Persistent model order reduction for complex dynamical systems using smooth orthogonal decomposition

Abstract Full-scale complex dynamic models are not effective for parametric studies due to the inherent constraints on available computational power and storage resources. A persistent reduced order model (ROM) that is robust, stable, and provides high-fidelity simulations for a relatively wide range of parameters and operating conditions can provide a solution to this problem. The fidelity of a new framework for persistent model order reduction of large and complex dynamical systems is investigated. The framework is validated using several numerical examples including a large linear system and two complex nonlinear systems with material and geometrical nonlinearities. While the framework is used for identifying the robust subspaces obtained from both proper and smooth orthogonal decompositions (POD and SOD, respectively), the results show that SOD outperforms POD in terms of stability, accuracy, and robustness.

[1]  G. Kerschen,et al.  Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems , 2008 .

[2]  A. Antoulas,et al.  Chapter 3 Model Order Reduction — Methods , Concepts and Properties , 2015 .

[3]  Christophe Pierre,et al.  Non-linear normal modes and invariant manifolds , 1991 .

[4]  Ioannis T. Georgiou,et al.  Advanced Proper Orthogonal Decomposition Tools: Using Reduced Order Models to Identify Normal Modes of Vibration and Slow Invariant Manifolds in the Dynamics of Planar Nonlinear Rods , 2005 .

[5]  Wenliang Zhou,et al.  Smooth orthogonal decomposition-based vibration mode identification , 2006 .

[6]  Christophe Pierre,et al.  Modal Reduction of a Nonlinear Rotating Beam Through Nonlinear Normal Modes , 2002 .

[7]  Hankel-norm model reduction for delayed fuzzy systems , 2015, The 27th Chinese Control and Decision Conference (2015 CCDC).

[8]  Jerrold E. Marsden,et al.  Empirical model reduction of controlled nonlinear systems , 1999, IFAC Proceedings Volumes.

[9]  Peter Benner,et al.  Dimension Reduction of Large-Scale Systems , 2005 .

[10]  Alexander F. Vakakis,et al.  Nonlinear normal modes, Part I: A useful framework for the structural dynamicist , 2009 .

[11]  C. Pierre,et al.  A NEW GALERKIN-BASED APPROACH FOR ACCURATE NON-LINEAR NORMAL MODES THROUGH INVARIANT MANIFOLDS , 2002 .

[12]  David Chelidze.  Identifying Robust Subspaces for Dynamically Consistent Reduced-Order Models , 2014 .

[13]  Victor M. Calo,et al.  Fast multiscale reservoir simulations using POD-DEIM model reduction , 2015, ANSS 2015.

[14]  Joel R. Phillips,et al.  Projection-based approaches for model reduction of weakly nonlinear, time-varying systems , 2003, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[15]  J. Marsden,et al.  A subspace approach to balanced truncation for model reduction of nonlinear control systems , 2002 .

[16]  Alexander F. Vakakis,et al.  Nonlinear Targeted Energy Transfer in Granular Chains , 2012 .

[17]  Shahab Ilbeigi,et al.  Reduced Order Models for Systems with Disparate Spatial and Temporal Scales , 2016 .

[18]  Shahab Ilbeigi,et al.  Model Order Reduction of Nonlinear Euler-Bernoulli Beam , 2016 .

[19]  Troy R. Smith,et al.  Low-Dimensional Modelling of Turbulence Using the Proper Orthogonal Decomposition: A Tutorial , 2005 .

[20]  Pierre Ladevèze,et al.  On the verification of model reduction methods based on the proper generalized decomposition , 2011 .

[21]  E. Kreuzer,et al.  Nonlinear System Analysis with Karhunen–Loève Transform , 2005 .

[22]  Fabrice Thouverez,et al.  Computing multiple periodic solutions of nonlinear vibration problems using the harmonic balance method and Groebner bases , 2015 .

[23]  Charbel Farhat,et al.  Projection‐based model reduction for contact problems , 2015, 1503.01000.

[24]  H. Abarbanel,et al.  Determining embedding dimension for phase-space reconstruction using a geometrical construction. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[25]  F. Chinesta,et al.  A Short Review in Model Order Reduction Based on Proper Generalized Decomposition , 2018 .

[26]  Peter Benner,et al.  Model Order Reduction for Coupled Problems: Survey , 2015 .

[27]  Peter Benner,et al.  Two-Sided Projection Methods for Nonlinear Model Order Reduction , 2015, SIAM J. Sci. Comput..

[28]  Efficient model order reduction for dynamic systems with local nonlinearities , 2014 .

[29]  P. Beran,et al.  Reduced-order modeling: new approaches for computational physics , 2004 .

[30]  Xingyuan Wang.  Construction of frequency-energy plots for nonlinear dynamical systems from time-series data , 2010 .

[31]  Jan G. Korvink,et al.  Computationally efficient and stable order reduction method for a large-scale model of MEMS piezoelectric energy harvester , 2014, 2014 15th International Conference on Thermal, Mechanical and Mulit-Physics Simulation and Experiments in Microelectronics and Microsystems (EuroSimE).

[32]  M. P. Païdoussis,et al.  Reduced-order models for nonlinear vibrations of cylindrical shells via the proper orthogonal decomposition method , 2003 .

[33]  H. Hetzler,et al.  A nonlinear model order reduction approach to the elastohydrodynamic problem , 2015 .

[34]  Muruhan Rathinam,et al.  A New Look at Proper Orthogonal Decomposition , 2003, SIAM J. Numer. Anal..

[35]  Jean-François Mercier,et al.  On the numerical computation of nonlinear normal modes for reduced-order modelling of conservative vibratory systems , 2013 .

[36]  G. Kerschen,et al.  The Method of Proper Orthogonal Decomposition for Dynamical Characterization and Order Reduction of Mechanical Systems: An Overview , 2005 .

[37]  Pascal Reuss,et al.  Towards Finite Element Model Updating Based on Nonlinear Normal Modes , 2016 .

[38]  Francisco Chinesta,et al.  On the deterministic solution of multidimensional parametric models using the Proper Generalized Decomposition , 2010, Math. Comput. Simul..

[39]  D. Broomhead,et al.  Dimensionality Reduction Using Secant-Based Projection Methods: The Induced Dynamics in Projected Systems , 2005 .

[40]  F. Marques,et al.  IDENTIFICATION AND PREDICTION OF UNSTEADY TRANSONIC AERODYNAMIC LOADS BY MULTI-LAYER FUNCTIONALS , 2001 .

[41]  Peter Benner,et al.  Model Order Reduction for Linear and Nonlinear Systems: A System-Theoretic Perspective , 2014, Archives of Computational Methods in Engineering.

[42]  J. Peraire,et al.  Balanced Model Reduction via the Proper Orthogonal Decomposition , 2002 .

[43]  A. Nouy.  A priori model reduction through Proper Generalized Decomposition for solving time-dependent partial differential equations , 2010 .

[44]  Gaëtan Kerschen,et al.  On the exploitation of chaos to build reduced-order models , 2003 .

[45]  Luís Miguel Silveira,et al.  Guaranteed passive balancing transformations for model order reduction , 2003, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[46]  Matthew S. Allen,et al.  Evaluation of Geometrically Nonlinear Reduced-Order Models with Nonlinear Normal Modes , 2015 .

[47]  Francisco Chinesta,et al.  Recent Advances and New Challenges in the Use of the Proper Generalized Decomposition for Solving Multidimensional Models , 2010 .

[48]  A. Nouy,et al.  Model order reduction based on proper generalized decomposition for the propagation of uncertainties in structural dynamics , 2012 .

[49]  A. Wynn,et al.  A method for normal-mode-based model reduction in nonlinear dynamics of slender structures , 2015 .

[50]  Marissa Condon,et al.  Empirical Balanced Truncation of Nonlinear Systems , 2004, J. Nonlinear Sci..

[51]  Roland W. Freund,et al.  Efficient linear circuit analysis by Pade´ approximation via the Lanczos process , 1994, EURO-DAC '94.

[52]  G. Sell,et al.  On the computation of inertial manifolds , 1988 .

[53]  Alexander F. Vakakis,et al.  Normal modes and localization in nonlinear systems , 1996 .

[54]  K. Glover.  All optimal Hankel-norm approximations of linear multivariable systems and their L, ∞ -error bounds† , 1984 .

[55]  Z. Bai.  Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems , 2002 .