Research Papers

Hyper-Reduction Over Nonlinear Manifolds for Large Nonlinear Mechanical Systems

[+] 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

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 4, 2018; final manuscript received March 29, 2019; published online May 15, 2019. Assoc. Editor: Zdravko Terze.

J. Comput. Nonlinear Dynam 14(8), 081008 (May 15, 2019) (11 pages) Paper No: CND-18-1538; doi: 10.1115/1.4043450 History: Received December 04, 2018; Revised March 29, 2019

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.

Copyright © 2019 by ASME
Topics: Manifolds
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Chaturantabut, S. , and Sorensen, D. C. , 2010, “ Nonlinear Model Reduction Via Discrete Empirical Interpolation,” SIAM J. Sci. Comput., 32(5), pp. 2737–2764. [CrossRef]
Tiso, P. , and Rixen, D. J. , 2013, “ Discrete Empirical Interpolation Method for Finite Element Structural Dynamics,” Topics in Nonlinear Dynamics, Volume 1, G. Kerschen, D. Adams, and A. Carrella, eds., Vol. 35, Springer, New York.
Chaturantabut, S. , Beattie, C. , and Gugercin, S. , 2016, “ Structure-Preserving Model Reduction for Nonlinear Port-Hamiltonian Systems,” SIAM J. Sci. Comput., 38(5), pp. B837–B865.
Barrault, M. , Maday, Y. , Nguyen, N. C. , and Patera, A. T. , 2004, “ An ‘Empirical Interpolation' Method: Application to Efficient Reduced-Basis Discretization of Partial Differential Equations,” C. R. Math. Acad. Sci. Paris, 339(9), pp. 667–672. [CrossRef]
Farhat, C. , Avery, P. , Chapman, T. , and Cortial, J. , 2014, “ Dimensional Reduction of Nonlinear Finite Element Dynamic Models With Finite Rotations and Energy-Based Mesh Sampling and Weighting for Computational Efficiency,” Int. J. Numer. Methods Eng., 98(9), pp. 625–662. [CrossRef]
Matthies, H. G. , and Meyer, M. , 2003, “ Nonlinear Galerkin Methods for the Model Reduction of Nonlinear Dynamical Systems,” Comput. Struct., 81(9), pp. 1277–1286. [CrossRef]
Farhat, C. , Chapman, T. , and Avery, P. , 2015, “ Structure-Preserving, Stability, and Accuracy Properties of the Energy-Conserving Sampling and Weighting Method for the Hyper Reduction of Nonlinear Finite Element Dynamic Models,” Int. J. Numer. Methods Eng., 102(5), pp. 1077–1110. [CrossRef]
Jain, S. , Tiso, P. , and Haller, G. , 2018, “ Exact Nonlinear Model Reduction for a von Kármán Beam: Slow-Fast Decomposition and Spectral Submanifolds,” J. Sound Vib., 423, pp. 195–211. [CrossRef]
Daniel Millán, D. , and Arroyo, M. , 2013, “ Nonlinear Manifold Learning for Model Reduction in Finite Elastodynamics,” Comput. Methods Appl. Mech. Eng., 261–262, pp. 118–131.
Peharz, R. , and Pernkopf, F. , 2012, “ Sparse Nonnegative Matrix Factorization With ℓ0-Constraints,” Neurocomputing, 80, pp. 38–46. [CrossRef] [PubMed]
Jain, S. , Tiso, P. , Rixen, D. J. , and Rutzmoser, J. B. , 2017, “ A Quadratic Manifold for Model Order Reduction of Nonlinear Structural Dynamics,” Comput. Struct., 188, pp. 80–94. [CrossRef]
Rutzmoser, J. B. , Rixen, D. J. , Tiso, P. , and Jain, S. , 2017, “ Generalization of Quadratic Manifolds for Reduced Order Modeling of Nonlinear Structural Dynamics,” Comput. Struct., 192, pp. 196–209. [CrossRef]
Lines, L. R. , and Treitel, S. , 1984, “ A Review of Least-Squares Inversion and Its Application to Geophysical Problems,” Geophys. Prospect., 32(2), pp. 159–186. [CrossRef]
van der Maaten, L. J. P., Postma, E. O., and van den Herik, H. J., 2009, “Dimensionality Reduction: A Comparative Review,” Tilburg University, Tilburg, The Netherlands, Report No. TiCC-TR 2009–005.
Ryckelynck, D. , 2005, “ A Priori Hyperreduction Method: An Adaptive Approach,” J. Comput. Phys., 201(1), pp. 346–366. [CrossRef]
Jain, S. , 2015, “ Model Order Reduction for Non-Linear Structural Dynamics,” Master Thesis, Delft University of Technology, Delft, The Netherlands.
Jain, S. , and Tiso, P. , 2018, “ Simulation-Free Hyper-Reduction for Geometrically Nonlinear Structural Dynamics: A Quadratic Manifold Lifting Approach,” ASME J. Comput. Nonlinear Dyn., 13(7), p. 071003. [CrossRef]
Mignolet, M. P. , Przekop, A. , Rizzi, S. A. , and Spottswood, S. M. , 2013, “ A Review of Indirect/Non-Intrusive Reduced Order Modeling of Nonlinear Geometric Structures,” J Sound Vib, 332(10), pp. 2437–2460. [CrossRef]
Barbič, J. , and James, D. L. , 2005, “ Real-Time Subspace Integration for St. Venant-Kirchhoff Deformable Models,” ACM Trans. Graph., 24(3), pp. 982–990. [CrossRef]
Everson, R. , and Sirovich, L. , 1995, “ Karhunen-Loeve Procedure for Gappy Data,” J. Opt. Soc. Am. A, 12(8), pp. 1657–1664. [CrossRef]
Haller, G. , and Ponsioen, S. , 2017, “ Exact Model Reduction by a Slow? Fast Decomposition of Nonlinear Mechanical Systems,” Nonlinear Dyn., 90(1), pp. 617–647. [CrossRef]
Carlberg, K. , Bou-Mosleh, C. , and Farhat, C. , 2011, “ Efficient Non-Linear Model Reduction Via a Least-Squares Petrov? Galerkin Projection and Compressive Tensor Approximations,” Int. J. Numer. Meth. Eng., 86(2), pp. 155–181. [CrossRef]
Carlberg, K. , Farhat, C. , Cortial, J. , and Amsallem, D. , 2013, “ The GNAT Method for Nonlinear Model Reduction: Effective Implementation and Application to Computational Fluid Dynamics and Turbulent Flows,” J. Comput. Phys., 242, pp. 623–647. [CrossRef]
Chapman, T. , Avery, P. , Collins, P. , and Farhat, C. , 2017, “ Accelerated Mesh Sampling for the Hyper Reduction of Nonlinear Computational Models,” Int. J. Numer. Meth. Eng., 109(12), pp. 1623–1654. [CrossRef]
Rega, G. , and Troger, H. , 2005, “ Dimension Reduction of Dynamical Systems: Methods, Models, Applications,” Nonlinear Dyn., 41(1–3), pp. 1–15. [CrossRef]
Ding, Q. , and Zhang, K. , 2012, “ Order Reduction and Nonlinear Behaviors of a Continuous Rotor System,” Nonlinear Dyn., 67(1), pp. 251–262. [CrossRef]
Lall, S. , Krysl, P. , and Marsden, J. , 2003, “ Structure-Preserving Model Reduction for Mechanical Systems,” Phys. D, 184(1–4), pp. 304–318. [CrossRef]
Brüls, O. , Duysinx, P. , and Golinval, J.-C. , 2007, “ The Global Modal Parameterization for Non-Linear Model-Order Reduction in Flexible Multibody Dynamics,” Int. J. Numer. Methods Eng., 69(5), pp. 948–977. [CrossRef]
Heirman, G. H. K. , Naets, F. , and Desmet, W. , 2011, “ A System-Level Model Reduction Technique for the Efficient Simulation of Flexible Multibody Systems,” Int. J. Numer. Methods Eng., 85(3), pp. 330–354. [CrossRef]
Aarts, R. G. K. M. , and Jonker, J. B. , 2002, “ Dynamic Simulation of Planar Flexible Link Manipulators Using Adaptive Modal Integration,” Multibody Syst. Dyn., 7(1), pp. 31–50. [CrossRef]
Rutzmoser, J. B. , and Rixen, D. J. , 2017, “ A Lean and Efficient Snapshot Generation Technique for the Hyper-Reduction of Nonlinear Structural Dynamics,” Comput. Methods Appl. Mech. Eng., 325, pp. 330–349. [CrossRef]
Géradin, M. , and Rixen, D. , 2015, Mechanical Vibrations: Theory and Application to Structural Dynamics, 3rd ed., Wiley, Chichester, UK.


Grahic Jump Location
Fig. 1

Model—I: (a) flat plate example—simply supported on sides A and D. The plate is L =40 mm long, H =20 mm wide, t =0.8 mm thick. The Young Modulus is E =70 GPa, the Poisson's ratio is ν = 0.33, and the density is ρ = 2700 Kg/m3. A uniform pressure is applied on the plate, according to the time history p(t)=P sin(ωt), where P =1 N/mm2 and ω = 2.097 × 104 rad/s. (b) The curved plate is discretized using flat, triangular shell elements. The FE mesh contains 400 elements and 1386 DOFs. The two opposite sides parallel to the y-axis are simply supported, a uniform pressure is applied on the curved surface, according to the time history.

Grahic Jump Location
Fig. 3

Model—II: (a) 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. (43); (b) The structure is meshed with triangular flat shell elements with 6DOFs per node and each with a thickness of 1.5 mm. The mesh contains n =135,770 DOFs, ne = 49,968 elements. For illustration purposes, the skin panels are removed, and the mesh is shown.

Grahic Jump Location
Fig. 2

The linear solution (dashed-blue) grows quickly to high amplitudes for the choice of resonant loading; the full solution (thick black) shows limited amplitudes due to the presence of geometrical nonlinearities; the ROM obtained using quadratic manifold (yellow) and hyper-reduced solution for this ROM using EECSW (dashed-red) capture the full solution accurately. Subfigure (b) highlights the inaccuracy of the POD ROM with m =2.

Grahic Jump Location
Fig. 4

(a)–(c): Comparison of full, linear, and various reduced responses at tip node two in x, y, and z directions. Note that the QM-based reduced solution (with five reduced unknowns) accurately captures the full system response, whereas the POD-based ROM with five modes fails to reproduce the nonlinear system response (POD-5).



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