please scroll down for article - university of bathstaff.bath.ac.uk › ... › papers ›...

PLEASE SCROLL DOWN FOR ARTICLE

This article was downloaded by: [University of Bath Library]On: 28 December 2008Access details: Access Details: [subscription number 773568399]Publisher Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Numerical Heat Transfer, Part B: FundamentalsPublication details, including instructions for authors and subscription information:http://www.informaworld.com/smpp/title~content=t713723316

On the Truly Meshless Solution of Heat Conduction Problems inHeterogeneous MediaJiannong Fang a; Gao-Feng Zhao b; Jian Zhao b; Aurèle Parriaux a

a Engineering and Environmental Geology Laboratory, Ecole Polytechnique Fédérale de Lausanne,Lausanne, Switzerland b Rock Mechanics Laboratory, Ecole Polytechnique Fédérale de Lausanne, Lausanne,Switzerland

Online Publication Date: 01 January 2009

To cite this Article Fang, Jiannong, Zhao, Gao-Feng, Zhao, Jian and Parriaux, Aurèle(2009)'On the Truly Meshless Solution of HeatConduction Problems in Heterogeneous Media',Numerical Heat Transfer, Part B: Fundamentals,55:1,1 — 13

To link to this Article: DOI: 10.1080/10407790802605067

URL: http://dx.doi.org/10.1080/10407790802605067

Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf

This article may be used for research, teaching and private study purposes. Any substantial orsystematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply ordistribution in any form to anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representation that the contentswill be complete or accurate or up to date. The accuracy of any instructions, formulae and drug dosesshould be independently verified with primary sources. The publisher shall not be liable for any loss,actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directlyor indirectly in connection with or arising out of the use of this material.

http://www.informaworld.com/smpp/title~content=t713723316

http://dx.doi.org/10.1080/10407790802605067

http://www.informaworld.com/terms-and-conditions-of-access.pdf

ON THE TRULY MESHLESS SOLUTION OF HEATCONDUCTION PROBLEMS IN HETEROGENEOUS MEDIA

Jiannong Fang1, Gao-Feng Zhao2, Jian Zhao2, andAurele Parriaux1

1Engineering and Environmental Geology Laboratory, Ecole PolytechniqueFederale de Lausanne, Lausanne, Switzerland2Rock Mechanics Laboratory, Ecole Polytechnique Federale de Lausanne,Lausanne, Switzerland

A truly meshless method based on the weighted least-squares (WLS) approximation and

the method of point collocation is proposed to solve heat conduction problems in hetero-

geneous media. It is shown that, in the case of strong heterogeneity, accurate and smooth

solutions for temperature and heat flux can be obtained by applying the WLS approxi-

mation in each homogeneous domain and using a double-stage WLS approximation

technique together with a proper neighbor selection criterion at each interface.

1. INTRODUCTION

In recent years, a large family of meshless methods with the aim of getting ridof mesh constraints has been developed for solving partial differential equations. Thebasic idea of meshless methods is to provide numerical solutions on a set of arbi-trarily distributed points without using any mesh to connect them. Compared tomesh generation, it is relatively simple to establish a point distribution and adaptit locally. The points are grouped together in ‘‘clouds’’ for which a local approxi-mation for the problem variables is written. Depending on the methodology usedto discretize the equations, meshless methods can be classified into two major cate-gories: meshless strong-form methods and meshless weak-form methods. Mostmeshless weak-form methods, such as the element-free Galerkin method [1], are‘‘meshless’’ only in terms of the numerical approximation of field variables, and theyhave to use a background mesh to do numerical integration of a weak form over theproblem domain, which is computationally expensive. Meshless strong-formmethods, such as the generalized finite-difference method [2], often use the pointcollocation method to satisfy governing partial differential equations and boundaryconditions. They are simple to implement and computationally efficient. Since theydo not need any background mesh, they are truly meshless methods.

The finite point method (FPM) is a truly meshless method proposed by O~nnateet al. [3]. The FPM uses the weighted least-squares (WLS) approximation within each

Received 4 April 2008; accepted 17 October 2008.

Address correspondence to Jiannong Fang, Ecole Polytechnique Federale de Lausanne,

ENAC-ICARE-GEOLEP, Station 18, CH-1015 Lausanne, Switzerland. E-mail: [email protected]

1

Numerical Heat Transfer, Part B, 55: 1–13, 2009

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7790 print=1521-0626 online

DOI: 10.1080/10407790802605067

Downloaded By: [University of Bath Library] At: 15:23 28 December 2008

point cloud, which can be easily constructed to have consistency of a desired order,and adopts the point collocation method to obtain discrete equations directly frompartial differential equations. Therefore, it is easy for numerical implementation,and boundary conditions can be implemented in a natural way by just prescribingboundary conditions on points placed on boundaries. We noted that the finite pointmethod is similar to the weighted least-squares collocation method proposed by Sadatand Prax [4] for solving fluid flow and heat transfer problems. The FPM has beenapplied and extended successfully to solve a range of problems, including convec-tive-diffusive transport [5], compressible flow [6], incompressible flow [7, 8], potentialflow [9], metal solidification [10], elasticity problems in structural mechanics [11], two-phase flows [12], and fluid–structure interactions [13]. Although quite successful inmany applications, the extension and validation of the FPM for problems involvingheterogeneous media remains a big challenge [11]. Sadat et al. [14] attempted to solvea heterogeneous heat conduction problem in a two-layer wall by using the weightedleast-squares collocation method and found that one has to use a double-stage WLSapproximation technique in which a second-order numerical derivation is obtainedfrom a sequence of two first-order numerical derivations by WLS and take intoaccount the neighbors of each neighbor of the calculation node in order to achievesufficient accuracy in the case of strong heterogeneity.

The present authors revisited the method of Sadat et al. [14] for heterogeneousheat conduction problems and found that fluctuations existed in the predicted resultsof temperature and heat flux in the case of strong heterogeneity. In this article wereport this finding and show that accurate and smooth solutions for temperatureand heat flux can be obtained by applying the usual WLS approximation techniquein each homogeneous domain and using the double-stage WLS approximationtechnique together with a proper neighbor selection criterion at each interface.

The article is organized as follows. In Section 2 the weighted least-squaresapproximation method is briefly described. Then, the double-stage WLS approxi-mation technique and the proposed method for solving heat conduction problemsin heterogeneous media are presented. Numerical examples are considered inSection 4 for the purpose of evaluating accuracy of different methods. The articleends with concluding remarks in Section 5.

NOMENCLATURE

a vector of unknowns

A matrix of computation

b vector of function differences

C matrix of computation

e vector of truncation errors

f scalar field

G matrix of final algebra equations

h searching radius

m number of neighbor points

M matrix of computation

n normal coordinate

N number of total points

p vector of monomials

q normal heat flux

Qx, Qy components of heat flux vector

s heat source

T temperature

w weight function

x position vector

x, y Cartesian coordinates

j thermal conductivity

Subscripts

i, j, k, n point index

a, b component index

2 J. FANG ET AL.


2. THE WEIGHTED LEAST-SQUARES METHOD

The weighted least-squares method gives an approximation of a function f(x)and its derivatives at a given point x by using only the discrete function values atthe neighbor points in the support domain of x (usually a ball in three dimensionsor a disk in two dimensions). An advantage of this method is that it does not requireregular distribution of points. In the following, we briefly explain the method in twodimensions (extension to three dimensions is straightforward).

Consider a Taylor expansion of f(xi) around x,

f ðxiÞ ¼ f ðxÞ þX2

a¼1

faðxÞðxia � xaÞ þ1

2

X2

a;b¼1

fabðxÞðxia � xaÞðxib � xbÞ þ ei ð1Þ

where ei is the truncation error in the Taylor series expansion (here only to second-order; higher-order expansions are, of course, possible), fa is the derivative withrespect to xa (the ath component of the position vector x), and fab is the derivativewith respect to xa and xb. The symbols xia and xib denote the ath and bth compo-nents of the position vector xi, respectively. From the given function values f (x)and f (xi) ði ¼1; 2; . . . ;mÞ, the unknowns fa and fab for a, b¼ 1, 2 (note that fab¼ fba)are computed by minimizing the error ei for i ¼ 1; 2; . . . ;m. Here m is the number ofneighbor points inside the support domain of x.

Using the Taylor expansion (1) repeatedly for i ¼ 1; 2; . . . ;m, the system ofequations for the five unknowns can be written as

e ¼Ma� b ð2Þ

with

e ¼ e1; e2; . . . ; em½ �T

a ¼ f1; f2; f11; f12; f22½ �T

b ¼ f x1ð Þ � f xð Þ; f x2ð Þ � f xð Þ; . . . ; f xmð Þ � f xð Þ½ �T

M ¼ p1; p2; . . . ; pm½ �T

where a is the vector containing the five unknowns and M is a matrix in which thevector pi is defined as

pi ¼ xi1�x1; xi2�x2;xi1�x1ð Þ2

2; xi1 � x1ð Þ xi2 � x2ð Þ; xi2�x2ð Þ2

2

" #T

ð3Þ

For m > 5, this system is overdetermined with respect to the five unknowns ina. This problem can be simply overcome by determining the unknown vector a by

TRULY MESHLESS HEAT CONDUCTION 3


minimizing the quadratic form

J ¼Xm

i¼1

wie2i ð4Þ

where wi ¼ wðxi � xÞ is the value of a weight function w at point xi. Standardminimization of J with respect to a gives

a ¼ C�1Ab ð5Þ

where

C ¼Xm

i¼1

wipipTi ð6Þ

A ¼ w1p1;w2p2; . . . ;wmpm½ � ð7Þ

In this article, we use a Gaussian weight function of the form

wðr; hÞ ¼ expð�er2=h2Þ if r � h0 else

�ð8Þ

where r ¼ kxi � xk and e is a positive constant chosen to be equal to 6.3 in ourcomputations. The size of the searching radius h determines m, the number ofneighboring points around x to be used for WLS approximation.

3. STEADY HEAT CONDUCTION IN A HETEROGENEOUS MEDIUMAND NUMERICAL METHODS

Two-dimensional steady heat conduction in a heterogeneous medium isgoverned by the Laplace equation as follows:

qqx

jqT

qx

� �þ qqy

jqT

qy

� �¼ s ð9Þ

where T is the temperature, j is the thermal conductivity of the medium, and s is thesource term. On the boundaries, Dirichlet boundary condition,

T � Tb ¼ 0 on Cd ð10Þ

and Neumann boundary condition,

jqT

qn� q ¼ 0 on Cn ð11Þ

are to be satisfied with the prescribed values of the temperature Tb and the normalheat flux q.

4 J. FANG ET AL.


To obtain the discretized equations, the point collocation method is applied, i.e.,the Laplace equation (9) and the boundary conditions (10)–(11) are forced to be satis-fied at each internal and boundary point, respectively. This gives the set of equations

½L�i ¼qqx

jqT

qx

� �� i

þ qqy

jqT

qy

� �� i

�si ¼ 0 in X ð12Þ

Tj � Tjb ¼ 0 on Cd ð13Þ

jqT

qn

� �k

� qk ¼ 0 on Cn ð14Þ

Expressing spatial derivatives occurring in the above set of equations in terms ofthe unknown temperatures at points by the WLS method leads to the final discretizedsystem of equations

G~hh ¼ ~ff ð15Þ

where G is the coefficient matrix, vector ~hh contains the unknown temperatures, and ~ffis a vector containing the contributions from the prescribed values si, T

jb, and qk.

How to express the two derivative terms in ½L�i in terms of the unknowntemperatures is an important issue. Taking the first term as an example, it can beexpanded as

qqx

jqT

qx

� �� i

¼ jq2T

qx2

" #i

þ qjqx

� �i

qT

qx

� �i

ð16Þ

Therefore, a natural way is to calculate the first derivative of j by WLS and express theother derivatives in terms of the unknown temperatures by WLS again. However, itwas shown [14] that this formulation implies important errors that increase with thedegree of heterogeneity. Hence, it will be excluded from our consideration. Sadatet al. [14] proposed another way by first representing the heat flux at each point interms of temperatures using WLS and then, based on these heat flux expressions, usingthe WLS approximation again for the first derivatives of the heat flux. Because thisapproximation procedure uses WLS twice in a sequence, we denote it the double-stageWLS approximation. The detailed implementation of the method is described below.

Taking again the first term in ½L�i as an example, it can be expressed as

qqx

jqT

qx

� �� i

¼XN

j¼1

aijQxj ð17Þ

where

Qxj ¼ j

@T

@x

� �j

ð18Þ



and aij are the coefficients in the WLS approximation of the first derivative withrespect to x at point i. Here N is the total number of points. Note that only the coeffi-cients corresponding to the neighboring points of i are nonzero. The term Qx

j isdetermined by

Qxj ¼ jj

XN

n¼1

ajnTn ð19Þ

where jj corresponds to the thermal conductivity at point j and ajn are thecoefficients in the WLS approximation of the first derivative with respect to x atpoint j. The discretization of the first term in ½L�i is finally written as

XN

j¼1

XN

n¼1

jjaijajnTn ð20Þ

This expression takes into account the neighbors of the neighbors of the pointconsidered. For a point on the interface separating two domains of different thermalconductivities, Sadat et al. [14] suggested that its thermal conductivity can be takenfrom one of the two domains. However, in the next section, we will show thatdifferent choice of thermal conductivity gives different results.

From our numerical tests, we found that the method of Sadat et al. using thedouble-stage WLS approximation described above yields much improved results.However, the simulated results exhibited prominent fluctuations in both temperatureand heat flux. One reason is that, in heterogeneous heat conduction problems, thefirst derivatives of the temperature are discontinuous at the interface separatingtwo domains of different thermal conductivities, so the WLS approximation of thefirst derivatives of the temperature for the points near the interface introduces largeerrors when the neighboring points are chosen as usual from both domains. A properneighbor selection criterion was proposed to solve the problem. That is, for a givenpoint, it only selects its neighboring points from among those that have the samethermal conductivity and those that are located on the interface. This criterion isonly applied at the first stage, where the first derivatives of the temperature areapproximated by WLS. Although introducing the criterion into the method of Sadatet al. improved the results in one-dimensional problems, it still produced fluctuationsin the results of two-dimensional problems. To remove the fluctuations completely,we propose a new method as follows. For a point belongings to a homogeneousdomain, Eq. (12) can be simplified as

½L�i ¼ jq2T

qx2þ q2T

qy2

" #i

� si ¼ 0 ð21Þ

The second-order derivatives in Eq. (21) are approximated by WLS together with theproper neighbor selection criterion introduced above. For a point on the interface,the double-stage WLS approximation is used to discretize Eq. (12). Again, to calcu-late heat fluxes at the first stage, the proper neighbor selection criterion is applied.

6 J. FANG ET AL.


Heat flux calculated on an interface point can have different values corresponding todifferent choices of the thermal conductivity value for that point. In order to obtaina unique solution, the averaged heat flux is adopted in the proposed method.

Hereafter, we will call the meshless point collocation method proposed bySadat et al. [14] the MPCM1 and the new method proposed in this article theMPCM2.

4. NUMERICAL TESTS

To evaluate the two methods, MPCM1 and MPCM2, a steady-state heat con-duction problem in a two-layered composite wall, as depicted in Figure 1, is firstsolved in this section. This test problem was proposed by Sadat et al. [14] to validatetheir method, i.e., MPCM1 here. The two layers have the same thickness but differ-ent conductivities of j1 and j2. The ratio of conductivities j1=j2 accounts for thedegree of heterogeneity of the wall. Prescribed temperatures are imposed on thetwo vertical surfaces, whereas the horizontal boundaries are assumed to be adiabatic.The computational domain was considered to be a square with dimensionlesslength 1. A total of 441 (21� 21) points were first evenly placed at equal distancesðDx ¼ Dy ¼ 0:05Þ over the problem domain, with 80 of them located right on thefour sides of the square (21 on each side), which are so-called boundary points,and 21 of them located right on the interface between the two layers, which areso-called interface points. Then, the positions of boundary and interface points werefixed while the positions of other points were shifted randomly to give an irregularpoint distribution, as shown in Figure 2. The searching radius h was set to 2:1Dx.Both the case with a source term ðs 6¼ 0Þ and the one without a source termðs ¼ 0Þ are considered. The stationary temperature and heat flux profiles along thex direction obtained by the MPCM1 and MPCM2 are compared to the exact

Figure 1. Two-dimensional heat conduction in a two-layered composite wall.



analytical solutions. The parameter values and the results are given in dimensionlessform.

Figures 3 and 4 show the results of temperature and heat flux for j1=j2 ¼ 0:01with s ¼ 0 and j2 fixed to 1. It can be seen that the MPCM1 provides acceptableresults for the heat conduction problem in the case of strong heterogeneity.

Figure 2. Point distribution used for space discretization.

Figure 3. Temperature profile for j1=j2¼ 0.01 and s¼ 0. Here MPCM1 (k1) and MPCM1 (k2) mean the

thermal conductivities on the interface points are set to k1 and k2, respectively. The small figure inside is an

enlarged plot for the region 0.5� x� 1.

8 J. FANG ET AL.


Nevertheless, the results predicted by the MPCM1, especially the heat flux results,exhibit prominent fluctuations, as evidenced in Figure 4. Moreover, the results ofthe MPCM1 are sensitive to the choice of the thermal conductivities on the interfacepoints. These problems are solved perfectly by the MPCM2, and the results obtainedby the MPCM2 match the exact solutions accurately. In Figures 5 and 6, we showthe results obtained by the MPCM1 using the same parameters but a finer and

Figure 4. Heat flux profile for j1=j2¼ 0.01 and s¼ 0. Here MPCM1 (k1) and MPCM1 (k2) mean the

thermal conductivities on the interface points are set to k1 and k2, respectively.

Figure 5. Temperature profile for j1=j2¼ 0.01 and s¼ 0, predicted by the MPCM1 using two

different grids.



regular grid of 1,681 (41� 41) points. Oscillations are still observed, but they becomesmaller.

For the case of s ¼�10, the results with j1=j2 ¼ 0:01 are shown in Figures 7and 8 for temperature and heat flux, respectively. Again, the MPCM1 yields fluctu-ating results for both temperature and heat flux, while the results of the MPCM2 arefluctuation-free and are very close to the analytical solutions.

Figure 6. Heat flux profile for j1=j2¼ 0.01 and s¼ 0, predicted by the MPCM1 using two different grids.

Figure 7. Temperature profile for j1=j2¼ 0.01 and s¼�10.

10 J. FANG ET AL.


The test problem solved above is actually one-dimensional. To further test theMPCM1 and MPCM2, we here consider a two-dimensional problem, as shown inFigure 9. The whole domain is a square with dimensionless length 1. The center partof the domain is a square with dimensionless length 0.5 and is made of a material(j1 ¼ 0:01) different from that of the outer part (j2 ¼ 1). A regular grid with a totalof 1,681 (41� 41) points is used in the simulation. Shown in Figures 10 and 11 arethe temperature profiles along a horizontal line and a vertical line, respectively.

Figure 8. Heat flux profile for j1=j2¼ 0.01 and s¼�10.

Figure 9. Two-dimensional heat conduction in a square domain with the center square made of a material

different from that of the outer part.



Results from a finite-element method are used for comparison. Again, the MPCM2outperforms the MPCM1 in terms of accuracy and smoothness.

5. CONCLUSIONS

A truly meshless method based on the WLS approximation and the pointcollocation approach has been presented for the numerical simulation of heat

Figure 10. Temperature profile along the horizontal line at y¼ 0.5 for j1=j2¼ 0.01 and s¼ 0.

Figure 11. Temperature profile along the vertical line at x¼ 0.875 for j1=j2¼ 0.01 and s¼ 0.

12 J. FANG ET AL.


conduction problems in heterogeneous media. The proposed method employs theWLS approximation for points inside the homogeneous domains and the double-stage WLS approximation for points at interfaces. In addition, the proper neighborselection criterion is introduced for the double-stage WLS approximation. It hasbeen shown that the proposed method is able to obtain accurate and smooth solu-tions in the case of strong heterogeneity. The proposed method can be easilyextended to transient problems.

REFERENCES

1. T. Belytschko, Y. Lu, and L. Gu, Element Free Galerkin Methods, Comput. Mech.,vol. 37, pp. 229–256, 1992.

2. T. Liszka and J. Orkisz, Finite Difference Method at Arbitrary Irregular Grids and ItsApplication in Applied Mechanics, Comput. Struct., vol. 11, pp. 83–95, 1979.

3. E. O~nnate, S. Idelsohn, O. Zienkiewicz, and R. Taylor, A Finite Point Method in Computa-tional Mechanics: Applications to Convective Transport and Fluid Flow, Int. J. Numer.Meth. Eng., vol. 39, pp. 3839–3866, 1996.

4. H. Sadat and C. Prax, Application of the Diffuse Approximation for Solving Fluid Flowand Heat Transfer Problems, Int. J. Heat Mass Transfer, vol. 39, pp. 214–218, 1996.

5. E. O~nnate, S. Idelsohn, O. Zienkiewicz, and R. Taylor, A Stabilized Finite Point Methodfor Analysis of Fluid Mechanics Problems, Comput. Meth. Appl. Mech. Eng., vol. 139,pp. 315–346, 1996.

6. R. Lohner, C. Sacco, E. O~nnate, and S. Idelsohn, A Finite Point Method for CompressibleFlow, Int. J. Numer. Meth. Eng., vol. 53, pp. 1765–1779, 2002.

7. E. O~nnate, S. Sacco, and C. Idelsohn, A Finite Point Method for Incompressible FlowProblems, Comput. Visual. Sci., vol. 3, pp. 67–75, 2000.

8. J. Fang and A. Parriaux, A Regularized Lagrangian Finite Point Method for the Simula-tion of Incompressible Viscous Flows, J. Comput. Phys., vol. 227, pp. 8894–8908, 2008.

9. E. Ortega, E. O~nnate, and S. Idelsohn, An Improved Finite Point Method for Tridimen-sional Potential Flows, Comput. Mech., vol. 40, pp. 949–963, 2007.

10. L. Zhang, Y. Rong, H. Shen, and T. Huang, Solidification Modeling in ContinuousCasting by Finite Point Method, J. Mater. Process. Technol., vol. 192, pp. 511–517, 2007.

11. E. O~nnate, F. Perazzo, and J. Miquel, A Finite Point Method for Elasticity Problems,Comput. Struct., vol. 79, pp. 2151–2163, 2001.

12. S. Tiwari and J. Kuhnert, Modeling of Two-Phase Flows with Surface Tension by FinitePointset Method (FPM), J. Comput. Appl. Math., vol. 203, pp. 376–386, 2007.

13. S. Tiwari, S. Antonov, D. Hietel, J. Kuhnert, F. Olawsky, and R. Wegener, A MeshfreeMethod for Simulations of Interactions between Fluids and Flexible Structures, inM. Griebel and M. Schweitzer (eds.), Meshfree Methods for Partial DifferentialEquations III, Vol. 57 of Lecture Notes in Computational Science and Engineering,pp. 249–264, Springer-Verlag, Berlin, 2007.

14. H. Sadat, N. Dubus, L. Gbahoue, and T. Sophy, On the Solution of HeterogeneousHeat Conduction Problems by a Diffuse Approximation Meshless Method, Numer. HeatTransfer B, vol. 50, pp. 491–498, 2006.



please scroll down for article - university of bathstaff.bath.ac.uk › ... › papers ›...

Documents