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

Journal Article · · SIAM Journal on Scientific Computing
DOI:https://doi.org/10.1137/140959602· OSTI ID:1140523
 [1];  [1];  [1]
  1. Sandia National Lab. (SNL-CA), Livermore, CA (United States)
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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.

Research Organization:
Sandia National Lab. (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; TRN: US1601161
Journal Information:
SIAM Journal on Scientific Computing, Vol. 37, Issue 2; ISSN 1064-8275
Publisher:
SIAMCopyright Statement
Country of Publication:
United States
Language:
English
Citation Metrics:
Cited by: 54 works
Citation information provided by
Web of Science

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Efficient structure-preserving model reduction for nonlinear mechanical systems with application to structural dynamics
  • Carlberg, Kevin; Tuminaro, Ray; Boggs, Paul
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Cited By (11)

Constrained sparse Galerkin regression journal January 2018
Structure-Preserving Model-Reduction of Dissipative Hamiltonian Systems journal February 2018
Sparse reduced-order modelling: sensor-based dynamics to full-state estimation journal April 2018
Modal Analysis of Fluid Flows: Applications and Outlook journal March 2020
A Reduced Generalized Multiscale Basis Method for Parametrized Groundwater Flow Problems in Heterogeneous Porous Media journal March 2019
Hyper-reduced order models for parametrized unsteady Navier-Stokes equations on domains with variable shape journal November 2019
Physically constrained data‐driven correction for reduced‐order modeling of fluid flows journal October 2018
High Performance Reduced Order Modeling Techniques Based on Optimal Energy Quadrature: Application to Geometrically Non-linear Multiscale Inelastic Material Modeling journal February 2018
Sparse reduced-order modeling : Sensor-based dynamics to full-state estimation text January 2017
Modal Analysis of Fluid Flows: Applications and Outlook preprint January 2019
Symplectic Model Reduction of Hamiltonian Systems journal January 2016

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Related Subjects

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