International Communications in Heat and Mass Transfer 33 (2006) 231–239www.elsevier.com/locate/ichmt

Meshless analysis of unsteady-state heat transfer in semi-infinitesolid with temperature-dependent thermal conductivity☆

Akhilendra Singh ⁎, Indra Vir Singh, Ravi Prakash

Department of Mechanical Engineering, Birla Institute of Technology and Science, Pilani-333031, Rajasthan, India

Available online 28 November 2005

Abstract

In this paper, meshless element-free Galerkin (EFG) method has been extended to obtain the numerical solution of linear andnon-linear heat transfer in semi-infinite solids. A model problem has been solved using constant and temperature-dependentthermal conductivities of the material. For the linearization of nonlinear equations, quasilinearization scheme is adopted to avoiditerations and for time integration, backward difference method has been used. The meshless formulation has been presented for anon-linear heat transfer in semi-infinite solids. The EFG results have been obtained for a model problem using cubic spline weightfunction. The results obtained by the EFG method are compared with those obtained by finite element and analytical methods.© 2005 Elsevier Ltd. All rights reserved.

Keywords: Meshless EFG method; Backward difference method; Variable thermal conductivity

1. Introduction

A number of engineering and science problems can be modeled as a heat transfer in semi-infinite medium. Thelinear problems of heat transfer in semi-infinite medium can be solved by various analytical methods includingseparation of variable, Duhamel's theorem, Green's function, Laplace transform, integral method and similaritytransformation [1–3]. Due to limitations of analytical solutions for nonlinear heat transfer problems, in the past, anumber of numerical methods were developed. Out of all the numerical methods developed so far, the finite elementmethod has been found the most general method not only to solve the problems of nonlinear heat transfer but also tosolve the various problems in different areas of engineering and sciences. In spite of its numerous advantages, it is notwell suited for certain class of problems such as crack propagation, dynamic impact problems, moving phaseboundaries, phase transformation, large deformations, modeling of multi-scale phenomena and non-linear thermalanalysis. To overcome these problems, a number of meshless methods have been developed in recent years. Theessential feature of these meshless methods is that they only require a set of nodes to construct the approximationfunctions. Among all the meshless methods, the EFG method has become quite popular due to its successfulapplicability in various fields of engineering such as fracture mechanics [4,5], wave and crack propagation [6,7],solid mechanics [8,9], non-destructive testing (NDT) [10], electromagnetic field [11], manufacturing process [12],vibration analysis [13] and linear heat transfer [14–18], etc.

☆ Communicated by A.R. Balakrishnan.⁎ Corresponding author.E-mail addresses: akhilendra.singh@gmail.com, singh_akhilendra@yahoo.co.in (A. Singh).

0735-1933/$ - see front matter © 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.icheatmasstransfer.2005.10.008

Nomenclature

aj(x) Non-constant coefficientsc Specific heat of the material, J/kg °Cdmax Scaling parameterk(T) Coefficient of thermal conductivity, W/m °Ck0 Reference thermal conductivity, W/m °Cm Number of terms in the basisn Number of nodes in the domain of influencen Number of time stepspj(x) Monomial basis functionq′ Heat flux, W/m2

r Normalized radiusT ˙= ∂T/∂tt Time, sΔt Time step-size, sTh(x) Moving least square approximantw(x−x1) Weight function used in MLS approximationw Weighting function used in weak formβ Coefficient of thermal conductivity variation, /°CΦI(x) Shape functionΩ Two-dimensional domainΓ Boundary of the domainρ Density of the material, kg/m3

232 A. Singh et al. / International Communications in Heat and Mass Transfer 33 (2006) 231–239

In the present work, the EFG method has been extended to obtain the numerical solution of nonlinear heat transferin semi-infinite solids with temperature-dependent thermal conductivity. Variational method has been used to obtainthe discrete equations. The meshless formulation has been given for a model problem of non-linear heat transfer. Theresults obtained by EFG method are compared with those obtained by finite element (ANSYS 8.0) and analyticalmethods [1,19].

2. The element-free Galerkin method

The EFG method utilizes the moving least square (MLS) approximants, which are constructed in terms of nodesonly. The MLS approximation consists of three components: a basis function, a weight function associated with eachnode and a set of coefficients that depends on node position. The unknown function T(x) is approximated by movingleast square (MLS) approximants Th(x) over the computational domain [14–18] as

ThðxÞ ¼Xmj¼1

pjðxÞajðxÞupT ðxÞaðxÞ; ð1Þ

where p(x) is a vector of complete basis functions and m is the number of terms in the basis. The unknown coefficientsa(x) at any given point x are determined by minimizing the weighted least square sum J

J ¼XnI¼1

w x� xIð Þ pT ðxÞaðxÞ � TI� �2

; ð2Þ

where TI is the nodal parameter at x=xI but these are not nodal values of Th(x=xI) because T

h(x) is an approximantsnot an interpolant; w(x−xI) is a non-zero weight function of node I at x and n is the number of nodes in the domain ofinfluence of x for which w(x−xI)≠0. The stationary value of J in Eq. (2) with respect to a(x) leads the following set of

233A. Singh et al. / International Communications in Heat and Mass Transfer 33 (2006) 231–239

linear equations

AðxÞaðxÞ ¼ BðxÞT ; ð3Þwhere A and B are given as

A ¼XnI¼1

w x� xIð Þp xIð ÞpT xIð Þ ð4Þ

BðxÞ ¼ w x� xIð Þp xIð Þ; w x� x2ð Þp x2ð Þ; N N N N N w x� xnð Þp xnð Þf g: ð5Þ

Substituting a(x) in Eq. (1), the MLS approximant is obtained as

ThðxÞ ¼XnI¼1

UI ðxÞTI ¼ FT ðxÞT; ð6Þ

where meshfree shape function ΦI(x) is defined by

FI ðxÞ ¼Xmj¼1

pjðxÞ A�1ðxÞBðxÞ� �jI¼ pTA�1BI : ð7Þ

2.1. Weight function description

The choice of weight function w(x−xI) affects the resulting approximation Th(xI) in EFG method. The weightfunction is non-zero over a small neighborhood of a node xI, called the support or domain of influence of nodeI. Since the smoothness and continuity of the shape function ΦI depend on smoothness and continuity of theweight function w(x−xI). Therefore, the selection of appropriate weight function is essential in the EFG method.The cubic spline weight function [18] used in present analysis can be written as a function of normalized radiusr as

w x� xIð Þuw rð Þ ¼

2

3� 4r 2 þ 4r 3 rV

1

243� 4r þ 4r 2 � 4

3r 3

12brV1

0 rN1

8>>><>>>:

9>>>=>>>;; ð8Þ

where ðr ÞI ¼tx� xIt

dmI, ‖x−xI‖ is the distance from a sampling point x to a node xI and dmI is the domain of

influence of node I. ðr xÞI ¼ tx�xItdmxI

, ðr yÞI ¼ tx�xItdmyI

, dmxI=dmaxcxI, dmyI=dmaxcyI, dmax=scaling parameter. cxI andcyI at node I, are the distances to the nearest neighbors. dmxI and dmyI are chosen such that the matrix is non-singular everywhere in the domain. The weight function at any given point is obtained as

wðx� xI Þ ¼ wðr xÞwðr yÞwðr zÞ ¼ wxwywz: ð9Þ

3. Numerical implementation

One-dimensional governing equation for unsteady-state heat transfer in semi-infinite solids with thermal conduc-tivity dependent of temperature is given as

A

Axk Tð ÞAT

Ax¼ qc

ATAt

; ð10aÞ

where kðTÞ ¼ k0ð1þ bTÞ: ð10bÞ

Fig. 1. Semi-infinite model for unsteady-steady heat transfer.

234 A. Singh et al. / International Communications in Heat and Mass Transfer 33 (2006) 231–239

The initial and boundary conditions are

ðiÞ Tðx;0Þ ¼ Ti

ðiiÞ � k Tð ÞATAx

����ð0;tÞ

¼ q Vg: ð10cÞ

The above problem has been modeled in 2-D domain as shown in Fig. 1. Therefore, the governing Eq. (10a)changes to

A

Axk Tð ÞAT

Axþ A

Ayk Tð ÞAT

Ay¼ qc

ATAt

: ð11aÞ

The initial and boundary conditions are

ðiÞ Tðx;y;0Þ ¼ Ti

ðiiÞ � k Tð ÞATAx

����ð0;y;tÞ

¼ q Vg: ð11bÞ

The weak form of the Eq. (11a) is obtained as

ZX

AwAx

k Tð ÞATAx

þ AwAy

k Tð ÞATAy

� qcwT ˙� �

dX ¼ 0: ð12Þ

Incorporating natural boundary conditions in Eq. (12), the functional I(T) is obtained as

I Tð Þ ¼ZX

12

ATAx

k Tð ÞATAx

þ ATAy

k Tð ÞATAy

� �dXþ

ZXqcT ˙ dXþ

ZC1

q VTdC: ð13Þ

235A. Singh et al. / International Communications in Heat and Mass Transfer 33 (2006) 231–239

Taking variation i.e. δI(T) of Eq. (13), it reduces to

dIT Tð Þ ¼ZX

ATAx

k Tð ÞdATAx

þ ATAy

k Tð ÞdATAy

� �dXþ

ZXqcT ˙ dTdXþ

ZC1

q VdTdC: ð14Þ

Since δT are arbitrary in Eq. (14), the following relations are obtained by using Eq. (6)

½KðTÞ�fTg þ ½C�fTg ¼ ffg ð15aÞwhere,

KIJ Tð Þ ¼ZX

AUI

Ax

� �T

k Tð Þ AUJ

Ax

� �þ AUI

Ay

� �T

k Tð Þ AUJ

Ay

� �" #dX; ð15bÞ

CIJ ¼ZXUT

I qcUJ dX; ð15cÞ

fI ¼ZCI

q VUI dC: ð15dÞ

Quasilinearization of Eq. (15a), the unconditionally stable implicit backward difference method [20] for timeapproximation can be written as

t KðTÞ½ �n þ Cb Tf gnþ1 ¼ ½RðTÞ�n� � ð16aÞ

where,

½RðTÞ�n ¼ ½C�fTgn þ Dtffg ð16bÞ

KðTÞ½ �n ¼ Dt½KðTÞ�n ð16cÞ

4. Numerical results and discussion

In this paper, numerical solution has been obtained for unsteady-state heat conduction in semi-infinite solids. Amodel problem has been solved by using constant and variable thermal conductivities of material. The thermalconductivity of the material is assumed to vary linearly with temperature. For the solution of a system of nonlinearequations, quasilinearization [20] scheme is adopted and for the time integration, unconditionally stable backwarddifference method has been used. The meshless EFG results are obtained using linear basis and cubic spline weight

Table 1Data for semi-infinite model problem

Parameter Value of the parameter

Length (L) 1 mWidth (W) 0.1 mDensity (ρ) 9000 kg/m3

Specific heat (c) 400 J/kg °CReference thermal conductivity (k0) 400 W/m °CCoefficient of thermal conductivity variation (β) 0.0005 /°CHeat flux q′ 105 W/m2

Initial temperature Ti 0

Table 2Comparison of EFG results with finite element and analytical methods at x=0 m for 22 nodes and Δt=1 s (Case I: constant thermal conductivity)

Time,t (s)

EFG FEM Analytical

Temperature (°C) % Error Temperature (°C) % Error Temperature (°C)

0 0.000 0.00 0.000 0.00 0.00010 7.123 24.25 4.957 47.28 9.40320 11.670 12.24 9.039 32.03 13.29830 15.031 7.71 12.477 23.39 16.28740 17.763 5.55 15.434 17.93 18.80650 20.118 4.32 18.028 14.26 21.02660 22.217 3.54 20.342 11.68 23.03370 24.131 3.00 22.437 9.81 24.87880 25.902 2.61 24.358 8.42 26.59690 27.559 2.30 26.137 7.34 28.209100 29.121 2.06 27.801 6.50 29.735

236 A. Singh et al. / International Communications in Heat and Mass Transfer 33 (2006) 231–239

function for two sets of nodes whereas FEM results are obtained using quadrilateral element (Plane 55, ANSYS 8.0)for the same sets of nodes. The results presented in tables are obtained for a time step (Δt) of 1 s at the location x=0whereas the results presented in figures are obtained for a time step of 5 s at the same location. The various parametersused for unsteady-state analysis of semi-infinite model as shown in Fig. 1 are tabulated in Table 1.

4.1. Case I: constant thermal conductivity

The thermal conductivity of the material is assumed to be constant i.e. k=k0. Table 2 shows a comparison of EFGresults with those obtained by finite element and analytical methods for 22 nodes. The error in EFG and FEM resultshas also been evaluated and is presented in Table 2. The maximum error in EFG and FEM results has been found to be24.25% and 47.28%, respectively, for 22 nodes. A similar comparison of EFG results with finite element andanalytical methods is presented in Table 3 for 42 nodes. From the results presented in Table 3, it has been observedthat the maximum error in EFG and FEM results has been reduced to 6.70% and 19.04%, respectively. Fig. 2 presentsa comparison of EFG results with those obtained by finite element and analytical methods for 22 and 42 nodes. Fromthe results presented in tables and figure, it is observed that the EFG results are more accurate as compared to FEMresults. Moreover, the EFG results start converging with the increase in number of nodes as well as with the refinementof time step size.

4.2. Case II: temperature-dependent thermal conductivity

In case of variable thermal conductivity, it is assumed that thermal conductivity of the material is varying linearlywith the temperature i.e. k=k0(1+βT). Table 4 shows a comparison of EFG results with those obtained by finite

able 3omparison of EFG results with finite element and analytical methods at x=0 for 42 nodes and Δt=1 s (Case I: constant thermal conductivity)

ime,(s)

EFG FEM Analytical

Temperature (°C) % Error Temperature (°C) % Error Temperature (°C)

0.000 0.00 0.000 0.00 0.0000 8.773 6.70 7.613 19.04 9.4030 12.883 3.12 12.096 9.04 13.2980 15.954 2.04 15.363 5.67 16.2870 18.521 1.51 18.032 4.12 18.8060 20.772 1.21 20.346 3.23 21.0260 22.802 1.00 22.419 2.67 23.0330 24.665 0.86 24.314 2.27 24.8780 26.397 0.72 26.071 1.97 26.5960 28.022 0.66 27.716 1.75 28.20900 29.557 0.60 29.269 1.57 29.735

TC

Tt

01234567891

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

30

AnalyticalEFGFEM

AnalyticalEFGFEM

Time (seconds)

Tem

pera

ture

(°C

)

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

30

Time (seconds)

Tem

pera

ture

(°C

)

42 nodes

22 nodes

Fig. 2. Comparison of EFG results with finite element and analytical methods at x=0 m for Δt=5 s (Case I: constant thermal conductivity).

237A. Singh et al. / International Communications in Heat and Mass Transfer 33 (2006) 231–239

element and analytical methods for 22 nodes. The maximum error in EFG and FEM results has been found to be25.83% and 48.36%, respectively, for 22 nodes. A similar comparison of EFG results with finite element andanalytical methods has been tabulated in Table 5 for 42 nodes. From the results presented in Table 5, it has beennoticed that the maximum error in EFG and FEM results has been reduced to 8.76% and 20.73%, respectively. Fig. 3presents a comparison of EFG results with those obtained by FEM and analytical methods for 22 and 42 nodes. Fromthe results presented in tables and figure, it is observed that the EFG results are more accurate as compared to FEM

Table 4Comparison of EFG results with finite element and analytical methods at x=0 for 22 nodes (Case II: temperature-dependent conductivity)

Time,t (s)

EFG FEM Analytical

Temperature (°C) % Error Temperature (°C) % Error Temperature (°C)

0 0.000 0.00 0.000 0.00 0.00010 7.120 25.83 4.957 48.36 9.59920 11.653 14.08 9.036 33.37 13.56230 14.993 9.67 12.469 24.88 16.59840 17.702 7.58 15.420 19.49 19.15350 20.030 6.41 18.005 15.87 21.40260 22.104 5.67 20.309 13.33 23.43370 23.992 5.17 22.392 11.49 25.29980 25.737 4.80 24.301 10.12 27.03490 27.367 4.52 26.068 9.05 28.663100 28.903 4.30 27.719 8.22 30.202

Table 5Comparison of EFG results with finite element and analytical methods at x=0 for 42 nodes (Case II: temperature-dependent conductivity)

Time,t (s)

EFG FEM Analytical

Temperature (°C) % Error Temperature (°C) % Error Temperature (°C)

0 0.000 0.00 0.000 0.00 0.00010 8.758 8.76 7.609 20.73 9.59920 12.843 5.30 12.082 10.91 13.56230 15.888 4.28 15.336 7.60 16.59840 18.428 3.78 17.993 6.06 19.15450 20.653 3.50 20.293 5.18 21.40260 22.656 3.32 22.353 4.61 23.43370 24.493 3.19 24.235 4.21 25.29980 26.199 3.09 25.978 3.91 27.03590 27.798 3.02 27.610 3.67 28.663100 29.307 2.96 29.150 3.48 30.202

238 A. Singh et al. / International Communications in Heat and Mass Transfer 33 (2006) 231–239

results. Moreover, with the increase in number of nodes and with the refinement of time step size, the EFG results startconverging.

5. Conclusion

In this paper, meshless EFG method has been successfully extended to solve linear and nonlinear heat transfer insemi-infinite solids. A model problem has been solved by taking both constant and temperature-dependent thermal

0 10 20 30 40 50

22 nodes

60 70 80 90 1000

5

10

15

20

25

30

AnalyticalEFGFEM

Time (seconds)

Tem

pera

ture

(°C

)

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

30

AnalyticalEFGFEM

Time (seconds)

Tem

pera

ture

(°C

)

42 nodes

Fig. 3. Comparison of EFG results with finite element and analytical methods at x=0 m for Δt=5 s (Case II: temperature-dependent conductivity).

239A. Singh et al. / International Communications in Heat and Mass Transfer 33 (2006) 231–239

conductivities. The thermal conductivity of the material is assumed to vary linearly with temperature. For thelinearization of a nonlinear system of equations, quasilinearization technique has been used and backward differencemethod is utilized for time integration. The EFG results have been obtained for linear and nonlinear system ofequations and are compared with those obtained by finite element and analytical methods. From the analysis, it hasbeen found that the EFG results are more accurate as compared to FEM results. The EFG results were found well-converged with the increase in number of nodes. The accuracy of EFG and FEM results increases with the decrease intime step size. For time integration, predictor-corrector method can also be used to enhance the accuracy of the EFGmethod. This work can be extended to solve the heat transfer problems involving strong nonlinearities in semi-infinitesystems.

References

[1] M.N. Özisik, Heat Conduction, Second ed., John Wiley and Sons, Singapore, 1993, pp. 338–339.[2] L.M. Jiji, Heat Conduction, first ed., Begell House, New York, 2000, pp. 106–107.[3] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, Second ed., Oxford University Press, London, 1959, pp. 353–354.[4] N. Sukumar, B. Moran, T. Black, T. Belytschko, An element-free Galerkin method for three-dimensional fracture mechanics, Computational

Mechanics 20 (1997) 170–175.[5] T. Belytschko, M. Tabbara, Dynamic fracture using element-free Galerkin methods, International Journal for Numerical Methods in

Engineering 39 (1996) 923–938.[6] Y.Y. Lu, T. Belytschko, M. Tabbara, Element-free Galerkin methods for wave propagation and dynamic fracture, Computer Methods in Applied

Mechanics and Engineering 126 (1995) 131–153.[7] T. Belytschko, Y.Y. Lu, L. Gu, Crack propagation by element-free Galerkin methods, Engineering Fracture Mechanics 51 (1995) 295–315.[8] P. Krysl, T. Belytschko, Analysis of thin plates by element-free Galerkin method, Computational Mechanics 17 (1996) 26–35.[9] P. Krysl, T. Belytschko, Analysis of thin shells by element-free Galerkin method, International Journal of Solids and Structures 33 (1996)

3057–3080.[10] L. Xuan, Z. Zeng, B. Shanker, L. Udpa, Meshless element-free Galerkin method in NDT applications, review of progress in quantitative

nondestructive evaluation, in: D.O. Thompson, D.E. Chimenti (Eds.), American Institute of Physics, 2002, pp. 1960–1967.[11] V. Cingoski, N. Miyamoto, H. Yamashita, Element-free Galerkin method for electromagnetic field computations, IEEE Transactions on

Magnetics 34 (1998) 3236–3239.[12] G. Li, T. Belytschko, Element-free Galerkin method for contact problems in metal forming analysis, Engineering Computations 18 (2001)

62–78.[13] X.L. Chen, G.R. Liu, S.P. Lim, An element free Galerkin method for the free vibration analysis of composite laminates of complicated shape,

Composite Structures 59 (2003) 279–289.[14] I.V. Singh, K. Sandeep, R. Prakash, The element free Galerkin method in three-dimensional steady state heat conduction, International Journal

of Computational Engineering Sciences 3 (2002) 291–303.[15] I.V. Singh, K. Sandeep, R. Prakash, Heat transfer analysis of two-dimensional fins using meshless element-free Galerkin method, Numerical

Heat Transfer. Part A 44 (2003) 73–84.[16] I.V. Singh, K. Sandeep, R. Prakash, Meshless EFG method in transient heat conduction problems, International Journal of Heat and Technology

21 (2003) 99–105.[17] I.V. Singh, K. Sandeep, R. Prakash, Application of meshless element free Galerkin method in two-dimensional heat conduction problems,

Computer Assisted Mechanics and Engineering Sciences 11 (2004) 264–265.[18] I.V. Singh, A numerical solution of composite heat transfer problems using meshless method, International Journal of Heat and Mass Transfer

47 (2004) 2123–2138.[19] J.P. Holman, Heat Transfer, ninth ed., McGraw-Hill, Singapore, 2002, pp. 136–137.[20] J.N. Reddy, D.K. Gartling, The Finite Element Method in Heat Transfer and Fluid Dynamics, second ed., CRC Press, USA, 2001, pp. 94–95.