Preserving Lagrangian Structure in Nonlinear Model Reduction with Application to Structural Dynamics
Abstract
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.
- Authors:
-
- Sandia National Lab. (SNL-CA), Livermore, CA (United States)
- Publication Date:
- Research Org.:
- Sandia National Lab. (SNL-CA), Livermore, CA (United States)
- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)
- OSTI Identifier:
- 1140523
- Report Number(s):
- SAND2014-0933J
Journal ID: ISSN 1064-8275; 498849; TRN: US1601161
- Grant/Contract Number:
- AC04-94AL85000
- Resource Type:
- Accepted Manuscript
- Journal Name:
- SIAM Journal on Scientific Computing
- Additional Journal Information:
- Journal Volume: 37; Journal Issue: 2; Journal ID: ISSN 1064-8275
- Publisher:
- SIAM
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 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
Citation Formats
Carlberg, Kevin, Tuminaro, Ray, and Boggs, Paul. Preserving Lagrangian Structure in Nonlinear Model Reduction with Application to Structural Dynamics. United States: N. p., 2015.
Web. doi:10.1137/140959602.
Carlberg, Kevin, Tuminaro, Ray, & Boggs, Paul. Preserving Lagrangian Structure in Nonlinear Model Reduction with Application to Structural Dynamics. United States. https://doi.org/10.1137/140959602
Carlberg, Kevin, Tuminaro, Ray, and Boggs, Paul. Wed .
"Preserving Lagrangian Structure in Nonlinear Model Reduction with Application to Structural Dynamics". United States. https://doi.org/10.1137/140959602. https://www.osti.gov/servlets/purl/1140523.
@article{osti_1140523,
title = {Preserving Lagrangian Structure in Nonlinear Model Reduction with Application to Structural Dynamics},
author = {Carlberg, Kevin and Tuminaro, Ray and Boggs, Paul},
abstractNote = {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.},
doi = {10.1137/140959602},
journal = {SIAM Journal on Scientific Computing},
number = 2,
volume = 37,
place = {United States},
year = {Wed Mar 11 00:00:00 EDT 2015},
month = {Wed Mar 11 00:00:00 EDT 2015}
}
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