Multilevel model reduction for uncertainty quantification in computational structural dynamics

This work deals with an extension of the reducedorder models (ROMs) that are classically constructed by modal analysis in linear structural dynamics for which the computational models are assumed to be uncertain. It is based on a multilevel projection strategy consisting in introducing three reduced-order bases that are obtained by using a spatial filtering methodology of local displacements. This filtering involves global shape functions for the kinetic energy. The proposed multilevel stochastic ROM is constructed by using the nonparametric probabilistic approach of uncertainties. It allows for affecting a specific level of uncertainties to each type of displacements associated with the corresponding vibration regime. The proposed methodology is applied to the computational model of an automobile structure, for which the multilevel stochastic ROM is identified with respect to experimental measurements. This identification is performed by solving a statistical inverse problem.

[1]  Jean-François Sigrist,et al.  Dynamic analysis of a tube bundle with fluid–structure interaction modelling using a homogenisation method , 2008 .

[2]  Habib N. Najm,et al.  Stochastic spectral methods for efficient Bayesian solution of inverse problems , 2005, J. Comput. Phys..

[3]  Christian Soize,et al.  Random uncertainties model in dynamic substructuring using a nonparametric probabilistic model , 2003 .

[4]  R. Lyon,et al.  Theory and Application of Statistical Energy Analysis , 2014 .

[5]  Christian Soize,et al.  Nonparametric stochastic modeling of linear systems with prescribed variance of several natural frequencies , 2008 .

[6]  Anas Batou,et al.  Global reduced-order model adapted to the low- and medium-frequency analysis of complex dynamical structures , 2015 .

[7]  Christian Soize,et al.  Structural partitioning of complex structures in the medium-frequency range. An application to an automotive vehicle , 2011 .

[8]  Christian Soize,et al.  Variational-Based Reduced-Order Model in Dynamic Substructuring of Coupled Structures Through a Dissipative Physical Interface: Recent Advances , 2014 .

[9]  Christian Soize,et al.  Réduction de modèle adaptée à la dynamique basse et moyenne fréquence des structures complexes , 2015 .

[10]  J. Beck,et al.  Updating Models and Their Uncertainties. I: Bayesian Statistical Framework , 1998 .

[11]  Stijn Donders,et al.  Subcomponent modelling of input parameters for statistical energy analysis by using a wave-based boundary condition , 2010 .

[12]  Christian Soize,et al.  Probabilistic approach for model and data uncertainties and its experimental identification in structural dynamics: Case of composite sandwich panels , 2006 .

[13]  J. L. Guyader,et al.  Modal sampling method for the vibration study of systems of high modal density , 1990 .

[14]  R. Craig.  A review of time-domain and frequency-domain component mode synthesis method , 1985 .

[15]  Damijan Markovic,et al.  Reduction of substructural interface degrees of freedom in flexibility‐based component mode synthesis , 2007 .

[16]  Louis Jezequel,et al.  HIGH-FREQUENCY RADIATION OF L-SHAPED PLATES BY A LOCAL ENERGY FLOW APPROACH , 2002 .

[17]  A. Sarkar,et al.  Mid-frequency structural dynamics with parameter uncertainty , 2001 .

[18]  Christian Soize,et al.  REDUCED-ORDER MODEL FOR THE DYNAMICAL ANALYSIS OF COMPLEX STRUCTURES WITH A HIGH MODAL DENSITY , 2014 .

[19]  Pierre Ladevèze,et al.  A new computational method for transient dynamics including the low‐ and the medium‐frequency ranges , 2005 .

[20]  R. Langley,et al.  A hybrid method for the vibration analysis of complex structural-acoustic systems , 1999 .

[21]  C. Farhat,et al.  The discontinuous enrichment method for elastic wave propagation in the medium‐frequency regime , 2006 .

[22]  Vincent Cotoni,et al.  A statistical energy analysis subsystem formulation using finite element and periodic structure theory , 2008 .

[23]  Geert Degrande,et al.  A non-parametric probabilistic model for ground-borne vibrations in buildings , 2006 .

[24]  Christian Soize,et al.  Reduced models in the medium frequency range for general dissipative structural-dynamics systems , 1998, European Journal of Mechanics - A/Solids.

[25]  Reuven Y. Rubinstein,et al.  Simulation and the Monte Carlo method , 1981, Wiley series in probability and mathematical statistics.

[26]  Klaus-Jürgen Bathe,et al.  On nonlinear dynamic analysis using substructuring and mode superposition , 1981 .

[27]  Yong-hwa Park,et al.  Partitioned Component Mode Synthesis via a Flexibility Approach , 2004 .

[28]  Christian Soize,et al.  Structural Acoustics and Vibration , 2001 .

[29]  Christian Soize,et al.  Structural-acoustic modeling of automotive vehicles in presence of uncertainties and experimental identification and validation. , 2008, The Journal of the Acoustical Society of America.

[30]  Christian Soize,et al.  Nonparametric stochastic modeling of structures with uncertain boundary conditions / coupling between substructures , 2013 .

[31]  Youngwon Hahn,et al.  Identification of global modeshape from a few nodal eigenvectors using simple free-form deformation , 2005, Engineering with Computers.

[32]  L. G. Jaeger,et al.  Dynamics of structures , 1990 .

[33]  P. Ladevèze,et al.  The variational theory of complex rays for the calculation of medium‐frequency vibrations , 2001 .

[34]  Leonard Meirovitch,et al.  Dynamics And Control Of Structures , 1990 .

[35]  Laurent Gagliardini,et al.  Reduced-Order Computational Model for Low-Frequency Dynamics of Automobiles , 2013 .

[36]  Daniel W. Apley,et al.  Improving Identifiability in Model Calibration Using Multiple Responses , 2012, DAC 2011.

[37]  Laurent Maxit,et al.  Extension of SEA model to subsystems with non-uniform modal energy distribution , 2003 .

[38]  D. Rixen,et al.  General Framework for Dynamic Substructuring: History, Review and Classification of Techniques , 2008 .

[39]  A. O'Hagan,et al.  Bayesian calibration of computer models , 2001 .

[40]  Christian Soize,et al.  Random vectors and random fields in high dimension. Parametric model-based representation, identification from data, and inverse problems , 2017 .

[41]  Christian Soize,et al.  Reduced-order computational model in nonlinear structural dynamics for structures having numerous local elastic modes in the low-frequency range. Application to fuel assemblies , 2013 .

[42]  HansenPer Christian.  The truncated SVD as a method for regularization , 1987 .

[43]  Charbel Farhat,et al.  On a component mode synthesis method and its application to incompatible substructures , 1994 .

[44]  P. Villon,et al.  Forced vibrations in the medium frequency range solved by a partition of unity method with local information , 2005 .

[45]  G. I. Schuëller,et al.  Developments in stochastic structural mechanics , 2006 .

[46]  Charbel Farhat,et al.  A nonparametric probabilistic approach for quantifying uncertainties in low‐dimensional and high‐dimensional nonlinear models , 2017 .

[47]  J. S. Przemieniecki.  Matrix Structural Analysis of Substructures , 1963 .

[48]  Zhaojun Bai,et al.  An Implementation and Evaluation of the AMLS Method for Sparse Eigenvalue Problems , 2008, TOMS.

[49]  Walter C. Hurty,et al.  Vibrations of Structural Systems by Component Mode Synthesis , 1960 .

[50]  Christian Soize,et al.  Stochastic reduced order computational model of structures having numerous local elastic modes in low frequency dynamics , 2013 .

[51]  Tiangang Cui,et al.  Data‐driven model reduction for the Bayesian solution of inverse problems , 2014, 1403.4290.

[52]  Christian Soize,et al.  Robust updating of uncertain damping models in structural dynamics for low- and medium-frequency ranges , 2008 .

[53]  Christian Soize,et al.  Random matrix models and nonparametric method for uncertainty quantification , 2016 .

[54]  Karen Willcox,et al.  Parameter and State Model Reduction for Large-Scale Statistical Inverse Problems , 2010, SIAM J. Sci. Comput..

[55]  O. L. Maître,et al.  Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics , 2010 .

[56]  D. Rixen,et al.  Generalized Methodology for Assembly and Reduction of Component Models for Dynamic Substructuring , 2011 .

[57]  R. Macneal.  A hybrid method of component mode synthesis , 1971 .

[58]  Melvin S. Anderson,et al.  Continuum Models for Beam- and Platelike Lattice Structures , 1978 .

[59]  L. Meirovitch,et al.  Rayleigh-Ritz Based Substructure Synthesis for Flexible Multibody Systems , 1991 .

[60]  K. Willcox,et al.  Interpolation among reduced‐order matrices to obtain parameterized models for design, optimization and probabilistic analysis , 2009 .

[61]  J A Sethian,et al.  A fast marching level set method for monotonically advancing fronts. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[62]  Christian Soize.  Medium frequency linear vibrations of anisotropic elastic structures , 1982 .

[63]  R. Langley,et al.  Vibro-acoustic analysis of complex systems , 2005 .

[64]  A. Le Bot.  ENERGY TRANSFER FOR HIGH FREQUENCIES IN BUILT-UP STRUCTURES , 2002 .

[65]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[66]  Christian Soize,et al.  Robust Design Optimization in Computational Mechanics , 2008 .

[67]  H. Morand.  A modal hybridization method for vibroacoustic studies at medium frequencies , 1992 .

[68]  James O. Berger,et al.  A Framework for Validation of Computer Models , 2007, Technometrics.

[69]  Laurent Gagliardini,et al.  Virtual SEA: Mid-Frequency Structure-Borne Noise Modeling Based on Finite Element Analysis , 2003 .

[70]  J A Sethian,et al.  Computing geodesic paths on manifolds. , 1998, Proceedings of the National Academy of Sciences of the United States of America.

[71]  M. Bampton,et al.  Coupling of substructures for dynamic analyses. , 1968 .

[72]  K. Bathe.  Finite Element Procedures , 1995 .

[73]  J. H. Argyris,et al.  The Analysis of Fuselages of Arbitrary Cross‐section and Taper: A DSIR Sponsored Research Programme on the Development and Application of the Matrix Force Method and the Digital Computer , 1961 .

[74]  Christian Soize,et al.  Dynamic Substructuring of Damped Structures Using Singular Value Decomposition , 1997 .

[75]  Antoine Legay,et al.  Performance of a restrained-interface substructuring FE model for reduction of structural-acoustic problems with poroelastic damping , 2011 .

[76]  David Moens,et al.  A fuzzy finite element procedure for the calculation of uncertain frequency-response functions of damped structures: Part 1—Procedure , 2005 .

[77]  Christian Soize,et al.  UNCERTAINTY QUANTIFICATION IN LOW-FREQUENCY DYNAMICS OF COMPLEX BEAM-LIKE STRUCTURES HAVING A HIGH-MODAL DENSITY , 2013, International Journal for Uncertainty Quantification.

[78]  Christian Soize,et al.  Post-buckling nonlinear static and dynamical analyses of uncertain cylindrical shells and experimental validation , 2014 .

[79]  P. A. Martin.  Advanced Computational Vibroacoustics: Reduced-Order Models and Uncertainty Quantification , 2015 .

[80]  Peter Göransson,et al.  A residue-based mode selection and sorting procedure for efficient poroelastic modeling in acoustic finite element applications. , 2013, The Journal of the Acoustical Society of America.

[81]  Mohamed Ichchou,et al.  A coupled BEM and energy flow method for mid-high frequency internal acoustic , 2010 .

[82]  A. L. Hale,et al.  On the Substructure Synthesis Method , 1981 .

[83]  Y. K. Cheung,et al.  Convergence Studies of Dynamic Analysis by Using the Finite Element Method With Lumped Mass Matrix , 1993 .

[84]  W. Hurty.  Dynamic Analysis of Structural Systems Using Component Modes , 1965 .

[85]  Earl H. Dowell,et al.  Parametric Random Vibration , 1985 .

[86]  Brian R. Mace,et al.  Uncertainty in structural dynamics , 2005 .

[87]  W. A. Benfield,et al.  Vibration Analysis of Structures by Component Mode Substitution , 1971 .

[88]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[89]  Andrew Y. T. Leung,et al.  Dynamic Stiffness and Substructures , 1993 .

[90]  Christian Soize,et al.  Multilevel reduced-order computational model in structural dynamics for the low- and medium-frequency ranges , 2015, Computers & Structures.

[91]  Charbel Farhat,et al.  A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime , 2003 .

[92]  Izhak Bucher,et al.  Left eigenvectors : Extraction from measurements and physical interpretation , 1997 .

[93]  R S Langley,et al.  On the diffuse field reciprocity relationship and vibrational energy variance in a random subsystem at high frequencies. , 2007, The Journal of the Acoustical Society of America.

[94]  B. Mace,et al.  A mode-based approach for the mid-frequency vibration analysis of coupled long- and short-wavelength structures , 2006 .

[95]  Vipin Kumar,et al.  A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs , 1998, SIAM J. Sci. Comput..

[96]  Mohamed Ichchou,et al.  PREDICTION OF VIBROACOUSTICS ENERGY USING A DISCRETIZED TRANSIENT LOCAL ENERGY APPROACH AND COMPARISON WITH TSEA , 2002 .

[97]  Christian Soize.  A model and numerical method in the medium frequency range for vibroacoustic predictions using the theory of structural fuzzy , 1992 .

[98]  Andrew J. Kurdila,et al.  『Fundamentals of Structural Dynamics』(私の一冊) , 2019, Journal of the Society of Mechanical Engineers.

[99]  Roger Ghanem,et al.  Reduced models for the medium-frequency dynamics of stochastic systems. , 2003, The Journal of the Acoustical Society of America.

[100]  G. Schuëller,et al.  Uncertain linear systems in dynamics: Retrospective and recent developments by stochastic approaches , 2009 .

[101]  Klaus-Jürgen Bathe,et al.  The subspace iteration method - Revisited , 2013 .

[102]  Richard B. Lehoucq,et al.  An Automated Multilevel Substructuring Method for Eigenspace Computation in Linear Elastodynamics , 2004, SIAM J. Sci. Comput..

[103]  Sara Casciati,et al.  Quantity vs. Quality in the Model Order Reduction (MOR) of a Linear System , 2014 .

[104]  Geert Lombaert,et al.  The Green’s functions of a vertically inhomogeneous soil with a random dynamic shear modulus , 2007 .

[105]  Vincent Cotoni,et al.  Response variance prediction in the statistical energy analysis of built-up systems. , 2004, The Journal of the Acoustical Society of America.

[106]  Houman Owhadi,et al.  Handbook of Uncertainty Quantification , 2017 .

[107]  Jean-Louis Guyader,et al.  Characterization and reduction of dynamic models of vibrating systems with high modal density , 2009 .

[108]  D. Rixen.  A dual Craig-Bampton method for dynamic substructuring , 2004 .

[109]  Christian Soize,et al.  Uncertain Dynamical Systems in the Medium-Frequency Range , 2003 .

[110]  B. Irons.  Structural eigenvalue problems - elimination of unwanted variables , 1965 .

[111]  S. Rubin.  Improved Component-Mode Representation for Structural Dynamic Analysis , 1975 .

[112]  Henry F. Inman,et al.  The overlapping coefficient as a measure of agreement between probability distributions and point estimation of the overlap of two normal densities , 1989 .

[113]  Noureddine Bouhaddi,et al.  A method for selecting master DOF in dynamic substructuring using the Guyan condensation method , 1992 .

[114]  Ted Belytschko,et al.  Flexural wave propagation behavior of lumped mass approximations , 1980 .

[115]  J. Z. Zhu,et al.  The finite element method , 1977 .

[116]  M. Jensen,et al.  HIGH CONVERGENCE ORDER FINITE ELEMENTS WITH LUMPED MASS MATRIX , 1996 .

[117]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[118]  E. Jaynes.  Information Theory and Statistical Mechanics , 1957 .

[119]  Christian Soize,et al.  Stochastic reduced order models for uncertain geometrically nonlinear dynamical systems , 2008, Computer Methods in Applied Mechanics and Engineering.

[120]  Daniel Kressner,et al.  Block variants of Hammarling's method for solving Lyapunov equations , 2008, TOMS.

[121]  Christian Soize,et al.  Random field representations for stochastic elliptic boundary value problems and statistical inverse problems , 2013, European Journal of Applied Mathematics.

[122]  Christian Soize,et al.  Stochastic reduced-order model in low-frequency dynamics in presence of numerous local elastic modes , 2011 .

[123]  Christian Soize.  A nonparametric model of random uncertainties for reduced matrix models in structural dynamics , 2000 .

[124]  Christian Soize,et al.  Multilevel stochastic reduced-order model in linear structural dynamics for complex structures , 2016 .

[125]  David Galbally,et al.  Non‐linear model reduction for uncertainty quantification in large‐scale inverse problems , 2009 .

[126]  Christian Soize.  Stochastic modeling of uncertainties in computational structural dynamics—Recent theoretical advances , 2013 .

[127]  J. Planchard,et al.  Vibrations of nuclear fuel assemblies: A simplified model , 1985 .

[128]  R. Guyan.  Reduction of stiffness and mass matrices , 1965 .

[129]  Karen Willcox,et al.  Parametric reduced-order models for probabilistic analysis of unsteady aerodynamic applications , 2007 .

[130]  Christian Soize,et al.  Uncertainty quantification in computational linear structural dynamics for viscoelastic composite structures , 2016 .

[131]  Roger G. Ghanem,et al.  Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure , 2005, SIAM J. Sci. Comput..