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Numerical Heat Transfer, Part A:Applications: An International Journal ofComputation and MethodologyPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/unht20

Two-Dimensional Rosenthal MovingHeat Source Analysis Using the MeshlessElement Free Galerkin MethodXuan-Tan Pham aa Department of Mechanical Engineering, cole de technologiesuprieure, Montral, Qubec, CanadaVersion of record first published: 28 Feb 2013.

To cite this article: Xuan-Tan Pham (2013): Two-Dimensional Rosenthal Moving Heat Source AnalysisUsing the Meshless Element Free Galerkin Method, Numerical Heat Transfer, Part A: Applications: AnInternational Journal of Computation and Methodology, 63:11, 807-823

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TWO-DIMENSIONAL ROSENTHAL MOVING HEATSOURCE ANALYSIS USING THE MESHLESS ELEMENTFREE GALERKIN METHOD

Xuan-Tan PhamDepartment of Mechanical Engineering, Ecole de technologie superieure,Montreal, Quebec, Canada

The quasi stationary-state solution of the two-dimensional Rosenthal equation for a moving

heat source using the meshless element free Galerkin method is studied in this article.

Node-based moving least square approximants are used to approximate the temperature

field. Essential boundary conditions are enforced by using Lagrange multipliers. A Gaussian

surface heat source is used for the modeling of the moving heat source. The results obtained

for a two-dimensional model are compared with the results of the finite-element method.

INTRODUCTION

Many manufacturing processes such as fusion welding, heat treatment, orforming by line heating involve the use of moving heat sources. The studies ofRosenthal on the heat distribution during arc welding and cutting [1, 2] are con-sidered as the earliest work to solve analytically the problem of moving heat sources.The works of Rosenthal established the foundation to solve this type of problem inmany research studies in this field later by both analytical and numerical methods[39]. The modeling of moving heat sources and deformation of work pieces usingthe finite element method (FEM) has been studied by many authors [1012]. To havea good prediction, this method requires a fine mesh for the heat sources as well aswhen the deformation is critical. Particularly when elements are distorted, remeshingis required to avoid the problem of singularity. Adaptive mesh refinement is gener-ally used to solve these problems. However, this method takes a lot of time forremeshing and projecting results on the new meshes. Meshless methods are newtrends and recently developed to avoid these difficulties by using node-basedapproximation rather than element-based approximation, such as the smoothedparticle hydrodynamic (SPH) for heat and fluid analysis [1315] as well as for solidmechanics [1618], the meshless local Petrov-Galerkin (MLPG) method [19], the dif-fuse finite element method [20], the meshless element free Galerkin (MEFG) method[21], the meshless finite-difference method [22], a space-time meshless method [23],

Received 28 September 2012; accepted 17 November 2012.

Address correspondence to Xuan-Tan Pham, Department of Mechanical Engineering, Ecole de

technologie superieure, 1100, rue Notre-Dame Ouest, Montreal, Quebec H3C1K3, Canada. E-mail:

tan.pham@etsmtl.ca

Numerical Heat Transfer, Part A, 63: 807823, 2013

Copyright # Taylor & Francis Group, LLCISSN: 1040-7782 print=1521-0634 online

DOI: 10.1080/10407782.2013.757089

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and so on. Among these methods, the MEFG method has been successfully used tosolve many different problems, particularly for heat transfer analysis. Singh et al.[24] studied the heat transfer of two-dimensional fins using the MEFG method.Three-dimensional heat transfer problems for orthotropic materials were also solvedusing the MEFG method by Singh [25]. A numerical solution for temperature-dependent thermal conductivity problems was obtained by Singh et al. [26] whenapplying the MEFG method. However, very little work on the modeling of materialprocessing involving moving heat sources by the MEFG method has been reported.In this article, the modeling of a two-dimensional Rosenthal moving heat sourceusing the MEFG method was performed. The moving least square approximant(MLS) was used in this study. Both regular and irregular meshes were employedto study the influence of the support domain on the approximation. Numericalresults of temperature distribution obtained by the MEFG method were comparedwith the solutions of the finite element analysis. Very good agreements between thesetwo methods were found in this study.

ROSENTHAL MOVING HEAT SOURCE FORMULATION

The heat transfer equation [27] is defined by

qcqhqt

Qr:q 1

where h is the temperature, t is the time, q is the density, c is the specific heat, Q is thevolume heat rate, and q is the heat flux vector of heat conduction in solids [28] deter-mined by the Fourier law as

q k: ~rrh 2

in which k is the heat conductivity coefficient tensor and ~rrh is the gradient of thetemperature. If we denote n the surface normal vector, the boundary conditions ofthe problem are as follows:

NOMENCLATURE

a(x) coefficient vector

c specific heat, J=kg=Cdmax scaling parameter

dmI size of the support domain of node I, mm

dI distance between point x and node I, mm

h convective heat transfer coefficient,

W=mm2=Ck thermal conductivity coefficient tensor,

W=mm=Cn surface normal vector

NI(s) Lagrange interpolant

p(x) complete monomial basis

q heat flux vector, W=mm2

Q volume heat rate, W=mm3

QG surface heat rate, W=mm2

t time, s

w test function

wI(x) weight function at node I

h temperature, Ch1 surrounding temperature, C

h_I nodal variable at node I

k Lagrange multiplierq density (kg=mm3)UI(x) MLS shape function at node I

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. For a prescribed temperature: h hh on Ch

. For a heat flux: q:n qq on Cq

. For a convection: q.n h(h h1) on Ch

where h is the heat convection coefficient, h is the temperature of the wall, and h1 isthe surrounding temperature.

For a thermally isotropic material, the heat equation of a moving heat sourceQ that moves along the x-axis with the speed v can be described in the moving coor-dinate system which attaches to the moving heat source as follows (see Appendix A):

qcqhqt

vqc qhqx

kr2hQ 0 3

For the quasi-stationary state, this equation, which is referred to as theRosenthal equation for a moving heat source, simply becomes the following:

vqc qhqx

kr2hQ 0 4

WEIGHTED RESIDUAL METHOD

Let w denote the test function, and the weighted residual formulation of theRosenthal equation can be written as follows:

ZV

wfvqc qhqx

kr2hQgdV 0 5

When choosing the test function w 0 on the essential boundary Ch, Eq. (5)reduces to

ZV

wkr2hdV ZCqn

wqndCZV

w;iqidV 6

where Cqn is the natural boundary for heat flux and=or convection. By replacingEq. (6) into Eq. (5), we get the weak form of the Rosenthal equation, as follows:

ZV

w;iqidV ZV

wvqcqhqx

dV ZCqn

wqndCZV

wQdV 7

MEFG METHOD

Moving Least Square Approximation

Given a set of N nodal values for the field function h_1

; h_2

; . . . ; h_N

at N nodesx1, x2,. . ., xN that are in the support domain, the moving least square (MLS) approx-imant for the values of the field function at these nodes is defined [20, 29] by

2-D ROSENTHAL ANALYSIS USING GALERKIN METHOD 809

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hhx; xI pTxI ax; I 1; 2; . . . ;N 8

where pT(x) [p1(x), p2(x), . . ., pm(x)] is a complete monomial basis and m is thenumber of terms in the basis, a(x) is the coefficient vector and h

_I hxI is thenodal parameter of the field variable at node I. The coefficient vector a(x) is deter-mined by minimizing a weight discrete L2 norm, which is defined as the followingequation:

Jax XNI1

wI xpT xIax h_I

2 9

where wI(x)w(x xI) is the positive weight function at node I, which quicklydecreases as the distance jjx xIjj increases to preserve the local character ofthe approximation. The minimum of Eq. (9) leads to the following set of linearequations:

ax A1x Bx h_

10

where

Ax XNI

wI xpxI pTxI 11

Bx w1px1 w2px2 . . .wNpxN 12

h_T h

_1h_2

. . . h_N 13

By substituting Eq. (10) into Eq.(8), the MLS approximant has the usualfollowing form:

h_hx pTx A1x Bx h

_

Uxh_

14

where the MLS shape function is defined by

Ux pTxA1xBx 15

whose index form is described as follows:

UI x Xmj1

Xmk1

pjxA1jk xBkI x 16

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Weight Function

Weight functions play an important role in the MEFG method. They should beconstructed according to some conditions as mentioned in references [21, 30]. Theweight function wI(x) should be nonzero over a small neighbourhood of xI, calledthe domain of influence of node I or also the compact support, in order to generatea set of sparse discrete equations. In this article, the cubic spline weight function [21,30] was used. This function is defined as follows:

wIx wx xI 23 4r2 4r3 ifr 0:543 4r 4r2 43r3 if 0:5 < r 10 if r > 1

81xN fh_

KtgNx1 24

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So,

h;i qhqxi

XNK1

qUKqxi

h_

K

< UK ;i >1xN h_

K

Nx1

with i 1; 2; 325

qi kqhqxi

kh;i kXNK1

UK;i h_

K k < UK;i > h_

K

with i 1; 2; 326

On the other hand, the test function is a virtual variation of the shape function, asfollows:

w dh XNK1

UKdh_

K < UK >1xN fdh_

KgNx1 27

Its derivatives are defined by

w;i dh;i XNK1

UK ;idh_

K < UK ;i >1xN dh_

K

Nx1

28

The Lagrange multiplier k should be an unknown function of the coordinates,which needs to also be interpolated using the nodes on the essential boundaries toobtain a discretized set of system equations; i.e.,

kx XNkI1

NI skI < NI s >1xNk fkIgNkx1; x 2 Ch 29

where Nk is the number of nodes used for this interpolation, s is the arc-length alongthe essential boundary, kI is the Lagrange multiplier at node I on the essentialboundary, and NI (s) can be a Lagrange interpolant used in the conventional finiteelement method.

Finally, the discretized form of Eq. (23) has the following form:

dh_TfKv Kk Khh

_

Gk qn qh Qrg dkT GT h_

qhh 0 30

where Kv;Kk;Kh; qn; qh; qhh;Qr are defined in Appendix B. Because dh_

and dk arearbitrary, the above equation can be satisfied only if

Kv Kk Khh_

Gk qn qh QrGT h

_

qhh

(31

2-D ROSENTHAL ANALYSIS USING GALERKIN METHOD 813

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APPLICATION FOR THE TWO-DIMENSIONAL CASE

2-D Linear Basis

The basis of a two-dimensional case has the following form:

< px >< 1 x y > 32

By replacing Eq. (32) into Eqs. (11) and (12), the following relations areobtained as

Ax XNI

wx xI 1 xI yIxI x

2I xIyI

yI xIyI y2I

24

35 33

Bx wx x11x1y1

8

ZV

fUI ;igqidV

< dh_

I >

ZV

fUI ;igNx1kij < UJ;j >1xN dV

24

35fh_Jg < dh_I > Kk fh_Jg

where (Kk)IJRV

UI, mkmnUJ, n dV.

2-D ROSENTHAL ANALYSIS USING GALERKIN METHOD 821

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First Convective Term

ZCh

wTqhhdC < dh_

I >

ZCh

fUIghhdC < dh_

I > ZCh

hfUIg < UJ >dCfh_

Jg

< dh_

I > Khfh_

Jg

where KhIJ RChhUIUJdC:

Second Convective Term

ZCh

wTq0hdC < dh_

I > ZCh

fUIg < UJ > fq0hJgdC dh_T

qh

where qhI RChUIUJq0hJdC:

Heat Flux Term

ZCq

wTq0ndC < dh_

I > ZCq

fUIg < UJ > fq0nJgdC dh_T

qn

where qnI RCqUIUJq0nJdC:

First Constraint Term

ZCh

dhTkdC < dh_

I > ZCh

fUIg < NJ > dCfkJg dh_T

Gk

where GIJ RChUINJdC:

Second Constraint Term

ZCh

dkT h hhdC < dkI > ZCh

fNIg < UJ > dCfh_

Jg < dkI >ZCh

fNIghhdC

dkT GT h_

qhh

where GTIJ RChNIUJdC and qhhI

RChNIhhdC:

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Heat Rate Term

ZV

wTQdV < dh_

I > ZV

fUIg < UJ > fQJgdV dh_T

Qr

where (Qr)IRV

UIUJQJdV.

2-D ROSENTHAL ANALYSIS USING GALERKIN METHOD 823

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