic Manifold Lifting Approach

[+] Author and Article Information
Shobhit Jain

Institute for Mechanical Systems,
ETH Zürich,
Leonhardstrasse 21,
Zürich 8092, Switzerland
e-mail: shjain@ethz.ch

Paolo Tiso

Institute for Mechanical Systems,
ETH Zürich,
Leonhardstrasse 21,
Zürich 8092, Switzerland
e-mail: ptiso@ethz.ch

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received October 19, 2017; final manuscript received April 10, 2018; published online May 28, 2018. Assoc. Editor: Mohammad Younis.

J. Comput. Nonlinear Dynam 13(7), 071003 (May 28, 2018) (12 pages) Paper No: CND-17-1462; doi: 10.1115/1.4040021 History: Received October 19, 2017; Revised April 10, 2018

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.

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Figures

Grahic Jump Location
Fig. 1

A schematic of the QM-based training set generation technique. Snapshots of the modal superposition solution, which lies in the linear modal subspace, are “lifted” on to the QM using Eq. (17).

Grahic Jump Location
Fig. 2

Model-I: (a) schematic of a slightly curved plate. The plate is L = 40 mm long, H = 20 mm wide, t = 0.8 mm thick, and curved with a radius R = 200.5 mm and height w = 1 mm. The Young modulus is E = 70 GPa, the Poisson's ratio is ν=0.33, and the density is ρ = 2700 Kg/mm3. (b) The curved plate is discretized using flat, triangular shell elements. The FE mesh contains 400 elements and 1386 DOFs. The two opposite sidesparallel to the y-axis are simply supported, a uniform pressure is applied on the curved surface, according to the time history p(t)=P[sin(ωt)+sin(πωt)], where P = 0.2 N/mm2 and ω=2.965×104 rad/s.

Grahic Jump Location
Fig. 3

The full nonlinear solution in comparison with the linearized solution shows that the applied loading invokes nonlinear behavior. The out-of-plane displacement (in the z direction) at the middle node of the mesh in Fig. 2 is shown.

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Fig. 4

Classical (hyper-)reduction: (a) the POD-1, POD-2, ECSW-POD responses and (b) the corresponding elements and their weights selected by the sNNLS Algorithm 1

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Fig. 5

Simulation-free (hyper-)reduction: (a) and (c) show that the modal-based reduction techniques L1 and L2, and their modal-based hyper-reduced counterparts are able to accurately capture the nonlinear response. (b) and (d) A small fraction of the total number elements was selected for hyper-reduction by the sNNLS algorithm, resulting in high online speed-ups as reported in Table 1.

Grahic Jump Location
Fig. 6

Model-II: a wing structure with NACA 0012 airfoil (length (L) = 5 m, width (W) = 0.9 m, height (H) = 0.1 m) stiffened with ribs along the longitudinal and lateral direction. The Young modulus is E = 70 GPa, the Poisson's ratio is ν=0.33, and the density is ρ = 2700 Kg/m3. The wing is cantilevered at one end. Uniform pressure is applied on the highlighted area, with a pulse load as given by Eq. (20) (shown in (a)). The structure is meshed with triangular flat shell elements with 6 DOFs per node and each with a thickness of 1.5 mm. The mesh contains n = 135770 DOFs, ne = 49968 elements. For illustration purposes, the skin panels are removed and the mesh is shown in (b).

Grahic Jump Location
Fig. 7

The comparison of the norm of the linear and the nonlinear internal force during a full nonlinear solution for model-II (cf., Fig. 6) showing that geometric nonlinearities are moderately excited by the applied loading

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