Preserving Lagrangian Structure in Nonlinear Model Reduction with Application to Structural Dynamics

Carlberg, Kevin; Tuminaro, Ray; Boggs, Paul

doi:10.1137/140959602

Preserving Lagrangian Structure in Nonlinear Model Reduction with Application to Structural Dynamics

Journal Article · Wed Mar 11 00:00:00 EDT 2015 · SIAM Journal on Scientific Computing

DOI:https://doi.org/10.1137/140959602· OSTI ID:1140523

Carlberg, Kevin ^[1]; Tuminaro, Ray ^[1]; Boggs, Paul ^[1]

Sandia National Lab. (SNL-CA), Livermore, CA (United States)

Our work proposes a model-reduction methodology that preserves Lagrangian structure and achieves computational efficiency in the presence of high-order nonlinearities and arbitrary parameter dependence. As such, the resulting reduced-order model retains key properties such as energy conservation and symplectic time-evolution maps. We focus on parameterized simple mechanical systems subjected to Rayleigh damping and external forces, and consider an application to nonlinear structural dynamics. To preserve structure, the method first approximates the system's “Lagrangian ingredients''---the Riemannian metric, the potential-energy function, the dissipation function, and the external force---and subsequently derives reduced-order equations of motion by applying the (forced) Euler--Lagrange equation with these quantities. Moreover, from the algebraic perspective, key contributions include two efficient techniques for approximating parameterized reduced matrices while preserving symmetry and positive definiteness: matrix gappy proper orthogonal decomposition and reduced-basis sparsification. Our results for a parameterized truss-structure problem demonstrate the practical importance of preserving Lagrangian structure and illustrate the proposed method's merits: it reduces computation time while maintaining high accuracy and stability, in contrast to existing nonlinear model-reduction techniques that do not preserve structure.

View Accepted Manuscript (DOE)

Research Organization:: Sandia National Laboratories (SNL-CA), Livermore, CA (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: AC04-94AL85000

OSTI ID:: 1140523

Report Number(s):: SAND2014--0933J; 498849

Journal Information:: SIAM Journal on Scientific Computing, Journal Name: SIAM Journal on Scientific Computing Journal Issue: 2 Vol. 37; ISSN 1064-8275

Publisher:: SIAMCopyright Statement

Country of Publication:: United States

Language:: English

References (17)

Non-linear model reduction for uncertainty quantification in large-scale inverse problems Galbally, D.; Fidkowski, K.; Willcox, K. International Journal for Numerical Methods in Engineering https://doi.org/10.1002/nme.2746	journal	January 2009
Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods Prud’homme, C.; Rovas, D. V.; Veroy, K. Journal of Fluids Engineering, Vol. 124, Issue 1 https://doi.org/10.1115/1.1448332	journal	November 2001
Nonlinear Model Reduction via Discrete Empirical Interpolation Chaturantabut, Saifon; Sorensen, Danny C. SIAM Journal on Scientific Computing, Vol. 32, Issue 5 https://doi.org/10.1137/090766498	journal	January 2010
Efficient structure-preserving model reduction for nonlinear mechanical systems with application to structural dynamics Carlberg, Kevin; Tuminaro, Ray; Boggs, Paul 53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference<BR>20th AIAA/ASME/AHS Adaptive Structures Conference<BR>14th AIAA https://doi.org/10.2514/6.2012-1969	conference	April 2012
Efficient non-linear model reduction via a least-squares Petrov-Galerkin projection and compressive tensor approximations Carlberg, Kevin; Bou-Mosleh, Charbel; Farhat, Charbel International Journal for Numerical Methods in Engineering, Vol. 86, Issue 2 https://doi.org/10.1002/nme.3050	journal	October 2010
Dimensional reduction of nonlinear finite element dynamic models with finite rotations and energy-based mesh sampling and weighting for computational efficiency: NONLINEAR MODEL REDUCTION AND ENERGY-CONSERVING SAMPLING AND WEIGHTING Farhat, Charbel; Avery, Philip; Chapman, Todd International Journal for Numerical Methods in Engineering, Vol. 98, Issue 9 https://doi.org/10.1002/nme.4668	journal	April 2014
Structure-preserving model reduction for mechanical systems Lall, Sanjay; Krysl, Petr; Marsden, Jerrold E. Physica D: Nonlinear Phenomena, Vol. 184, Issue 1-4 https://doi.org/10.1016/S0167-2789(03)00227-6	journal	October 2003
An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations Barrault, Maxime; Maday, Yvon; Nguyen, Ngoc Cuong Comptes Rendus Mathematique, Vol. 339, Issue 9 https://doi.org/10.1016/j.crma.2004.08.006	journal	November 2004
A priori hyperreduction method: an adaptive approach Ryckelynck, D. Journal of Computational Physics, Vol. 202, Issue 1 https://doi.org/10.1016/j.jcp.2004.07.015	journal	January 2005
The GNAT method for nonlinear model reduction: Effective implementation and application to computational fluid dynamics and turbulent flows Carlberg, Kevin; Farhat, Charbel; Cortial, Julien Journal of Computational Physics, Vol. 242 https://doi.org/10.1016/j.jcp.2013.02.028	journal	June 2013
Discrete mechanics and variational integrators Marsden, J. E.; West, M. Acta Numerica, Vol. 10 https://doi.org/10.1017/S096249290100006X	journal	May 2001
A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations Grepl, Martin A.; Patera, Anthony T. ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 39, Issue 1 https://doi.org/10.1051/m2an:2005006	journal	January 2005
Missing Point Estimation in Models Described by Proper Orthogonal Decomposition Astrid, Patricia; Weiland, Siep; Willcox, Karen IEEE Transactions on Automatic Control, Vol. 53, Issue 10 https://doi.org/10.1109/TAC.2008.2006102	journal	November 2008
Reduced Basis Approximation for Nonlinear Parametrized Evolution Equations based on Empirical Operator Interpolation Drohmann, Martin; Haasdonk, Bernard; Ohlberger, Mario SIAM Journal on Scientific Computing, Vol. 34, Issue 2 https://doi.org/10.1137/10081157X	journal	January 2012
Low-Rank Matrix Completion by Riemannian Optimization Vandereycken, Bart SIAM Journal on Optimization, Vol. 23, Issue 2 https://doi.org/10.1137/110845768	journal	January 2013
A Posteriori Error Estimation for DEIM Reduced Nonlinear Dynamical Systems Wirtz, D.; Sorensen, D. C.; Haasdonk, B. SIAM Journal on Scientific Computing, Vol. 36, Issue 2 https://doi.org/10.1137/120899042	journal	January 2014
Karhunen–Loève procedure for gappy data Everson, R.; Sirovich, L. Journal of the Optical Society of America A, Vol. 12, Issue 8 https://doi.org/10.1364/JOSAA.12.001657	journal	January 1995

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Symplectic Model Reduction of Hamiltonian Systems Peng, Liqian; Mohseni, Kamran SIAM Journal on Scientific Computing, Vol. 38, Issue 1 https://doi.org/10.1137/140978922	journal	January 2016
Physically constrained data‐driven correction for reduced‐order modeling of fluid flows Mohebujjaman, M.; Rebholz, L. G.; Iliescu, T. International Journal for Numerical Methods in Fluids, Vol. 89, Issue 3 https://doi.org/10.1002/fld.4684	journal	October 2018
Hyper-reduced order models for parametrized unsteady Navier-Stokes equations on domains with variable shape Santo, Niccolò Dal; Manzoni, Andrea Advances in Computational Mathematics, Vol. 45, Issue 5-6 https://doi.org/10.1007/s10444-019-09722-9	journal	November 2019
Structure-Preserving Model-Reduction of Dissipative Hamiltonian Systems Maboudi Afkham, Babak; Hesthaven, Jan S. Journal of Scientific Computing, Vol. 81, Issue 1 https://doi.org/10.1007/s10915-018-0653-6	journal	February 2018
High Performance Reduced Order Modeling Techniques Based on Optimal Energy Quadrature: Application to Geometrically Non-linear Multiscale Inelastic Material Modeling Caicedo, Manuel; Mroginski, Javier L.; Toro, Sebastian Archives of Computational Methods in Engineering, Vol. 26, Issue 4 https://doi.org/10.1007/s11831-018-9258-3	journal	February 2018
Constrained sparse Galerkin regression Loiseau, Jean-Christophe; Brunton, Steven L. Journal of Fluid Mechanics, Vol. 838 https://doi.org/10.1017/jfm.2017.823	journal	January 2018
Sparse reduced-order modelling: sensor-based dynamics to full-state estimation Loiseau, Jean-Christophe; Noack, Bernd R.; Brunton, Steven L. Journal of Fluid Mechanics, Vol. 844 https://doi.org/10.1017/jfm.2018.147	journal	April 2018
A Reduced Generalized Multiscale Basis Method for Parametrized Groundwater Flow Problems in Heterogeneous Porous Media He, Xinguang; Li, Qiuqi; Jiang, Lijian Water Resources Research, Vol. 55, Issue 3 https://doi.org/10.1029/2018wr023954	journal	March 2019
Modal Analysis of Fluid Flows: Applications and Outlook Taira, Kunihiko; Hemati, Maziar S.; Brunton, Steven L. AIAA Journal, Vol. 58, Issue 3 https://doi.org/10.2514/1.j058462	journal	March 2020
Sparse reduced-order modeling : Sensor-based dynamics to full-state estimation Loiseau, Jean-Christophe; Noack, Bernd R.; Brunton, Steven L. arXiv https://doi.org/10.48550/arxiv.1706.03531	text	January 2017
Modal Analysis of Fluid Flows: Applications and Outlook Taira, Kunihiko; Hemati, Maziar S.; Brunton, Steven L. arXiv https://doi.org/10.48550/arxiv.1903.05750	preprint	January 2019

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Related Subjects

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
97 MATHEMATICS AND COMPUTING
Hamiltonian dynamics
Lagrangian dynamics
matrix symmetry
nonlinear model reduction
positive definiteness
structural dynamics
structure preservation

Preserving Lagrangian Structure in Nonlinear Model Reduction with Application to Structural Dynamics

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References (17)

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