Engineering Analysis with Boundary Elements 35 (2011) 729–734

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Engineering Analysis with Boundary Elements

0955-79

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/enganabound

Boundary knot method for heat conduction in nonlinear functionallygraded material

Zhuo-Jia Fu a,b, Wen Chen a,n, Qing-Hua Qin b

a Center for Numerical Simulation Software in Engineering and Sciences, Department of Engineering Mechanics, Hohai University, Nanjing, Jiangsu, PR Chinab School of Engineering, Building 32, Australian National University, Canberra ACT 0200, Australia

a r t i c l e i n f o

Article history:

Received 13 August 2010

Accepted 11 December 2010

Keywords:

Boundary knot method

Kirchhoff transformation

Nonlinear functionally graded material

Heat conduction

Meshless

97/$ - see front matter & 2010 Elsevier Ltd. A

016/j.enganabound.2010.11.013

esponding author.

ail address: chenwen@hhu.edu.cn (W. Chen).

a b s t r a c t

This paper firstly derives the nonsingular general solution of heat conduction in nonlinear functionally

graded materials (FGMs), and then presents boundary knot method (BKM) in conjunction with Kirchhoff

transformation and various variable transformations in the solution of nonlinear FGM problems. The

proposed BKM is mathematically simple, easy-to-program, meshless, high accurate and integration-free,

and avoids the controversial fictitious boundary in the method of fundamental solution (MFS). Numerical

experiments demonstrate the efficiency and accuracy of the present scheme in the solution of heat

conduction in two different types of nonlinear FGMs.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Functionally graded materials (FGMs) are a new generation ofcomposite materials whose microstructure varies from one mate-rial to another with a specific gradient. In particular, ‘‘a smoothtransition region between a pure ceramic and pure metal wouldresult in a material that combines the desirable high temperatureproperties and thermal resistance of a ceramic, with the fracturetoughness of a metal’’ [1]. In virtue of their excellent behaviors,FGMs have become more and more popular in material engineeringand have featured in a wide range of engineering applications(e.g., thermal barrier materials [2], optical materials [3], electronicmaterials [4] and ever biomaterials [5])

During the past decades extensive studies have been carried outon developing numerical methods for analyzing the thermal beha-vior of FGMs, for example, the finite element method (FEM) [6], theboundary element method (BEM) [7,8], the meshless local boundaryintegral equation method (LBIE) [9], the meshless local Petrov–Galerkin method (MLPG) [10–13] and the method of fundamentalsolution (MFS) [14–16]. However, the conventional FEM is inefficientfor handling materials whose physical property varies continuously;BEM needs to treat the singular or hyper-singular integrals [17,18],which is mathematically complex and requires additional comput-ing costs. It is worth noting that, with the exception of mesh-basedFEM and BEM, the other above-mentioned methods are classified tothe meshless method. Among these meshless methods, LBIE and

ll rights reserved.

MLPG belong to the category of weak-formulation, and MFS belongsto the category of strong-formulation.

This study focuses on strong-formulation meshless methodsdue to their inherent merits on easy-to-program and integration-free. The MFS distributes the boundary knots on fictitious boundary[19] outside the physical domain to avoid the singularities offundamental solutions, and selecting the appropriate fictitiousboundary plays a vital role for the accuracy and reliability of theMFS solution, however, it is still arbitrary and tricky task, largelybased on experiences.

Later, Chen and Tanaka [20] develops an improved method,boundary knot method (BKM), which used the nonsingular generalsolution instead of the singular fundamental solution and thuscircumvents the controversial artificial boundary in the MFS. Thisstudy first derives the nonsingular general solution of heat con-duction in FGM, and then applies the BKM in conjunction with theKirchhoff transformation to heat transfer problems with nonlinearthermal conductivity. A brief outline of the paper is as follows:Section 2 describes the BKM coupled with Kirchhoff transformationfor heat conduction in nonlinear FGM, followed by Section 3 topresent and discuss the numerical efficiency and accuracy of theproposed approach in two typical examples. Finally some conclu-sions are summarized in Section 4.

2. Boundary knot method for nonlinear functionally gradedmaterial

Consider a heat conduction problem in an anisotropic hetero-geneous nonlinear FGM, occupying a 2D arbitrary shaped region

Z.-J. Fu et al. / Engineering Analysis with Boundary Elements 35 (2011) 729–734730

O�R2 bounded by its boundary G, and in the absence of heatsources. Its governing differential equation is stated as

X2

i,j ¼ 1

@

@xiKij x,Tð Þ

@TðxÞ

@xj

� �¼ 0, xAO ð1Þ

with the following boundary conditions.Dirichlet/essential condition:

TðxÞ ¼ T , xAGD ð2aÞ

Neumann/natural condition:

qðxÞ ¼�X2

i,j ¼ 1

Kij@TðxÞ

@xjniðxÞ ¼ q, xAGN ð2bÞ

Robin/convective condition:

qðxÞ ¼ heðTðxÞ�T1Þ, xAGR ð2cÞ

where T is the temperature, G¼GDþGNþGR and K ¼

fKijðx,TÞg1r i, jr2 denotes the thermal conductivity matrix whichsatisfies the symmetry (K12 ¼ K21) and positive-definite(DK ¼ detðKÞ ¼ K11K22�K2

1240) conditions. nif g the outward unitnormal vector at boundary xA@O,he the heat conduction coefficientand T1 the environmental temperature.

In this study, we assume the heat conductor is an exponentiallyfunctionally graded material such that its thermal conductivity canbe expressed by

Kij x,Tð Þ ¼ a Tð ÞKijeP2

i ¼ 12bixi , x¼ x1,x2ð ÞAO ð3Þ

in which a Tð Þ40,K ¼ fKijg1r i,jr2 is a symmetric positive definitematrix, and the values are all real constants. b1 and b2 denoteconstants of material property characteristics.

By employing the Kirchhoff transformation

fðTÞ ¼Z

aðTÞdT ð4Þ

Eqs. (1) and (2) can be reduced as the following form:

X2

i,j ¼ 1

Kij@2FT ðxÞ

@xi@xjþ2biKij

@FT ðxÞ

@xj

!0@

1Ae

P2

i ¼ 1

2bixi

¼ 0, xAO ð5Þ

FT ðxÞ ¼fðTÞ, xAGD ð6aÞ

qðxÞ ¼�X2

i,j ¼ 1

Kij@TðxÞ

@xjniðxÞ ¼�e

P2

i ¼ 12bixi

X2

i,j ¼ 1

Kij@FT ðxÞ

@xjniðxÞ ¼ q, xAGN

ð6bÞ

qðxÞ ¼ he FT ðxÞ�jðT1Þ� �

, xAGR ð6cÞ

where FT ðxÞ ¼jðTðxÞÞand the inverse Kirchhoff transformation

TðxÞ ¼j�1 FT ðxÞð Þ ð7Þ

And then we derive the nonsingular general solution of Eq. (5)by two-step variable transformations:

Step 1: To simplify the expression of Eqs. (5), let FT ¼

Ce�P2

i ¼ 1biðxiþ siÞ. Eq. (5) can then be rewritten as follows:

X2

i,j ¼ 1

Kij@CðxÞ@xi@xj

�l2CðxÞ

0@

1AeP2

i ¼ 1bi xiþ sið Þ

¼ 0, xAO ð8Þ

where l¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP2

i ¼ 1

P2j ¼ 1 biKijbj

q. Since e

P2

i ¼ 1bi xiþ sið Þ40. The Trefftz

functions of Eq. (8) are equal to those of anisotropic modifiedHelmholtz equation.

Step 2: To transform the anisotropic Eq. (8) into isotropic one, we set

y1

y2

!¼

1=ffiffiffiffiffiffiffiffiK11

p0

�K12=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiK11DK

q ffiffiffiffiffiffiffiffiK11

p=ffiffiffiffiffiffiffiDK

q0B@

1CA x1

x2

!ð9Þ

where DK ¼ det K� �¼ K11K22�K

2

1240.It follows from Eq. (8) that

X2

i ¼ 1

@2C yð Þ

@yi@yi�l2C yð Þ

!¼ 0, yAO ð10Þ

Therefore, Eq. (10) is the isotropic modified Helmholtz equation,the corresponding nonsingular solution can be found in [20]. Thenthe nonsingular solution of Eq. (8) can be obtained by using inversetransformation (9),

uGðx,sÞ ¼�1

2pffiffiffiffiffiffiffiDK

q I0ðlRÞ ð11Þ

in which R¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP2

i ¼ 1

P2j ¼ 1 riK

�1

ij rj

q,r1 ¼ x1�s1,r2 ¼ x2�s2, where-

x,sare collocation points and source points, respectively, and I0

denotes the zero-order modified Bessel function of first kind.Finally, by implementing the variable transformation

FT ¼Ce�P2

i ¼ 1bi xiþ sið Þ, the nonsingular solution of Eq. (5) is in

the following form:

uGðx,sÞ ¼�I0ðlRÞ

2pffiffiffiffiffiffiffiD

K

q e�P2

i ¼ 1bi xiþ sið Þ

ð12Þ

It is worth noting that the source points are placed on thephysical boundary by using the present nonsingular generalsolution uG.

In the boundary knot method, the solution of Eqs. (5) and (6) isapproximated by a linear combination of general solutions with theunknown expansion coefficients as shown below:

F xð Þ ¼XN

i ¼ 1

aiuG x,sið Þ ð13Þ

where faig are the unknown coefficients determined by boundaryconditions. AfterFðxÞ is obtained, the temperature solution T to Eqs.(1) and (2) can be obtained using Eq. (7).

The heat flux can then be given by

qðxÞ ¼XN

i ¼ 1

aiQ ðx,siÞ ð14Þ

in which

Q ðx,siÞ ¼X2

i,j ¼ 1

Kij@uGðx,siÞ

@xjniðxÞe

P2

i ¼ 12bixi

¼eP2

i ¼ 12biri

2pffiffiffiffiffiffiffiDK

q �lR

I1ðlRÞX2

i ¼ 1

niðxÞriþ I0ðlRÞX2

i ¼ 1

X2

j ¼ 1

niðxÞKijbj

0@

1Að15Þ

where I1 denotes the first-order modified Bessel function offirst kind.

In view of the general solution satisfying the governing Eq. (5), apriori, the presented method only needs boundary discretization tosatisfy boundary conditions

Aa¼ b ð16Þ

in which

A¼

uGðxj,siÞ

Q ðxj,siÞ

Q ðxj,siÞ�heuðxj,siÞ

0B@

1CA ð17aÞ

Z.-J. Fu et al. / Engineering Analysis with Boundary Elements 35 (2011) 729–734 731

a¼ a1,a2,. . .,aNð ÞT

ð17bÞ

b¼

j Tj

� �qj

�hej T1ð Þj

� �0BBB@

1CCCA ð17cÞ

where source points siA@O, i¼ 1,2,. . .,N, coinciding with thecollocation points xjA@O, j¼ 1,2,. . .,M. In other words, unlike theMFS, the BKM places the source points and collocation points on thesame set of boundary knots, and M¼N.

3. Numerical results and discussions

In this section, the efficiency, accuracy and convergence of theproposed BKM are assessed by considering two heat conductionproblems in functionally graded materials (FGMs). The perfor-mance of the proposed method is compared with MFS solution andanalytical solution. Rerr(w) and Nerr(w) defined below representaverage relative errorand normalized error, respectively:

RerrðwÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

NT

XNT

i ¼ 1

wðiÞ�wðiÞ

wðiÞ

2

vuut ð18Þ

NerrðwÞ ¼wðiÞ�wðiÞ max

1r irNTwðiÞ , ð19Þ

where u ið Þ and u ið Þ are the analytical and numerical solutions at xi,respectively, and NT denotes the total number of uniform testpoints in the interest domain. Unless otherwise specified, NT istaken to be 100 in all following numerical cases.

Example 1. Consider the heat transfer in a nonlinear exponentialheterogeneous FGM [16] whose coefficients of heat conduction are

Fig. 1. (a) The condition number of the interpolation matrix and accuracy variation of (b)

of boundary knots by BKM and MFS with different fictitious boundary parameters (d¼

defined by Eq. (3) with aðTÞ ¼ eT . This example always occurs inhigh-temperature environments. Using Kirchhoff transformation,we can obtain FT ¼ eT , T ¼j�1ðFT Þ ¼ lnðFT Þ.

Let us consider an orthotropic material in the squareO¼ ð�1,1Þ�

ð�1,1Þ in which K ¼2 0

0 1

� �and b1 ¼ 0,b2 ¼ 1. The analytical

solution is

TðxÞ ¼ ln

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�Tx=Tr

2Tr

rsinhðTrÞe�Ty

!ð20aÞ

FT ðxÞ ¼ eTðxÞ ð20bÞ

where Tx¼ x1=ffiffiffi2p�1, Ty¼ x2, Tr¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTx2þTy2

p.

Fig. 1(a) shows the condition number of interpolation matrix inExample 1 with respect to the number of boundary knots by usingBKM and MFS with different fictitious boundary parameters. Thecondition number Cond in Fig. 1(a) is defined as the ratio of the largestand smallest singular value. It is observed that with increasingboundary points, the condition numbers of both the BKM and theMFS grow rapidly, which downplay these two methods. This ill-conditioned matrix problem is always found in the other collocationtechniques, such as the Trefftz method [21] and the Kansa method[22]. There are several ways to handle this ill-conditioning problem,including the domain decomposition method [23], preconditioningtechnique based on approximate cardinal basis function, the fastmultiple method [24] and regularization methods [25] such as thetruncated singular value decomposition (TSVD).

This study will use the TSVD to mitigate the effect of badconditioning in the BKM and MFS solutions, and the generalizedcross-validation (GCV) function choice criterion is employed toestimate an appropriate regularization parameter of the TSVD.Our computations use the MATLAB SVD code developed byHansen [25].

temperature, heat flux in (c) x1 and (d) x2 directions of Example 1 against the number

2 and 4).

Fig. 2. Isolines of normalized errors of temperature by 20 boundary nodes BKM in

Example 1.

Fig. 3. Isolines of normalized errors of heat flux in the x1 direction by 20 boundary

nodes BKM in Example 1.

Fig. 4. Isolines of normalized errors of heat flux in the x2 direction by 20 boundary

nodes BKM in Example 1.

Z.-J. Fu et al. / Engineering Analysis with Boundary Elements 35 (2011) 729–734732

By implementing BKM and MFS coupled with the TSVD tosolve the ill-conditioning matrix system, Fig. 1(b)–(d) shows thenumerical accuracy variation of temperature, heat flux in the x1 andx2 directions, respectively, against the number of boundary collo-cation points N. As compared with MFS in Example 1, in general, theBKM has roughly similar degrees of accuracy compared with theMFS in heat flux fields. It can be found from Fig. 1 that the BKMyields more accurate solution than MFS with few knots, however,with further increasing knots, the BKM solution can not improvethe accuracy better than MFS. This may result from that the BKMinterpolation matrix becomes much worse than MFS.

Figs. 2–4 show the distribution of normalized errors of tem-perature and heat flux in the x1 and x2 directions, respectively, byusing 20 boundary knots BKM in Example 1. It can be observed thatthe BKM results are in good agreement with the analytical solution.Nevertheless the BKM solution errors tend to become worse fromthe central to the boundary-adjacent regions, especially at bound-ary corners.

Example 2. This example considers another type of nonlinearexponential heterogeneous FGM in the same geometryO¼ ð�1,1Þ � ð�1,1Þ. In practice, the dependence of the thermalconductivity on the temperature may be chosen as linear,i.e., aðTÞ ¼ 1þmT , where m is a constant. By using Kirchhofftransformation, we can obtain FT ¼ Tþðm=2ÞT2, T ¼j�1ðFT Þ ¼

�1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ2mFT

p=m.

The analytical solution in this example is

TðxÞ ¼�1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ2mFT ðxÞ

pm

ð21aÞ

FT ðxÞ ¼ el TxþTyð Þ=t�P2

i ¼ 1bixi ð21bÞ

in which

t¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK11

ffiffiffiffiffiffiffiDK

q�K12

K11

0@

1A

2

þ2K12

ffiffiffiffiffiffiffiDK

q�K12

K11

0@

1AþK22

vuuuut

Tx¼x1

ffiffiffiffiffiffiffiD

K

qK11

, Ty¼�x1K12

K11

þx2

where

K ¼1 0:25

0:25 3

� �

and b1 ¼ 0:1,b2 ¼ 0:8, andm¼ 1=4.As in Example 1, Fig. 5(a) shows that the condition numbers of

BKM and MFS interpolation matrices in Example 2 increase rapidlywith the increasing boundary knots. Fig. 5(b)–(d) shows theconvergent rate of temperature and heat flow in Example 2 byusing BKM and MFS coupled with the TSVD. From these figures, itcan be seen that the BKM has better performance with fewinterpolation knots than MFS. It is noted that the BKM solutionaccuracy improves evidently with modestly increasing boundaryknots, but enhances slowly with a relatively large number of nodescharacterized by visible oscillations, due to the severely ill-condi-tioned matrix.

On the other hand, the fictitious boundary in the MFS affects itsnumerical accuracy and stability in a remarkable way. It can beobserved from the above figures that the MFS with a largerparameter d (d¼4), which characterizes the distance between

Fig. 6. Isolines of normalized errors of temperature by 16 boundary nodes BKM in

Example 2.

Fig. 5. (a)The condition number of the interpolation matrix and accuracy variation of (b) temperature, heat flux in (c) x1 and (d) x2 directions of Example 1 against the number

of boundary knots by BKM and MFS with different fictitious boundary parameters (d¼2 and 4).

Fig. 7. Isolines of normalized errors of heat flux in the x1 direction by 16 boundary

nodes BKM in Example 2.

Z.-J. Fu et al. / Engineering Analysis with Boundary Elements 35 (2011) 729–734 733

the fictitious and real boundaries, can obtain more accuratesolution than with small parameter d¼2 in this example. However,in some cases the placement of the fictitious boundary far fromthe physical domain can lead to numerical instability or everwrong solutions in the MFS [26]. In practical applications, thedetermination of the fictitious boundary is still quite tricky andoften troublesome, especially in multi-connected and irregulardomain problems. Therefore, the proposed method has the advan-tage over the MFS in that no fictitious boundary is required at all.

Figs. 6–8 represent the distribution of normalized errors oftemperature and heat flux in the x1 and x2 directions, respectively,by using 16 boundary knots BKM in Example 2. It can be found thatthe proposed method provides very accurate approximations of thetemperature and heat flux fields. As in Example 1, the errors atboundary-adjacent region are also worse than the central region. Itis noted that this phenomena is always found in the othercollocation techniques, such as the MFS, the Trefftz method andthe Kansa method.

Fig. 8. Isolines of normalized errors of heat flux in the x2 direction by 16 boundary

nodes BKM in Example 2.

Z.-J. Fu et al. / Engineering Analysis with Boundary Elements 35 (2011) 729–734734

4. Conclusions

This paper presents the nonsingular general solution fortwo-dimensional heat conduction problems in exponentialFGMs by way of the Kirchhoff transformation and coordinatetransformations. The boundary knot method in conjunction withthe truncated singular value decomposition is used for heatconduction analysis in nonlinear FGMs. Numerical demonstrationsshow that the proposed BKM is a competitive boundary collocationnumerical method for the solution of heat conduction in nonlinearFGMs, which is mathematically simple, easy-to-program, mesh-less, high accurate and integration-free, and avoids the controver-sial fictitious boundary in the MFS. Future extension of theproposed method can be made to cases of three-dimensionalcomposite materials [27] and transient heat transfer problems inFGMs [10,28,29].

Acknowledgements

The work described in this paper was supported by NationalBasic Research Program of China (973 Project no. 2010CB832702),and the R&D Special Fund for Public Welfare Industry (Hydro-dynamics, Project no. 201101014) and Foundation for Open Projectof the State Key Laboratory of Structural Analysis for IndustrialEquipment (Project no. GZ0902). The first author would like tothank the Fundamental Research Funds for the Central Universities(Project no. 2009B03214) and the China Scholarship Council (CSC)for financial support.

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