Polynomial chaos expansion in structural dynamics: Accelerating the convergence of the first two statistical moment sequences

Abstract Polynomial chaos solution for the frequency response of linear non-proportionally damped dynamic systems has been considered. It has been observed that for lightly damped systems the convergence of the solution can be very poor in the vicinity of the deterministic resonance frequencies. To address this, Aitken׳s transformation and its generalizations are suggested. The proposed approach is successfully applied to the sequences defined by the first two moments of the responses, and this process significantly accelerates the polynomial chaos convergence. In particular, a 2-dof system with respectively 1 and 2 parameter uncertainties has been studied. The first two moments of the frequency response were calculated by Monte Carlo simulation, polynomial chaos expansion and Aitken׳s transformation of the polynomial chaos expansion. Whereas 200 polynomials are required to have a good agreement with Monte Carlo results around the deterministic eigenfrequencies, less than 50 polynomials transformed by the Aitken׳s method are enough. This latter result is improved if a generalization of Aitken׳s method (recursive Aitken׳s transformation, Shank׳s transformation) is applied. With the proposed convergence acceleration, polynomial chaos may be reconsidered as an efficient method to estimate the first two moments of a random dynamic response.

[1]  Huajiang Ouyang,et al.  Statistics of complex eigenvalues in friction-induced vibration , 2015 .

[2]  Jeremy E. Oakley,et al.  Bayesian Analysis of Computer Model Outputs , 2002 .

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

[4]  Sondipon Adhikari,et al.  Dynamic analysis of stochastic structural systems using frequency adaptive spectral functions , 2015 .

[5]  Sondipon Adhikari,et al.  Polynomial Chaos Expansion and Steady-State Response of a Class of Random Dynamical Systems , 2015 .

[6]  G. Schuëller,et al.  On advanced Monte Carlo simulation procedures in stochastic structural dynamics , 1997 .

[7]  George Em Karniadakis,et al.  Adaptive multi-element polynomial chaos with discrete measure , 2015 .

[8]  Sondipon Adhikari,et al.  Stochastic free vibration analysis of angle-ply composite plates – A RS-HDMR approach , 2015 .

[9]  H. Rabitz,et al.  General foundations of high‐dimensional model representations , 1999 .

[10]  森山 昌彦,et al.  「確率有限要素法」(Stochastic Finite Element Method) , 1985 .

[11]  Mircea Grigoriu,et al.  Convergence properties of polynomial chaos approximations for L2 random variables. , 2007 .

[12]  Humberto Contreras,et al.  The stochastic finite-element method , 1980 .

[13]  Mircea Grigoriu,et al.  On the accuracy of the polynomial chaos approximation for random variables and stationary stochastic processes. , 2003 .

[14]  Bruno Sudret,et al.  Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach , 2008 .

[15]  G. Stefanou.  The stochastic finite element method: Past, present and future , 2009 .

[16]  Ilya M. Sobol,et al.  Theorems and examples on high dimensional model representation , 2003, Reliab. Eng. Syst. Saf..

[17]  Peter Graves-Morris,et al.  Padé Approximants Second Edition: Contents , 1996 .

[18]  Alex H. Barbat,et al.  Monte Carlo techniques in computational stochastic mechanics , 1998 .

[19]  G. Schuëller,et al.  Uncertainty analysis of complex structural systems , 2009 .

[20]  S. Adhikari,et al.  Transient Response of Structural Dynamic Systems with Parametric Uncertainty , 2014 .

[21]  Manolis Papadrakakis,et al.  Robust and efficient methods for stochastic finite element analysis using Monte Carlo simulation , 1996 .

[22]  Y. J. Ren,et al.  Stochastic FEM based on local averages of random vector fields , 1992 .

[23]  Claude Brezinski,et al.  Extrapolation algorithms and Pade´ approximations: a historical survey , 1996 .

[24]  Masanobu Shinozuka,et al.  Neumann Expansion for Stochastic Finite Element Analysis , 1988 .

[25]  Roger Ghanem,et al.  Convergence acceleration of polynomial chaos solutions via sequence transformation , 2014 .

[26]  Claude Brezinski,et al.  Convergence acceleration during the 20th century , 2000 .

[27]  Marc C. Kennedy,et al.  Case studies in Gaussian process modelling of computer codes , 2006, Reliab. Eng. Syst. Saf..

[28]  Joe Wiart,et al.  A new surrogate modeling technique combining Kriging and polynomial chaos expansions - Application to uncertainty analysis in computational dosimetry , 2015, J. Comput. Phys..

[29]  Sondipon Adhikari,et al.  Stochastic structural dynamic analysis using Bayesian emulators , 2013 .

[30]  J. Sinou,et al.  Influence of Polynomial Chaos expansion order on an uncertain asymmetric rotor system response , 2015 .

[31]  Bruno Sudret,et al.  Adaptive sparse polynomial chaos expansion based on least angle regression , 2011, J. Comput. Phys..

[32]  A. OHagan,et al.  Bayesian analysis of computer code outputs: A tutorial , 2006, Reliab. Eng. Syst. Saf..