Browse > Article

http://dx.doi.org/10.12989/csm.2012.1.4.345
###

Fuzzy finite element method for solving uncertain heat conduction problems |

Chakraverty, S.
(Department of Mathematics, National Institute of Technology)
Nayak, S. (Department of Mathematics, National Institute of Technology) |

Publication Information

Abstract

In this article we have presented a unique representation for interval arithmetic. The traditional interval arithmetic is transformed into crisp by symbolic parameterization. Then the proposed interval arithmetic is extended for fuzzy numbers and this fuzzy arithmetic is used as a tool for uncertain finite element method. In general, the fuzzy finite element converts the governing differential equations into fuzzy algebraic equations. Fuzzy algebraic equations either give a fuzzy eigenvalue problem or a fuzzy system of linear equations. The proposed methods have been used to solve a test problem namely heat conduction problem along with fuzzy finite element method to see the efficacy and powerfulness of the methodology. As such a coupled set of fuzzy linear equations are obtained. These coupled fuzzy linear equations have been solved by two techniques such as by fuzzy iteration method and fuzzy eigenvalue method. Obtained results are compared and it has seen that the proposed methods are reliable and may be applicable to other heat conduction problems too.

Keywords

finite element method; uncertainty; interval arithmetic; fuzzy number; fuzzy finite element method;

Citations & Related Records

- Reference

1 | Neumaier, A. (1990), Interval methods for systems of equations, Cambridge University Press, New York. |

2 | Nicolai, B.M. and De Baerdemaeker, J. (1993), "Computation of heat conduction in materials with random variable thermo physical properties", Int. J. Numer. Meth. Eng., 36(3), 523-536. DOI |

3 | Nicolai, B.M., Scheerlinck, N., Verboven, P. and De Baerdemaeker, J. (2000), "Stochastic perturbation analysis of thermal food processes with random eld parameters", Trans. ASAE, 43,131-138. DOI |

4 | Nicolai, B.M., Verboven, P., Scheerlinck, N. and De Baerdemaeker, J. (1999), "Numerical analysis of the propagation of random parameter uctuations in time and space during thermal food processes", J. Food Eng., 38(3), 259-278. |

5 | Nicolai, B.M., Verlinden, B., Beuselinck, A., Jancsok, P., Quenon, V., Scheerlinck, N.,Verboven, P. and De Baerdemaeker, J. (1999), "Propagation of stochastic temperature uctuations in refrigerated fruits", Int. J. Refrig., 22(2), 81-90. DOI ScienceOn |

6 | Panahi, A., Allahviranloo, T. and Rouhparvar, H. (2008), "Solving fuzzy linear systems of equations", ROMAI J., 4(1), 207-214. |

7 | Peterson, R.B. (1999), "Numerical modeling of conduction effects in microscale counterflow heat exchangers", Microscale Therm. Eng., 3(1),17-30. DOI ScienceOn |

8 | Senthilkumar, P. and Rajendran, G. (2011), "New approach to solve symmetric fully fuzzy linear systems", Sadhana, 36(6), 933-940. DOI ScienceOn |

9 | Varga, S., Oliveira, J. and Oliveira, F. (2000), "Influence of the variability of processing factors on the F-value distribution in batch retorts", J. Food Eng., 44(3), 155-161. DOI ScienceOn |

10 | Vijayalakshmi, V. and Sattanathan, R. (2011), "ST decomposition method for solving fully fuzzy linear systems using gauss jordan for trapezoidal fuzzy matrices", Forum Math., 6(45), 2245- 2254. |

11 | Wang, J., Wolfe, R.R. and Hayakawa, K. (1991), "Thermal process lethality variability in conduction heated foods", J. Food Sci., 56(5), 1424-1428. DOI |

12 | Yang, H.Q., Yao, H. and Jones, J.D. (1993), "Calculating functions on fuzzy numbers", Fuzzy Set. Syst., 55(3), 273-283. DOI ScienceOn |

13 | Zadeh, L.A. (1965), "Fuzzy Sets, information and control", 8, 338-353. DOI |

14 | Dong, W. and Shah, H. (1987), "Vertex method for computing functions of fuzzy variables", Fuzzy Set. Syst., 24(1), 65-78. DOI ScienceOn |

15 | Nicolai, B.M., Egea, J.A., Scheerlinck, N., Banga, J.R. and Datta, A.K. (2011), "Fuzzy finite element analysis of heat conduction problems with uncertain parameters", J. Food Eng., 103(1),38-46. DOI ScienceOn |

16 | Bondarev, V.A. (1997), "Variational method for solving non-linear problems of unsteady-state heat conduction", Int. J. Heat Mass Tran., 40(14), 3487-3495. DOI ScienceOn |

17 | Carlslaw, H.S. and Jaeger, J.C. (1986), Conduction of Heat in Solids, 2nd Ed., Oxford University Press, USA. |

18 | Caro-Corrales, J., Cronin, K., Abodayeh, K., Gutierrez-Lopez, G. and Ordorica-Falomir, C. (2002), "Analysis of random variability in biscuit cooling", J. Food Eng., 54(2), 147-156. DOI ScienceOn |

19 | Demir, A.D., Baucour, P., Cronin, K. and Abodayeh, K. (2003), "Analysis of temperature variability during the thermal processing of hazelnuts", Innov. Food Sci. Emerg. Technol., 4(1), 69-84. DOI ScienceOn |

20 | Deng, Z.S. and Liu, J. (2002), "Monte Carlo method to solve multidimensional bioheat transfer problem", Numer. Heat Tr. B. Fund., 42(6), 543-567. DOI ScienceOn |

21 | Dong, W.M. and Wong F.S. (1987)," Fuzzy weighted average and implementation of the extension principle", Fuzzy Set Syst., 21(2), 183-199. DOI ScienceOn |

22 | Wilson, E.L. and Nickell, R.E. (1966), "Application of the finite element method to heat fonduction analysis", Nuclear Eng. Design, 4, 276-286, North-Holland Publishing Comp., Amsterdam. DOI ScienceOn |

23 | Onate, E., Zarate, F. and Idelsohn, S.R. (2006), "Finite element formulation for convective-diffusive problems with sharp gradients using finite calculus", Comput. Method. Appl. M., 195(13-16), 1793-1825. DOI ScienceOn |

24 | Halder, A., Datta, A.K. and Geedipalli, S.S.R. (2007), "Uncertainty in thermal process calculations due to variability in firstorder and Weibull parameters", J. Food Sci., 72(4), 155-167. DOI |

25 | Hanss, M. (2002), "The transformation method for the simulation and analysis of systems with uncertain parameters", Fuzzy Set. Syst., 130(3), 277-289. DOI ScienceOn |

26 | Laguerre, O. and Flick, D. (2010), "Temperature prediction in domestic refrigerators: deterministic and stochastic approaches", Int. J. Refrig., 33(1), 41-51. DOI ScienceOn |

27 | Iijima, K. (2004), "Numerical solution of backward heat conduction problems by a high order lattice-free finite difference method", J. Chinese Inst. Eng., 27(4), 611-620. DOI |

28 | Klir, G.J. (1997), "Fuzzy arithmetic with requisite constraints", Fuzzy Set. Syst., 91(2),165-175. DOI ScienceOn |

29 | Kulpa, Z., Pownuk, A. and Skalna, I. (1998), "Analysis of linear mechanical structures with uncertainties by means of interval methods", Comput. Mech. Eng. Sci., 5, 443-477. |

30 | Ling X., Keanini R.G. and Cherukuri, H.P. (2003), "A non-iterative finite element method for inverse heat conduction problems", Int. J. Numer. Meth.Eng., 56(9),1315-1334. DOI ScienceOn |

31 | Liu, J.Y., Minkowycz, W.J. and Cheng, P. (1986), "Conjugated, mixed convection-conduction heat transfer along a cylindrical fin in a porous medium", Int. J. Heat Mass Tran., 29(5),769-775. DOI ScienceOn |

32 | Liu, K.C. and Cheng, P.J. (2006), "Numerical analysis for dual-phase-lag heat conduction in layered films", Numer. Heat Tr. A. Appl., 49(6), 589-606. DOI ScienceOn |

33 | Igboekwe, M.U. and Achi, N.J. (2011), "Finite difference method of modelling groundwater flow", J. Water Res. Protection, 3, 192-198. DOI |

34 | Matinfar, M., Nasseri, S.H. and Sohrabi, M. (2008), "Solving fuzzy linear system of equations by using householder decomposition method ", Appl. Math. Sci., 2(52), 2569 -2575. |

35 | Monte, F. de (2000), "Transient heat conduction in one-dimensional composite slab. A 'natural' analytic approach", Int. J. Heat Mass Tran., 43(19), 3607-3619. DOI ScienceOn |

36 | Muhanna, R.L. and Mullen, R.L. (2001), "Uncertainty in mechanics problems - interval - based approach", J. Eng. Mech. - ASCE, 127(6), 557-556. DOI ScienceOn |

37 | Muhieddine, M., Canot, E. and March, R. (2009), "Various approaches for solving problems in heat conduction with phase change", IJFV Int. J. On Finite, 6(1). |