Fuzzy interval perturbation method for uncertain heat conduction problem with interval and fuzzy parameters

Summary This paper proposes a fuzzy interval perturbation method (FIPM) and a modified fuzzy interval perturbation method (MFIPM) for the hybrid uncertain temperature field prediction involving both interval and fuzzy parameters in material properties and boundary conditions. Interval variables are used to quantify the non-probabilistic uncertainty with limited information, whereas fuzzy variables are used to represent the uncertainty associated with the expert opinions. The level-cut method is introduced to decompose the fuzzy parameters into interval variables. FIPM approximates the interval matrix inverse by the first-order Neumann series, while MFIPM improves the accuracy by considering higher-order terms of the Neumann series. The membership functions of the interval temperature field are eventually derived using the fuzzy decomposition theorem. Three numerical examples are provided to demonstrate the feasibility and effectiveness of the proposed methods for solving heat conduction problems with hybrid uncertain parameters, pure interval parameters, and pure fuzzy parameters, respectively. Copyright © 2015 John Wiley & Sons, Ltd.

[1]  D. Xiu,et al.  A new stochastic approach to transient heat conduction modeling with uncertainty , 2003 .

[2]  Haitian Yang,et al.  A numerical method to estimate temperature intervals for transient convection–diffusion heat transfer problems , 2013 .

[3]  Fabio Nobile,et al.  A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data , 2010, SIAM Rev..

[4]  Dejie Yu,et al.  An interval random perturbation method for structural‐acoustic system with hybrid uncertain parameters , 2014 .

[5]  Snehashish Chakraverty,et al.  Numerical solution of interval and fuzzy system of linear equations , 2011 .

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

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

[8]  Xiaoping Du,et al.  Reliability sensitivity analysis with random and interval variables , 2009 .

[9]  Zdeněk Kala.  Fuzzy Sets Theory in Comparison with Stochastic Methods to Analyse Nonlinear Behaviour of a Steel Member under Compression , 2005 .

[10]  Di Wu,et al.  Numerical analysis of uncertain temperature field by stochastic finite difference method , 2014 .

[11]  R. Ghanem,et al.  An invariant subspace‐based approach to the random eigenvalue problem of systems with clustered spectrum , 2012 .

[12]  I. Elishakoff,et al.  Convex models of uncertainty in applied mechanics , 1990 .

[13]  Michael F. Modest,et al.  Backward Monte Carlo Simulations in Radiative Heat Transfer , 2003 .

[14]  Kankanhalli N. Seetharamu,et al.  Finite element modelling of flow, heat and mass transfer in fluid saturated porous media , 2002 .

[15]  Singiresu S. Rao,et al.  Analysis of uncertain structural systems using interval analysis , 1997 .

[16]  G. Muscolino,et al.  Interval analysis of structures with uncertain-but-bounded axial stiffness , 2011 .

[17]  M. Kaminski.  Probabilistic entropy in homogenization of the periodic fiber‐reinforced composites with random elastic parameters , 2012 .

[18]  I. Elishakoff,et al.  Antioptimization of structures with large uncertain-but-non-random parameters via interval analysis , 1998 .

[19]  Lotfi A. Zadeh,et al.  Fuzzy Sets , 1996, Inf. Control..

[20]  Zhiping Qiu,et al.  Fuzzy finite difference method for heat conduction analysis with uncertain parameters , 2014 .

[21]  David Moens,et al.  Recent advances in non-probabilistic approaches for non-deterministic dynamic finite element analysis , 2006 .

[22]  A. Neumaier.  Interval methods for systems of equations , 1990 .

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

[24]  Hong-Zhong Huang,et al.  Perturbation finite element method of structural analysis under fuzzy environments , 2005, Eng. Appl. Artif. Intell..

[25]  B. Blackwell,et al.  A technique for uncertainty analysis for inverse heat conduction problems , 2010 .

[26]  Michał Kleiber,et al.  Stochastic finite element modelling in linear transient heat transfer , 1997 .

[27]  G H Huang,et al.  IFRP: a hybrid interval-parameter fuzzy robust programming approach for waste management planning under uncertainty. , 2007, Journal of environmental management.

[28]  Zhiping Qiu,et al.  Hybrid probabilistic fuzzy and non-probabilistic model of structural reliability , 2010, Comput. Ind. Eng..

[29]  Z. Qiu,et al.  Hybrid uncertain analysis for steady-state heat conduction with random and interval parameters , 2015 .

[30]  Julio R. Banga,et al.  Fuzzy finite element analysis of heat conduction problems with uncertain parameters , 2011 .

[31]  M. Tootkaboni,et al.  A multi‐scale spectral stochastic method for homogenization of multi‐phase periodic composites with random material properties , 2010 .

[32]  C. Jiang,et al.  A Hybrid Reliability Approach Based on Probability and Interval for Uncertain Structures , 2012 .