Interval and random analysis for structure–acoustic systems with large uncertain-but-bounded parameters

Abstract For the response analysis of the structure–acoustic system with uncertain-but-bounded parameters, three bounded uncertain models are introduced. One is the bounded random model in which all of the uncertain-but-bounded parameters are described as bounded random variables with well defined probability distribution. The second one is the interval model in which all of the uncertain-but-bounded parameters are described as interval variables due to the limited information. The third one is the bounded hybrid uncertain model in which the interval variables and the bounded random variables exist simultaneously. Based on the parametric Gegenbauer polynomial, which is formulated for bounded random model recently, the Gegenbauer Series Expansion Method (GSEM) is developed for the response prediction of the structure–acoustic system under these three bounded uncertain models. Within GSEM, the response of these three bounded uncertain models of the structure–acoustic system can be approximated by the unified Gegenbauer Series with different values of polynomial parameter. Then, the interval and random analysis for these three bounded uncertain models of the structure–acoustic system are conducted on the basis of Gegenbauer series. Owing to the orthogonal property of Gegenbauer polynomial, the analytical solution of the expectation and variance of Gegenbauer series with respect to the bounded random variables can be readily obtained. The bounds of Gegenbauer series with respect to the interval variables are determined by the Monte Carlo simulation. The GSEM is applied to solve a shell structure–acoustic system under these three bounded uncertain models. Inspired by the convergence behavior of GSEM, the relative improvement criterion is established to estimate the required retained order of Gegenbauer Series for large uncertain problems. The results on numerical examples show that GSEM with the estimated retained order can achieve a prescribed accuracy and good efficiency for structure–acoustic systems with large uncertain-but-bounded parameters.

[1]  Donald J. Nefske,et al.  Structural-acoustic finite element analysis of the automobile passenger compartment: A review of current practice , 1982 .

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

[3]  David Moens,et al.  A survey of non-probabilistic uncertainty treatment in finite element analysis , 2005 .

[4]  I. Elishakoff,et al.  Bounds of eigenvalues for structures with an interval description of uncertain-but-non-random parameters , 1996 .

[5]  G. Roeck,et al.  Improving interval analysis in finite element calculations by means of affine arithmetic , 2010 .

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

[7]  Dejie Yu,et al.  Hybrid uncertain analysis for structural–acoustic problem with random and interval parameters , 2013 .

[8]  Dejie Yu,et al.  Hybrid uncertain analysis of acoustic field with interval random parameters , 2013 .

[9]  A. Nouy.  Recent Developments in Spectral Stochastic Methods for the Numerical Solution of Stochastic Partial Differential Equations , 2009 .

[10]  Wei Gao.  Interval natural frequency and mode shape analysis for truss structures with interval parameters , 2006 .

[11]  Xiaoting Rui,et al.  Perturbation Finite Element Transfer Matrix Method for Random Eigenvalue Problems of Uncertain Structures , 2012 .

[12]  C. Jiang,et al.  Structural reliability analysis based on random distributions with interval parameters , 2011 .

[13]  Z. Qiu,et al.  The static displacement and the stress analysis of structures with bounded uncertainties using the vertex solution theorem , 2007 .

[14]  A. Sofi,et al.  Reliability analysis of structures with interval uncertainties under stationary stochastic excitations , 2016 .

[15]  Adrian Sandu,et al.  Modeling Multibody Systems with Uncertainties. Part I: Theoretical and Computational Aspects , 2006 .

[16]  A. Sofi,et al.  Stochastic analysis of structures with uncertain-but-bounded parameters via improved interval analysis , 2012 .

[17]  Adrian Sandu,et al.  Modeling multibody systems with uncertainties. Part II: Numerical applications , 2006 .

[18]  Liping Chen,et al.  A Chebyshev interval method for nonlinear dynamic systems under uncertainty , 2013 .

[19]  Z. Luo,et al.  A new uncertain analysis method and its application in vehicle dynamics , 2015 .

[20]  N. Wiener.  The Homogeneous Chaos , 1938 .

[21]  Xiaojun Wang,et al.  Uncertainty propagation in SEA for structural–acoustic coupled systems with non-deterministic parameters , 2014 .

[22]  Dejie Yu,et al.  Interval and subinterval perturbation methods for a structural-acoustic system with interval parameters , 2013 .

[23]  Dejie Yu,et al.  Modified Interval Perturbation Finite Element Method for a Structural-Acoustic System With Interval Parameters , 2013 .

[24]  I. Elishakoff,et al.  Combination of probabilistic and convex models of uncertainty when scarce knowledge is present on acoustic excitation parameters , 1993 .

[25]  Xiaolei Leng,et al.  Evolutionary earthquake response of uncertain structure with bounded random parameter , 2004 .

[26]  F. Tin-Loi,et al.  Hybrid probabilistic interval analysis of bar structures with uncertainty using a mixed perturbation Monte-Carlo method , 2011 .

[27]  Kok-Kwang Phoon,et al.  Some numerical experiences on convergence criteria for iterative finite element solvers , 2009 .

[28]  Dejie Yu,et al.  Dynamic load identification for stochastic structures based on Gegenbauer polynomial approximation and regularization method , 2015 .

[30]  M. K. Shoushtari,et al.  Preliminary scoping study of some neutronic aspects of new shim safety rods for a typical 5 MW research reactor by Monte Carlo simulation , 2009 .

[31]  W. Gao.  Interval Finite Element Analysis using Interval Factor Method , 2007 .

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

[33]  Liping Chen,et al.  Interval uncertain method for multibody mechanical systems using Chebyshev inclusion functions , 2013 .

[34]  Zhan Kang,et al.  Perturbation-based stochastic FE analysis and robust design of inelastic deformation processes , 2006 .

[35]  Dejie Yu,et al.  Modified sub-interval perturbation finite element method for 2D acoustic field prediction with large uncertain-but-bounded parameters , 2012 .

[36]  George Deodatis,et al.  A simple and efficient methodology to approximate a general non-Gaussian stationary stochastic vector process by a translation process with applications in wind velocity simulation , 2013 .

[37]  M. Hanss,et al.  Review: Non-probabilistic finite element analysis for parametric uncertainty treatment in applied mechanics: Recent advances , 2011 .

[38]  G. Alefeld,et al.  Interval analysis: theory and applications , 2000 .

[39]  Pol D. Spanos,et al.  A multiscale Monte Carlo finite element method for determining mechanical properties of polymer nanocomposites , 2008 .

[40]  Zhiping Qiu,et al.  The reliability analysis of probabilistic and interval hybrid structural system , 2010 .

[41]  Baizhan Xia,et al.  Hybrid uncertain analysis for the prediction of exterior acoustic field with interval and random parameters , 2014 .

[42]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[43]  Giuseppe Muscolino,et al.  RESPONSE STATISTICS OF LINEAR STRUCTURES WITH UNCERTAIN-BUT-BOUNDED PARAMETERS UNDER GAUSSIAN STOCHASTIC INPUT , 2011 .