1. International Journal of Advanced Research in Engineering RESEARCH IN ENGINEERING
INTERNATIONAL JOURNAL OF ADVANCED and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online)TECHNOLOGY (IJARET) pp. 06-15, © IAEME
AND Volume 5, Issue 2, February (2014),
ISSN 0976 - 6480 (Print)
ISSN 0976 - 6499 (Online)
Volume 5, Issue 2, February (2014), pp. 06-15
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IJARET
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FUZZY FINITE ELEMENT ANALYSIS OF A CONDUCTION HEAT
TRANSFER PROBLEM
Ajeet Kumar Rai*,
Gargi Jaiswal**
*Department of Mechanical Engineering, SHIATS-DU Allahabad, India
**Department of Applied Mechanics, IIT Delhi, India
ABSTRACT
In the heat transfer analysis, the material properties are taken as crisp values. Since the input
data are imprecise, no matter what techniques are used, the solution will not be reliable. In fuzzy
finite element heat transfer analysis, the material properties are considered fuzzy parameters in order
to take uncertainty into account. In the present study, a circular rod made up of iron has been
considered. The end of the bar is insulated and heat transfer is taking place through the periphery.
Here variation in material properties are considered as fuzzy and this requires the consideration of
complex interval or fuzzy arithmetic in the analysis and the problem is discretized into finite number
of elements and interval / fuzzy arithmetic is applied to solve this problem. The values of
temperature at different points (x = 0, x = 1.25, x = 2.5, x = 3.75, x = 5) of a fin is calculated by
using different methods. Corresponding results are given and compared with the known results in the
special cases.
Keywords: Finite Element Method, Triangular Fuzzy Number & Alpha Cut.
INTRODUCTION
The fuzzy finite element method combines the well-established finite element method with
the concept of fuzzy numbers, the latter being a special case of a fuzzy set. The advantage of using
fuzzy numbers instead of real numbers lies in the incorporation of uncertainty (on material
properties, parameters, geometry, initial conditions, etc.) in the finite element analysis.
In the fuzzy finite element method uncertain geometrical, material and loading parameters are
treated as fuzzy values. The modeling of uncertain parameters as fuzzy values is necessary when it is
not possible to uniquely and reliably specify these parameters either deterministically or
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2. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 2, February (2014), pp. 06-15, © IAEME
stochastically. Often in such cases, only a limited number of samples are available or the
reproduction conditions for generating sample elements vary. The parameters possess informal or
lexical uncertainty, which may be modeled as fuzziness. Physical parameters possessing fuzziness
with regard to external loading or material, geometrical and model parameters may occur at all points
of a structure.
An exhaustive work has been reported by various scholars showing the use of fuzzy logic
combining with the results of FEM. Nicolai, De Baerdemaeker [1] has used finite element
perturbation method for heat conduction problem with uncertain physical parameters. They found the
temperature in heat conduction problem for randomly varying parameters with respect to time. B.M,
Nicolai [2] has used a method for the direct computation of mean values and variances of the
temperature in conduction-heated objects with random variable thermo physical properties. Method
is based on a Taylor expansion of the finite element formulation of the heat conduction equation and
offers a powerful alternative to the computationally expensive Monte Carlo method. Both steadystate and transient problems are considered. The simulations indicate that the variability of the
thermo physical properties may cause a considerable variability of the temperature within the heated
object. Rao S.S, Sawyer J.P. [3] has applied the concepts of fuzzy logic to a static finite element
analysis. The basic concepts of the theory of fuzzy logic are described. Later, the static finite element
analysis will be extended to dynamic analyses. Naidoo P. [4] has studied the application of
intelligent technology on a multi-variable dynamical system, where a fuzzy logic control algorithm
was implemented to test the performance in temperature control. Jose et al. [5] has used four
different global optimization algorithms for interval finite element analysis of (non)linear heat
conduction problems: (i) sequential quadratic programming (SQP), (ii) a scatter search method
(SSm), (iii) the vertex algorithm, and (iv) the response surface method (RSM). Their performance
was compared based on a thermal sterilization problem and a food freezing problem. The RSM fuzzy
finite element method was identified as the fastest algorithm among all the tested methods. It was
shown that uncertain parameters may cause large uncertainties in the process variables. Majumadar
Sarangam and Chakraverty S. [6] have proposed new methods to handle fuzzy system of linear
equations. The material properties are actually uncertain and considered to vary in an interval or as
fuzzy. And in that case; complex interval arithmetic or fuzzy arithmetic was considered in the
analysis.
Identification of the Problem
Fig 1. Schematic diagram of a circular rod used as a fin
In the present communication calculation of temperature at different points (x = 0 , x = 1.25 ,
x = 2.5 , x = 3.75 , x = 5) of a fin is done by using analytical method, Finite difference method, and
Finite element method. The end of the fin is insulated and heat is convected through periphery.
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3. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 2, February (2014), pp. 06-15, © IAEME
Diameter was taken as 2 cm, length as 5 cm, heat transfer coefficient as 5 W/cm2K, and thermal
conductivity as 70 W/cm-K. One dimensional steady state condition was assumed to solve the
problem. Corresponding results are given and compared with the known result in special cases.
MATERIALS AND METHODS
1. Analytical method
2. Finite difference method (FDM)
Qleft + Qright + qm .A. x = 0
Where, qm is the energy generation at node m, and m = 1, 2,3……m-1.
3. Finite element method (FEM)
Steps:
(i). Discretization of the problem
Divide in small sub domain/ element. Each and every element has unique number.
(ii). Finite element approximation
………………..+
Where T1, T2 = temperature at respective nodes of the element
N1, N2 = Shape functions
(iii). Elemental equation for heat transfer
(iv). Assembly (global equation)
=
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4. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 2, February (2014), pp. 06-15, © IAEME
(v). Imposition of boundary conditions
MODIFIED MATRIX (vi). Nodal solution
Solving the matrix obtained by using MATLAB.
Here the problem is solved by fem firstly by increasing nodes and secondly by increasing
element. Six cases will arise with two different number of nodes (i.e. 2 and 3) and three different
number of elements ( i.e. 1, 2 and 3).
Each case is solved and comparison of all results is done with the exact solution.
4. Fuzzy finite element method
In this method the value of thermal conductivity K is considered as TFN [ 69, 70, 71] and
heat transfer coefficient h is considered as TFN[ 4, 5,6 ] . The corresponding values of temperature in
this interval are plotted with the help of fuzzy.
Triangular Fuzzy Number (TFN) and alpha cut
Let X denote a universal set. Then, the membership function
by which a fuzzy set A is
where [0,1] denotes the interval of real numbers from 0
usually defined as the form ,
to 1. Such a function is called a membership function and the set defined by it is called a fuzzy set. A
fuzzy number is a convex, normalized fuzzy set
which is piecewise continuous and has the
functional value
(x) =1 , where at precisely one element. Different types of fuzzy numbers are
there. These are triangular fuzzy number, trapezoidal fuzzy number and Gaussian fuzzy number etc.
Here we have discussed the said problem using triangular fuzzy number only. The membership
function for triangular fuzzy number is as below.
Fig.2 Triangular Fuzzy Number (TFN) [a, b, c]
Alpha cut is an important concept of fuzzy set. Given
triangular fuzzy number may be written as [a, b, c] may be written as
9
then the alpha cut for above
].

5. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 2, February (2014), pp. 06-15, © IAEME
RESULTS AND DISCUSSION
S.NO.
DISCRETE POINTS
TEMPERATURE
ANALYTICAL
FDM
FEM
2N1E
2N2E
2N3E
3N1E
3N2E
3N3E
1
At x= 0
T1(given )
140
140
140
140
140
140
140
140
2
At x= 1.25
T2
104.54
131.8075
119.622
110.89
106.673
106.489
104.555
105.6
3
At x = 2.5
T3
83.75
124.3149
99.245
81.78
85.116
83.6753
83.822
84.6459
4
At x =3.75
T4
72.91
117.4077
78.867
74.59
73.1395
71.55
72.9751
74.54
5
At x = 5.00
T5
69.55
107.787
58.49
67.4
68.53
70.1321
69.6341
70.6384
Temperature in celsius
T at x =1.25
150
100
50
T at x =1.25
0
EXACT
FDM
2N 1 E
2N 2 E
2N 3 E
3N 1 E
3N 2 E
3N 3 E
Different methods
Fig.3 Value of temperature at x=1.25, T2
Temperature in Celsius
T at x= 2.5
150
100
50
T at x= 2.5
0
exact
FDM
2N 1E
2N 2E
2N 3E
3N 1E
3N 2E
Different methods
Fig.4 Value of temperature at x=2.5, T3
10
3N 3E

6. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 2, February (2014), pp. 06-15, © IAEME
T at x =3.75
Temperature in celsius
150
100
50
T at x =3.75
0
exact
FDM
2N 1E
2N 2E
2N 3E
3N 1E
3N 2E
3N 3E
Different methods
Fig.5 Value of temperature at x=3.75, T4
Temperature in celsius
T at x = 5
150
100
50
T at x = 5
0
exact
FDM
2N 1E
2N 2E
2N 3E
3N 1E
3N 2E
3N 3E
Different methods
Fig.6. Value of temperature at x= 5, T5
Representation of FEM Results by Fuzzy Plot
For Temperature at point 2 . T2
Alpha
values
Fig7. 2 noded 1 element
Fig.8. 2 noded 2 element
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7. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 2, February (2014), pp. 06-15, © IAEME
Alpha
values
Fig.9. 2 noded 3 element
Fig. 10. 3 noded 1 element
Fig.12. 3 noded 3 element
Fig.11. 3 noded 2 element
Alpha
values
For temperature at point 3 T3
Alpha
values
Fig.12. 2 noded 1 element
Fig.13. 2 noded 2 element
Fig.14. 2 noded 3 element
Fig.15. 3 noded 2 element
Alpha
values
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8. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 2, February (2014), pp. 06-15, © IAEME
For temperature at point 4 T4
Alpha
values
Fig.15. 2 noded 1 element
Fig.15. 2 noded 2 element
Fig.15. 2 noded 3 element
Fig.15. 3 noded 1 element
Fig.15. 3 noded 2 element
Fig.15. 3 noded 2 element
Alpha
values
Alpha
values
For temperature at point 5 T5
Alpha
values
2 noded 2 element
2 noded 1element
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9. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 2, February (2014), pp. 06-15, © IAEME
Alpha
values
3 noded 1 element
2 noded 3 element
Alpha
values
3 noded 2 element
3 noded 3 element
Here heat transfer problem is solved by Finite Difference Method and Finite Element
Method. FDM shows a large variation in comparison to the exact solution or analytical solution.
FEM results are near to the exact solution. In FEM we have taken 6 cases, by increasing the element
and by increasing the node. By increasing the node we get better result. But results of increasing the
number of elements are very close to the exact solution. So, it is clear that while increasing the node
we get approximate solutions to the exact solution. By increasing node, we get less complication to
solve the problem.
Considering the uncertain parameters as fuzzy we have presented the result of the said
problem in fuzzy plot. When the value of the alpha becomes zero the fuzzy results changes to
interval form & for the value of alpha as one, the result changes into crisp form. We get a series of
narrow & peak distribution of temperatures which reflect better solution for the said problem & are
very close to the solution obtained from the traditional finite element method with crisp parameters.
CONCLUSION
The present work an attempt has been made to find the solution of a one dimensional steady
state heat conduction problem by using different techniques with and without consideration of
uncertainty in the material properties. Results obtained through different techniques are compared
and the effect of varying number of nodes and number of elements on the results obtained by FEM is
also studied. Uncertain parameters are considered as fuzzy and results are presented in triangular
fuzzy plot.
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10. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 2, February (2014), pp. 06-15, © IAEME
REFERENCES
1)
De Baerdemaeker, J., Computation of heat conduction in materials with random variable
thermophysical properties. Int. J. Numer. Methods Eng. 36,(1993), 523–536.
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Fuzzy Finite Element Analysis of Heat Conduction Problems with Uncertain Parameters,
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3) Rao, S. S., Sawyer, P., (1995) "Fuzzy Finite Element Approach for Analysis of Imprecisely
Defined Systems", AIAA Journal, v. 33, n. 12, pp. 2364-2370.
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Finite Element Analysis of Heat Conduction Problems with Uncertain Parameters, Journal of
Food Engineering 103 (2011) 38–46.
6) Majumadar Sarangam and Chakraverty S. may (2012) fuzzy finite element method for one
dimensional steady state heat conduction problem.
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