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International Journal of Heat and Mass Transfer 50 (2007) 1212–1219

Technical Note

Meshless element free Galerkin method for unsteady nonlinearheat transfer problems

Akhilendra Singh a,*, Indra Vir Singh b, Ravi Prakash a

a Department of Mechanical Engineering, Birla Institute of Technology and Science, Pilani 333031, Rajasthan, Indiab Department of Mechanical Systems Engineering, Shinshu University, 4-17-1 Wakasato, Nagano City 380-8553, Japan

Received 15 July 2005; received in revised form 16 August 2006Available online 21 November 2006

Abstract

In this paper, meshless element free Galerkin (EFG) method has been extended to obtain the numerical solution of nonlinear,unsteady heat transfer problems with temperature dependent material properties. The thermal conductivity, specific heat and densityof the material are assumed to vary linearly with the temperature. Quasi-linearization scheme has been used to obtain the nonlinear solu-tion whereas backward difference method is used for the time integration. The essential boundary conditions have been enforced byLagrange multiplier technique. The meshless formulation has been presented for a nonlinear 3-D heat transfer problem. In 1-D, theresults obtained by EFG method are compared with those obtained by finite element and analytical methods whereas in 2-D and3-D, the results are compared with those obtained by finite element method.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Meshless EFG method; Backward difference method; Temperature dependent material properties; Finite element method; Nonlinear heattransfer

1. Introduction

Analysis of nonlinear transient heat transfer problems isvery important in practice. It is difficult to find analyticalsolution for such problems, so the only choice left isapproximate numerical solution. A variety of numericaltechniques are available to solve these problems.

An FEM based numerical iterative method has beenused by Donea and Giuliani [1] to solve steady-state non-linear heat transfer problems in two-dimensional structureswith temperature dependent thermal conductivities andradiative heat transfer. Bathe and Khoshgoftaar [2] haveobtained an FEM based numerical solution for the nonlin-ear steady-state and transient heat transfer problems inwhich they considered the convection and radiation bound-

0017-9310/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijheatmasstransfer.2006.08.039

* Corresponding author. Tel.: +91 1596 245073x225; fax: +91 1596244183.

E-mail addresses: akhilendra.singh@gmail.com (A. Singh), iv_singh@yahoo.com (I.V. Singh).

ary conditions. Ling and Surana [3] used the p-version leastsquare finite element method for obtaining the numericalsolution of axisymmetric heat conduction problems withtemperature-dependent thermal conductivities. Yang [4]developed an FEM based time integration algorithm forthe solution of nonlinear heat transfer problems. Homoto-phy analysis method has been improved and applied byLiao [5] to solve strongly nonlinear heat transfer problems.Transient heat conduction and radiation heat transferproblems with variable thermal conductivity have beensolved by Talukdar and Mishra [6] using discrete transferand implicit schemes. Bondarev [7] has used variationalcalculation method to solve unsteady state strong nonlin-ear problems in heat conduction. General boundary ele-ment method has been used by Liao [8] to solvenonlinear heat transfer problems governed by hyperbolicheat conduction equation. Skerget and Alujevic [9] usesboundary element method to solve nonlinear transient heattransfer in reactor solids with convection and radiation onsurfaces. The further use of general boundary element

Nomenclature

C(T) specific heat, J/kg �CC0 reference specific heat, J/kg �Cdmax scaling parameterk(T) thermal conductivity, W/m �Ck0 reference thermal conductivity, W/m �Cm number of terms in the basis functionn number of nodes in the domain of influence�n number of iterationspj(x) monomial basis functionq heat flux, W/m2

_Q rate of internal heat generation per unit volume,W/m3

�r normalized radius_T oT

ott time, sDt time step-size, s

Th(x) moving least square approximantV three-dimensional domainw(x � xI) weight function used in MLS approximation�w weighting function used in weak formulation

Greek symbols

b1 coefficient of thermal conductivity variation, �Cb2 coefficient of specific heat variation, �Cb3 coefficient of density variation, �CX two-dimensional domainUI(x) shape functionk Lagrangian multiplierq(T) density, kg/m3

q0 reference density, kg/m3

A. Singh et al. / International Journal of Heat and Mass Transfer 50 (2007) 1212–1219 1213

method has been done by Liao and Chwang [10] to solvestrongly nonlinear heat transfer problems. A low orderspectral method has been used by Siddique and Khayat[11] to solve nonlinear heat conduction problems with peri-odic boundary conditions and periodic geometry. Theprecise time integration (PTI) method has been introducedby Chen et al. [12] and applied to linear and nonlineartransient heat conduction problems with temperaturedependent thermal conductivity and predictor-correctoralgorithm is employed to solve the nonlinear equations,etc. Out of all the numerical methods developed so far,finite element method has been found the most generalmethod not only to solve the problems of nonlinear heattransfer but also to solve the various problems in differentareas of engineering and sciences. In spite of its numerousadvantages, it is not well-suited for certain class of prob-lems such as crack propagation, dynamic impact problems,moving phase boundaries, phase transformation, largedeformations, modeling of multi-scale phenomena, andnonlinear thermal analysis. To overcome these problems,a number of meshless methods have been developed in lasttwo decades. These methods have the following advantagesover FEM [13]:

� Only nodal data is required for the interpolation of fieldvariables.� No mesh or elements are involved in the discretization

process.� Node insertion or elimination is quite easier than FEM.� No need of tedious and time consuming re-meshing

process.� No volumetric locking problem due to unavailability of

elements.� Selection of basis function is more flexible than FEM.� Complex geometries and moving domain problems can

be easily handled.

� Smooth shape functions are used based on localapproximation.� Good accuracy and high convergence rate can be

achieved.� Post-processing for the results is quite smooth.

Although most of the meshless methods have high com-putational cost as compared to FEM, but above advanta-ges of meshless methods over FEM have motivated us toextend the application of meshless method in unsteadystate nonlinear heat transfer. In the present work, elementfree Galerkin (EFG) method has been used due to its goodaccuracy, ease in formulation, and wide range of applica-tions [14,15], and to start with, simple geometries have beenchosen in 1-D, 2-D and 3-D. For nonlinear simulation, it isassumed that material properties i.e. thermal conductivity,specific heat and density are varying linearly with tempera-ture. Lagrange multiplier method has been used to enforcethe essential boundary conditions. The meshless formu-lation has been given for a 3-D model problem. The dis-crete equations are obtained using variational principleapproach. In 1-D problem, the results obtained by EFGmethod are compared with those obtained by finite element[ANSYS 8.0] and analytical methods [16], whereas in caseof 2-D and 3-D problems, the results are compared withthose obtained by finite element method [ANSYS 8.0].

2. Review of element free Galerkin method

The EFG method utilizes the moving least square(MLS) approximants, which are constructed in terms ofnodes only. The MLS approximation consists of threecomponents: a basis function, a weight function associatedwith each node, and a set of coefficients that depends onnode position. Using MLS approximation, an unknown

3Sz

6S

1214 A. Singh et al. / International Journal of Heat and Mass Transfer 50 (2007) 1212–1219

temperature function T(x) is approximated by Th(x)[14,15]:

T hðxÞ ¼Xm

j¼1

pjðxÞajðxÞ � pTðxÞaðxÞ ð1Þ

where pT(x) = [1, x,y,z], aT(x) = [a1(x),a2(x), a3(x),a4(x)],m is the number of terms in the basis function. At any givenpoint x, the unknown coefficients a(x) are determined byminimizing the weighted discrete L2-norm J:

J ¼Xn

I¼1

wðx� xIÞ½pTðxÞaðxÞ � T I �2 ð2Þ

where TI is the nodal parameter at x = xI, it is not thenodal value of Th(x = xI) since Th(x) is an approximantnot an interpolant; w(x � xI) is a non-zero weight functionof node I at x and n is the number of nodes in the domainof influence of x for which w(x � xI) 6¼ 0. The stationaryvalue of J in Eq. (2) with respect to a(x) leads to the follow-ing set of linear equations:

AðxÞaðxÞ ¼ BðxÞT ð3Þ

where A and B are given as

A¼Xn

I¼1

wðx�xIÞpðxIÞpTðxIÞ ð4Þ

BðxÞ ¼ fwðx�x1Þpðx1Þ;wðx�x2Þpðx2Þ; . . . ;wðx�xnÞpðxnÞgð5Þ

Substituting a(x) in Eq. (1), the MLS approximant is ob-tained as

T hðxÞ ¼Xn

I¼1

UIðxÞT I ¼ UTðxÞT ð6Þ

where

UTðxÞ ¼ fU1ðxÞ;U2ðxÞ;U3ðxÞ; . . . ;UnðxÞg ð7ÞTT ¼ ½T 1; T 2; T 3; . . . ; T n� ð8Þ

The mesh free shape function UI(x) is defined as

UIðxÞ ¼Xm

j¼1

pjðxÞðA�1ðxÞBðxÞÞjI ¼ pTA�1BI ð9Þ

y

x

W

H

L

1S

4S5S

2S

L = 1 mW = 1 mH = 1 m

1ST = 200 °C

t = 100 s

1ST

V

Δ

Fig. 1. Three-dimensional model.

2.1. The weight function

The weight function is non-zero over a small neighbor-hood of a node xI called the support or domain of influenceof node I. The exponential weight function [15] used inpresent analysis can be written as a function of normalizedradius �r as

wð�rÞ ¼ 100��r 0 6 �r 6 1

0 �r > 1

� �ð10Þ

where �rI ¼ kx� xIk=dmI and kx � xIk, is the distancefrom a sampling point x to a node xI, and dmI is the domainof influence of node I. ð�rxÞI ¼ kx� xIk=dmxI , ð�ryÞI ¼

ky � yIk=dmyI , ð�rzÞI ¼ kz� zIk=dmzI , dmxI = dmaxcxI, dmyI =dmaxcyI, dmzI = dmaxczI, dmax = scaling parameter. cxI, cyI

and czI at node I, are the distances to the nearest neighbors.The weight function at any given point is obtained as

wðx� xIÞ ¼ wð�rxÞwð�ryÞwð�rzÞ ¼ wxwywz ð11Þ

where wð�rxÞ, wð�ryÞ and wð�rzÞ can be calculated by replacing�r by �rx, �ry and �rz in the expression of wð�rÞ.

2.2. Enforcement of essential boundary conditions

Lacking of Kronecker delta property in EFG shapefunctions UI poses some difficulty in the imposition ofessential boundary conditions. For that different numericaltechniques have been proposed to enforce the essentialboundary conditions such as Lagrange multiplier method[14,15], modified variational principle approach [17], cou-pling with finite element method [18], penalty approach[19], full transformation technique [20], etc. In the presentwork, Lagrange multiplier method has been used due toits accuracy [21,22].

In 3-D, Lagrange multiplier k is expressed as

kðxÞ ¼ NIðaÞkI ; x 2 Si ð12aÞdkðxÞ ¼ NIðaÞdkI ; x 2 Si ð12bÞ

where NI(a) is a Lagrange interpolant, and a is the areaalong the essential boundaries.

3. Numerical formulation

In this section, meshless numerical formulation has beenpresented for a general three-dimensional (3-D) nonlinear,unsteady heat transfer problem (Fig. 1). A general form ofenergy equation for three-dimensional heat transfer in

A. Singh et al. / International Journal of Heat and Mass Transfer 50 (2007) 1212–1219 1215

isotropic materials with temperature dependent materialproperties is given as

r � fkðT ÞrTg þ _Q ¼ qðT ÞCðT Þ _T ð13aÞwith the following initial conditions and boundaryconditions

T ðx; 0Þ ¼ T i on V ð13bÞT ðx; tÞ ¼ T S1

x 2 S1 ð13cÞqðx; tÞ ¼ hðT � T1Þ x 2 Sj ð13dÞ

where k(T) = k0(1 + b1T), q(T) = q0(1 + b2T), C(T) =C0(1 � b3T), j = 2,3, . . . , 6.

The weak form of Eq. (13a) is obtained asZVr�w � fkðT ÞrTgdV �

ZV

�w _QdV

þZ

V�wqðT ÞCðT Þ _T dV þ

X6

j¼2

ZSj

�whðT � T1ÞdS ¼ 0

ð14Þ

The functional P(T) can be written as

PðT Þ ¼ 1

2

ZVrT � fkðT ÞrTgdV �

ZV

T _QdV

þZ

VqðT ÞCðT ÞT _T dV þ

X6

j¼2

ZSj

hT 2

2dS

�X6

j¼2

ZSj

hTT1dS ð15Þ

Enforcing essential boundary conditions using Lagrangemultiplier method, the modified functional P�ðT Þ has beenobtained as

P�ðT Þ ¼ 1

2

ZVrT :fkðT ÞrTgdV �

ZV

T _QdV

þZ

VqðT ÞCðT ÞT _T dV þ

X6

j¼2

ZSj

hT 2

2dS

�X6

j¼2

ZSj

hTT1dS þZ

S1

kðT � T S1ÞdS ð16Þ

Taking variation i.e. dP�ðT Þ of Eq. (16), it reduces to

dP�ðT Þ ¼Z

VrT � fkðT ÞdrT gdV �

ZV

_QdT dV

þZ

VqðT ÞCðT Þ _TdT dV þ

X6

j¼2

ZSj

hTdT dS

�X6

j¼2

ZSj

hT1dT dS þZ

S1

kdT dS

þZ

S1

dkðT � T S1ÞdS ð17Þ

Since dT and dk are arbitrary in preceding equation, thefollowing relations are obtained using Eq. (6)

½KðTÞ�fTg þ ½MðTÞ�f _Tg þ ½G�fkg ¼ ffg ð18aÞ½GT�fTg ¼ fqg ð18bÞ

where

KIJ ðT Þ ¼X

l¼x;y;z

ZV

UI ;l kðT ÞUJ ;l dV þX6

j¼2

ZSj

hUTI UJ dS ð19aÞ

MIJ ðT Þ ¼Z

VqðT ÞCðT ÞUT

I UJ dV ð19bÞ

fI ¼Z

V

_QUIdV þX6

j¼2

ZSj

hT1UIdS ð19cÞ

GIK ¼Z

S1

UINKdS and qK ¼Z

S1

T S1N KdS ð19dÞ

Applying quasi-linearization, and unconditionally stable im-plicit backward difference approach [23] for time approxi-mation, Eq. (18) can be written as

ð20Þ

where ½RðTÞ��n ¼ ½MðTÞ��nfTg�n þ Dtffg and ½KðTÞ��n ¼Dt½KðTÞ��n.

4. Numerical results and discussion

In the present work, a general three-dimensional formu-lation has been provided for a model problem in the previ-ous section. Numerical results have been obtained fornonlinear, unsteady state heat transfer problems with tem-perature dependent material properties, and it has beenassumed that the material parameters namely thermal con-ductivity, specific heat and density vary linearly with tem-perature. Quasi-linearization scheme is adopted for thesolution of nonlinear equations, and unconditionally-stablebackward difference method has been used for the timeintegration. The EFG results are obtained using linearbasis for exponential weight function, whereas FEM resultsare obtained by ANSYS package. The following data hasbeen used in the present simulations:

Thermal conductivity, k(T) = k0(1 + b1T), where k0 =400 W/m �C, b1 = 0.0001/�C.Specific heat, C(T) = C0(1 + b2T), where C0 = 300 J/kg �C, b2 = 0.0001/�C.Density, q(T) = q0(1 � b3T) where, q0 = 9000 kg/m3,b3 = 0.000001/�C.Heat transfer coefficient, h = 100 W/m2 �C.Surrounding fluid temperature, T1 = 20 �C.Initial temperature, Ti = 0 �C.

4.1. One-dimensional (1-D) analysis

Three-dimensional formulation has been used for thesimulation of one-dimensional, semi-infinite model heat

1216 A. Singh et al. / International Journal of Heat and Mass Transfer 50 (2007) 1212–1219

transfer problem, and it has been assumed that the variationof temperature in semi-infinite solids is one-dimensional butthe problem has been modeled in two-dimensional domainas shown in Fig. 2, hence terms containing z have beendropped from 3-D governing equation.

Model data, initial and boundary conditions are alsopresented in Fig. 2. The EFG results have been obtained

Initially at

qx

Tk

x

=∂−∂ =0

yq

L

W

Fig. 2. Semi-infinite model for u

0 0.1 0.2 0.3 0.4 00

10

20

30

40

50

60

70

80

90

100

110

t=50 s

t=250 s

t=500 s

t=750 s

t=1000 s

x

T, °

C

Fig. 3. Comparison of EFG results with an

using uniform nodal distribution scheme for 42 nodes,whereas FEM results have also been obtained using uni-form nodal distribution for the same number of nodesusing 20 elements. Fig. 3 shows a comparison of EFGresults with those obtained by finite element and analyticalmethods along the length of model for the various values oftime. From the results presented in Fig. 3, it has been

x

iTT =

x

insulated edge

insulated edge

L = 1 m W = 0.1 mq = 105 W/m2

t = 1 s Δ

nsteady-state heat transfer.

.5 0.6 0.7 0.8 0.9 1

analytical

EFGFEM

, m

alytical and FEM results along length.

A. Singh et al. / International Journal of Heat and Mass Transfer 50 (2007) 1212–1219 1217

observed that the results obtained by EFG method are ingood agreement with those obtained by finite elementand analytical methods.

4.2. Two-dimensional (2-D) analysis

For 2-D simulation, we consider heat conduction in aunit square as shown in Fig. 4. The left edge of the unitsquare has been subjected to a constant temperature T1,and top and right edges are exposed to convection with sur-

W

y

xL

Ω

Initially at iTT =

1TT =

convection edge

insulated edge

L = 1 mW = 1 m

1T = 200 °C

t = 100 s Δ

Fig. 4. Two-dimensional model.

0 0.1 0.2 0.3 0.40

20

40

60

80

100

120

140

160

180

t=500 s

t=1000 s

t=2500 s

t=5000 s

t=10000 s

T, °

C

y

Fig. 5. Comparison of EFG results w

rounding fluid temperature T1 while bottom edge is keptinsulated.

3-D formulation has been utilized for 2-D simulation bydropping the terms associated with variable z. FEM resultshave been obtained by ANSYS package [Version 8.0] usingfour node quadrilateral element [Plane 55] for a uniformmesh with 121 nodes (100 elements), and EFG results havebeen obtained using uniform nodal distribution scheme for121 nodes. A comparison of EFG results with thoseobtained by finite element method has been presented inFig. 5 at (x/L = 0.5,y) for t = 500 s, 1000 s, 2500 s, 5000 sand 10,000 s. From the results presented in Fig. 5, it canbe concluded that the results obtained by EFG methodare almost same as those obtained by finite element method.

4.3. Three-dimensional (3-D) analysis

Consider a unit cube (as shown in Fig. 1) for 3-Dunsteady-state heat transfer with an initial temperatureTi. The left surface of the cube has been subjected to a con-stant temperature T S1

, and other surfaces are subjected toconvection boundary conditions with surrounding fluidtemperature T1. The unsteady state EFG results areobtained using uniform nodal distribution scheme for 729nodes, whereas FEM results are obtained by ANSYS pack-age using 512 elements (3-D brick element, Solid 70) withuniform nodal data distribution scheme for the same num-ber of nodes. A comparison of EFG results with FEMresults is presented in Fig. 6 along a line at (x/L = 0.5,y/W = 0.5,z) for t = 500 s, 1000 s, 2500 s, 5000 s and

0.5 0.6 0.7 0.8 0.9 1

EFG

FEM

, m

ith FEM results at (x/L = 0.5,y).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 120

40

60

80

100

120

140

160

EFG

FEM

t=10000 s

t=5000 s

t=2500 s

t=1000 s

t=500 s

T, °

C

z, m

Fig. 6. Comparison of EFG results with FEM results at (x/L = 0.5, y/W = 0.5,z).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

EFG

FEM

t=10000 s

t=5000 s

t=2500 s

t=1000 s

t=500 s

T, °

C

z, m

Fig. 7. Comparison of EFG results with FEM results at (x/L = 1, y/W = 0.5,z).

1218 A. Singh et al. / International Journal of Heat and Mass Transfer 50 (2007) 1212–1219

10,000 s. A similar comparison of EFG results with FEMresults is presented in Fig. 7 for the same number of nodes

along another line at (x/L = 1, y/W = 0.5, z). From theresults presented in Figs. 6 and 7, it has been observed

A. Singh et al. / International Journal of Heat and Mass Transfer 50 (2007) 1212–1219 1219

that the EFG results are in good agreement with FEMresults.

5. Conclusions

In this work, meshless EFG method has been success-fully extended to obtain the numerical solution of unsteadystate nonlinear heat transfer problems with temperaturedependent material properties. It was assumed that thethermal conductivity, specific heat and density vary linearlywith temperature. For the linearization of nonlinear equa-tions, quasi-linearization technique was used, whereas fortime integration, backward difference method was utilized.In 1-D, the results obtained by EFG method are comparedwith those obtained by finite element and analytical meth-ods, whereas in 2-D and 3-D, the EFG results are com-pared with FEM results. From the above analysis, it canbe concluded that the results obtained by EFG methodare in good agreement with those obtained by FEM resultsin 1-D, 2-D and 3-D. In future, this work can be extendedfor the nonlinear heat transfer analysis of the problemshaving complicated geometries.

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