boundary element analysis of thermo-elastic problems with non-uniform heat sources

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The Journal of Strain Analysis for Engineering

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DOI: 10.1177/030932471004500803

2010 45: 605The Journal of Strain Analysis for Engineering DesignM Mohammadi, M R Hematiyan and M H Aliabadi

Boundary element analysis of thermo-elastic problems with non-uniform heat sources

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Boundary element analysis of thermo-elastic problemswith non-uniform heat sources

M Mohammadi1, M R Hematiyan1, and M H Aliabadi2*1Department of Mechanical Engineering, School of Engineering, Shiraz University, Shiraz, Iran2Department of Aeronautics, Imperial College, London, UK

The manuscript was received on 19 November 2009 and was accepted after revision for publication on 12 April 2010.

DOI: 10.1243/03093247JSA620

Abstract: A boundary element formulation for the analysis of two-dimensional and three-dimensional steady state thermo-elastic problems involving arbitrary non-uniform heat

sources is presented. All domain integrals are expressed in terms of a heat source functioninstead of a temperature function. The proposed method alleviates the difficulty associatedwith finding the temperature distribution throughout the domain. A simple, yet robust, methodreferred to as the Cartesian transformation method (CTM) is developed that allows boundary-only evaluation of domain integrals. Unlike other transformation methods the CTM does notrequire the introduction of a domain point for the transformation. Domain heat sourcesdefined either over the whole domain or over a specific part of the domain can be treated by theproposed method. Three examples including different forms of heat sources are analysed toshow the validity and efficiency of the presented methods.

Keywords: boundary element method, thermo-elasticity, non-uniform heat source,boundary-only formulation

1 INTRODUCTION

In some thermo-elastic applications with electrical

heating, internal chemical reactions, or laser heating,

a non-uniform heat source must be considered for

accurate modelling of the problem. Only a few simple

thermo-elastic problems involving non-uniform heat

sources can be solved analytically. To solve most

practical thermo-elastic problems, however, numer-

ical methods need to be employed. The boundary

element method (BEM) is an efficient and accuratenumerical technique for the analysis of thermo-

elastic phenomena. Presenting a BEM formulation

without domain discretization for the analysis of

thermo-elastic problems with non-uniform heat

sources is very attractive, especially from a modelling

point of view. Such a formulation is also a useful tool

for optimization in thermo-elastic systems [1].

Domain integrals that include a temperature rise

function appear in the conventional BEM formula-

tion of thermo-elasticity. Most researchers have

assumed that the temperature rise function is known

throughout the domain and have attempted to

evaluate the corresponding domain integrals with

or without domain discretization [27]. When the

temperature function is harmonic, i.e. satisfies the

steady state heat conduction equation without a

heat source, the domain integral can be exactly

transformed into a boundary integral [8]; however,

in cases with a more complicated form of tempera-

ture function, special techniques must be applied. If

the displacement and stress integral equations of

thermo-elasticity can be obtained in a form in which

domain integrals are expressed in terms of a heat

source function instead of temperature, then the

problem can be analysed with no need to find the

temperature distribution throughout the domain.

There are a few studies in the literature on the

BEM analysis of thermo-elastic problems involving

heat sources. Shiah and Tan [9, 10] presented a BEM

formulation for two-dimensional (2D) anisotropicsteady state thermo-elasticity involving uniform heat

*Corresponding author: Department of Aeronautics, Imperial

College, London, Prince Consort Road, South Kensington, London

SW7 2BY, UK.email: [email protected]

605

JSA620 J. Strain Analysis Vol. 45

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sources. They derived the Somigliana identity of the

interior strain and transformed the produced do-

main integrals exactly into boundary integrals. Shiah

and Lin [11] and Shiah and Huang [12] derived the

displacement and stress integral equations for 2D

anisotropic thermo-elasticity involving non-uniform

heat sources using the multiple reciprocity method.

They transformed domain integrals into an infinite

series of boundary integrals and considered several

first terms of the series in their computations.

On the other hand, Ochiai [13] presented a BEM

formulation for three-dimensional (3D) thermo-

elasticity involving non-uniform heat sources by

the triple-reciprocity method. A thermo-elastic dis-

placement potential was used to derive the stress

integral equation. In the triple-reciprocity method,

the forcing function is interpolated using additionalboundary integral equations and consideration of

some internal points [1315].

In this paper, a new BEM formulation for the

analysis of 2D and 3D steady state thermo-elastic

problems involving arbitrary non-uniform heat

sources is presented. Domain integrals in the

conventional displacement and stress integral equa-

tions of thermo-elasticity with arbitrary temperature

function are, respectively, weakly and strongly

singular [5, 7, 16]. However, in the present formula-

tion in which the problem is formulated in terms of a

heat source function instead of a temperaturefunction, these domain integrals are, respectively,

regular and weakly singular and therefore, can be

evaluated more efficiently. Domain integrals related

to heat source functions are efficiently evaluated by

boundary-only discretization using a new method.

There is no need to define any internal point to

transform the domain integrals into boundary

integrals. Domain heat sources defined over the

whole domain or over a specific part of the domain

can be treated by the proposed method.

This paper is organized as follows. In section 2, the

Cartesian transformation method, which is a me-

thod for the evaluation of domain integrals without

domain discretization, is described. A general for-

mula for differentiating a singular domain integral

with respect to source point coordinates is described

in section 3. The formula is subsequently used to

derive the stress integral equation by differentiating

the displacement integral equation. The temperature

and displacement integral equations of thermo-

elasticity involving domain heat sources are derived

in section 4. The stress integral equation for internal

points is derived in section 5. The methods for theevaluation of the domain integrals with boundary-

only discretization for various kinds of heat sources

are described in section 6. Some numerical examples

are presented in section 7, and finally concluding

remarks are made in section 8.

2 CARTESIAN TRANSFORMATION METHOD FORTHE EVALUATION OF DOMAIN INTEGRALS

Various methods have been developed for the

evaluation of the domain integrals in the BEM. Rizzo

and Shippy [2] and Cruse et al. [17] used a

divergence theorem to convert domain integrals into

boundary integrals for a limited range of body forces.

The Galerkin vector method was presented by

Danson [18]. A method for exact transformation of

domain integrals into boundary integrals by employ-

ing particular integrals was proposed by Pape and

Banerjee [19]. The dual reciprocity method (DRM)

[20] is a well-known method for the treatment of

domain integrals. In this method, the body force

function is approximated by a series of prescribed

basis functions. The prescribed basis functions in

the DRM must be defined in a radial form such that

it is possible to find corresponding particular

solutions. In the multiple reciprocity method

(MRM) [21], the reciprocity theorem is employed

several times and the domain integrals are trans-

formed into boundary integrals using integration byparts. The MRM can be employed only for a limited

number of problems. Another technique for the

evaluation of domain integrals with boundary-only

discretization is the radial integration method [22].

In this method, the domain integral is transformed

into a double integral consisting of a boundary

integral and a radial integral. Once the radial integral

is evaluated analytically, the domain integral can be

transformed exactly to the boundary.

In this section, the Cartesian transformation

method (CTM), which is a general and robust

method for the treatment of the domain integrals,is described. The method is very simple in both

concept and implementation and can be used for the

evaluation of regular and singular domain integrals

with boundary-only discretization. The CTM will be

used in this paper for the evaluation of the domain

integrals in the temperature, displacement, and

stress integral equations of thermo-elasticity.

Suppose that the following integral is to be com-

puted

I~V

h x dV 1

606 M Mohammadi, M R Hematiyan, and M H Aliabadi

J. Strain Analysis Vol. 45 JSA620



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where x represents a point within the domain V or

on its boundary C. This domain integral can be

transformed into a boundary integral using Greens

theorem which in general form, for a 2D or 3D

domain, can be expressed as follows [23]

V

Lp x Lxi

dV~

C

p x ni dC 2

where ni is the ith component of the unit outward

normal vector n. This theorem can also be applied to

multiply connected domains. For example, for the

multiply connected domain shown in Fig. 1, equa-

tion (2) can be expressed as follows

V

Lp x Lxi

dV~

C0,C1,C2

p x ni dC

~

C0

p x ni dCzC1

p x ni dC

z

C2

p x ni dC 3

Using equation (2), the integral in equation (1) be-

comes

I~

C

Hi x ni dC, i~1, 2, or 3 no sum on i 4

where

Hi x ~

h x dxi, i~1, 2, or 3 5

It should be noted that the derivative of Hi with

respect to xi is h. The one dimensional (1D) integral

in equation (5) can be evaluated analytically for

a wide variety of domain integrals [24]. When the

integrand h is very complicated, the integral in

equation (5) can be evaluated numerically as

Hi x ~xi

a

h x dxi i~1, 2, or 3 6

where a is an arbitrary constant. In cases with acomplicated integrand, the 1D integral in equation

(6) is evaluated numerically. By this approach a

mesh-free and adaptive technique for the evaluation

of domain integrals can be developed [25]. The

method can also be applied for cases in which the

integrand h is defined by some discrete data points

[26, 27]. Since the transformation of the domain

integral is carried out in the Cartesian coordinate

system, the method is called the CTM [26].

When the integrand in the domain integral in

equation (1) has a singularity (weak or higher order)

at point j, the integral can be written as follows

I~

V{Ve

h j, x dV 7

where Ve is an infinitesimal spherical (circular)

region of radius e around the singular point

(Fig. 2). Using Greens theorem, the domain integral

in equation (7) can be transformed to a boundary

integral, as follows

I~ C, Ce

Hi j, x

ni dC, i~1, 2, or 3 no sum on i 8

where Ce represents the boundary of the domain Ve.

The boundary integral in equation (8) over Cevanishes for weakly singular integrals. This integral

also has a constant value for strongly singular

integrands, and results in constant and divergent

terms for higher-order singularities. Equation (8) can

also be written as

Fig. 1 A multiply connected domain Fig. 2 An infinitesimal domain excluded from themain domain

BEM analysis of thermoelastic problems 607




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I~

C, Ce

h j, x dxi

!ni dC,

i~1, 2, or 3 no sum on i 9

which includes an outer boundary integral and aninner line integral, independent of the problem

geometry.

3 DIFFERENTIATING A SINGULAR DOMAININTEGRAL WITH RESPECT TO THE SOURCEPOINT COORDINATES

In the BEM formulation of some problems it is

required to find the derivative of a singular domain

integral with respect to the source point coordinates.

In this section, a general formula for differentiating2D and 3D singular domain integrals is described. In

section 5, this formula will be used to derive the

stress integral equation from the displacement

integral equation of thermo-elasticity.

Consider the following singular domain integral

I~

V{Ve

y x f j, x dV 10

where y is a regular and f is a singular function. It is

assumed that f consists of constants or terms in the

form of r or r,i where r is the Euclidian distancebetween the singular point j and the field point x.

The domain Ve is an infinitesimal spherical (circular)

region around the singular point. The integration

domain includes the main boundary C and the

infinitesimal boundaryCe. Assume that it is required

to find the derivative of this singular integral with

respect to the singular point coordinate ji. It should

be noted that any change in the location of the

source point results in a change in the integration

domain (Fig. 3). The derivative of the domain

integral in equation (10) with respect to, for example,

j1, can be written as follows

LI

Lj1~ lim

c?0

1

cI j1zc, j2, j3 {I j1, j2, j3

& '11

where c is an infinitesimal distance in the x1-direc-

tion and

I j1, j2, j3 ~V{Ve

y x f j1, j2, j3, x dV 12

I j1zc, j2, j3 ~V{Ve

y x f j1zc, j2, j3 , x dV 13

the Ve and Ve~Ve j1zc, j2, j3

in equations (12) and

(13) represent the infinitesimal region around thesource point before and after differential change of

the location of the source point. By following the

standard procedures for differentiating the singular

integrals [8, 28], the following formula is obtained

L

Lji

V{Ve

y(x)f j, x dV~V{Ve

y x Lf j, x Lji

dV

zy j Ce

f j, x ni dC 14

A point worth mentioning here is that the order ofthe singularity ofLf j, x =Lji is one more than that off j, x . This means that differentiating a singulardomain integral with respect to the source point

location, produces a higher-order singular domain

integral and a boundary integral which should be

evaluated over an infinitesimal boundary around the

source point. This boundary integral can be eval-

uated analytically.

4 THE TEMPERATURE AND DISPLACEMENT

INTEGRAL EQUATIONS OF THERMO-ELASTICITY INVOLVING NON-UNIFORM HEATSOURCES

The governing equations of steady state thermo-

elasticity can be expressed as

kH,ii x zs x ~0 15

sij,j x zbi x ~0 16

whereH

is the temperature rise, k is the thermalconductivity, sis the heat source function, sij are theFig. 3 A differential change of source point location





6/24

components of the stress tensor, bi are the compo-

nents of the body force vector, and x is a point in the

domain.

Strain components eij can be expressed in terms of

the stress components as follows

eij~1

E1zn sij{ndijskk

zaHdij 17

where Eis Youngs modulus, n is Poisson ratio, and a

is the coefficient of linear thermal expansion.

The temperature integral equation corresponding

to equation (15) can be written as follows [28, 29]

C j H j ~C

H j, x LH x

Ln{H x LH

j, x Ln

!dC

z 1k

V{Ve

H j, x s x dV 18

where j (with coordinates ji) and x (with coordinates

xi) are, respectively, the source point and the field

point, and C is a coefficient related to the local

geometry of the source point. Ve is an infinitesimal

sphere (circle) with radius e around the source point,

as shown in Fig. 2. Since the domain integral in

equation (18) is weakly singular, V{Ve can be

replaced with V. H is the fundamental solution ofthe Laplace problem and LH=Ln is its normal

derivative. The fundamental solution can be writtenas follows [28, 29]

H

~1

2pln

1

r

, for 2D problems 19

H

~1

4pr, for 3D problems 20

By considering the relations ri~xi{ji, Lr=Ln~r,ini,

and Lr=Lxi~r,i~ri=r, the normal derivative of the

fundamental solution can be written as follows

LH

Ln~

{1

2prr,ini, for 2D problems 21

LH

Ln~

{1

4pr2r,ini, for 3D problems 22

where ris the Euclidian distance between the source

point and the field point. The method for the

evaluation of the domain integral in equation (18)

will be described in section 6.

The displacement integral equation of thermo-

elasticity with bi~0 can be written as follows [8]

Cij j uj j ~C

Uij j, x tj x {Tij j, x uj x h i

dC

zEa

1{2n

V{Ve

H x Uik,k j, x dV 23

where Cij represent the coefficient matrix of the free

term, uj and tj are, respectively, components of

displacement and traction, and Uij and Tij are the

Kelvin fundamental solutions for displacement and

traction, respectively, and can be expressed as

Uij~1

8pG 1{n

3{4n dij ln 1

rzr,ir,j

!24

Tij~{1

4p 1{n rLr

Ln1{2n dijz2r,ir,j

&

{ 1{2n r,inj{r,jni 25

for 2D plane strain problems and

Uij~1

16pG 1{n r 3{4n dijzr,ir,j 26

Tij~{1

8p 1{n r2Lr

Ln1{2n dijz3r,ir,j

&

{ 1{2n r,inj{r,jni 27

for 3D problems. Gin these equations represents the

shear modulus of the rigidity. It should be noted that

a plane stress problem with material constants E, n,

and a can be analysed as a plane strain problem, by

replacing the material constants with E, n, and a as

follows [30]

E~1z2n

1zn 2 E,n~

n

1zn, a~

1zn

1z2na 28

The integrand of the domain integral in equation

(23) is currently expressed in terms of temperature.

Now this domain integral will be converted to a

domain integral in terms of heat source function.

The Uik,k term in the domain integral in equation(23) can be found using equations (24) and (26) as

follows

Uik,k~{ 1{2n

4pG 1{n r,ir , for 2D problems 29





7/24

Uik,k~{ 1{2n 8pG 1{n

r,ir2


The last term in equation (23) will in future be

denoted as IIi

IIi~Ea

1{2n

V{Ve


Substituting Uik,k, obtained from either equation(29) or equation (30), into equation (31) results, after

some mathematical manipulations, in the following

expression

IIi~a 1zn

8p

1{n

V{VeH x

+

2vi j, x

dV

32

where vi is given as follows

vi~{L

Lxir2 ln r

~{ri 2 ln rz1 ,

for 2D problems 33

vi~L

Lxir ~r,i, for 3D problems 34

It should be noted that +2vi~{4r,i=r for 2D prob-lems, and +2vi~{2r,i

r2 for 3D problems (see

appendix 1).

Now using Greens second identity in the form of

V{Ve

H+2vi dV~

V{Ve

vi+2HdV

z

C, Ce

HLvi

Ln{vi

LH

Ln

dC 35

and noting that +2H~{s x

=k (equation (15)), then

equation (32) can be rewritten as follows

IIi~a 1zn

8p 1{n {V{Ve

vis x

kdV

z

C

HLvi

Ln{vi

LH

Ln

dC

!36

vi in equation (36) is a regular function; therefore,

V{Ve can be replaced with V in the domain integral.

For cases where the source point is located on the

boundaryLvi=Ln~v

i,kn

kin equation (36) is a weakly

singular function and can be expressed as follows

Lvi

Ln~{ 2 ln rz1 niz2r,i Lr

Ln

!,

for 2D problems 37

LviLn

~ 1r

ni{r,i LrLn


It should be noted that the boundary integral over Cein equation (35) vanishes and thus does not appear

in equation (36). Substituting the right-hand side of

equation (36) into equation (23) leads to

Cij j uj j ~C

Uij j, x tj x {Tij j, x uj x h i

dC

za 1zn

8p 1{n

C

HLvi

Ln{vi

LH

Ln

dC

{a 1zn

8p 1{n kV{Ve

vis x dV 39

The domain integral in equation (23) has a singu-

larity of order r{1 and r{2 for 2D and 3D problems

respectively; however, the domain integral in equa-

tion (39) is regular. The method for the evaluation of

the domain integral in equation (39) without domain

discretization will be described in section 6.

In order to find the displacements using equation

(39), the temperature and its gradient at the

boundary should be known. Therefore, the heatconduction problem should be solved using equa-

tion (18) prior to the thermo-elastic analysis.

5 STRESS INTEGRAL EQUATIONS OF THERMO-ELASTICITY INVOLVING NON-UNIFORM HEATSOURCE

In order to derive the stress integral equations,

derivatives of the displacement integral equation at

internal points are found by the formula described in

section 3 and these derivatives are subsequentlyused to find strains. Stresses are then found using

the generalized Hookes law.

The derivative of the displacement components

with respect to the source point coordinates for an

internal point j can be found using equation (23).

Since for internal points Cijuj~ui, it is possible to

write that

L

Ljjui j ~

C

LUik j, x Ljj

tk x {LTik j, x

Ljjuk x

!dC

z

Ea

1{2n L

Ljj

V{Ve






8/24

The last term in equation (40) contains the derivative

of a singular domain integral. This term with the aid

of equation (14), can be written as

L

LjjV{Ve

H(x)Uik,k j, x dV

~

V{Ve

H x LUik,k j, x Ljj

dV

zH j Ce

Uik,k j, x nj dC 41

The last integral in equation (41), which is defined

over Ce, can be evaluated analytically. For 2D pro-

blems Uik,k is given by equation (29) and dC~rdh,r1~r cos h, and r2~r sin h in the polar coordinate

system. Since nj~{r,j over Ce, it is possible to writethat

H j Ce

Uik,k j,x nj dC

~H j 1{2n 4pG 1{n

2p0

r,ir

r,j rdh

~1{2n

4G 1{n H j dij 42

For 3D problems, Uik,k is given by equation (30) and

the following relationships apply in the spherical co-ordinate system:dC~r2 sinwdwdh, r1~r sinw cos h,

r2~r sinw sin h, and r3~r cosw. Therefore, it can be

written that

H j Ce

Uik,k j, x nj dC

~H j 1{2n 8pG 1{n

2p0

p0

r,ir2

r,j r2 sinwdwdh

~1{2n

6G 1{n

H j dij 43

A useful formula for the derivation of equations (42)

and (43) is

Ce

r,ir,j dC~1

3p cz2 ecdij 44

where c~1 for 2D problems and c~2 for 3D

problems.

Substituting equations (42) and (43) into equation

(41), and the resulting relation into equation (40),

and noting that LUikLjj~{U

ik,j

, LTikLjj~{T

ik,j

,

and LUik,k.Ljj~{Uik,kj , leads to

L

Ljjui j ~

C

{Uik,jtk x zTik,juk x h i

dC

{Ea

1{2n V{Ve

H x Uik,kj j, x dV

za 1zn b 1{n H j dij 45

where b~2 for 2D problems and b~3 for 3D prob-

lems.

The integrand of the domain integral in equation

(45) is expressed in terms of temperature. A

procedure will now be presented that converts this

domain integral into an integral in terms of the heat

source function. The Uik,kj term in the domainintegral in equation (45) can be found using

equations (29) and (30). This can be expressed asfollows (see appendix 2)

Uik,kj~{ 1{2n

16pG 1{n +2 r2 ln r

,ij

h i,

for 2D problems 46

Uik,kj~1{2n

16pG 1{n +2 r,ij


Denoting the second term on the right-hand side ofequation (45) by IIIij

IIIij ~Ea

1{2n V{Ve

H x Uik,kj j, x dV 48

and using Greens second identity over the region

V{Ve (with boundaries C and Ce) and noting that

+2H~{s x =k, leads to

IIIij ~a 1zn

8p 1{n

| {

V{Ve

lijs x

kdVz

C

HLlij

Ln{lij

LH

Ln

dC

z

Ce

HLlij

Ln{lij

LH

Ln

dC

!49

where

lij~{ r2 ln r

,ij

~{ 2 ln rz1 dijz2r,ir,j 50

Llij

Ln ~2

r 2r,ir,j{dij Lr

Ln { r,injzr,jni ! 51





9/24

for 2D problems, and

lij~r,ij~1

rdij{r,ir,j 52

Llij

Ln~

1

r23r,ir,j{dij Lr

Ln{ r,injzr,jni

! 53

for 3D problems.

The boundary integral over Ce in equation (49) can

be evaluated analytically. It can show thatCelij LH=Ln dC~0 and the other integral

IIIij ~CeH Llij

Ln

dC can be evaluated analytically

as follows. Since L=Ln~{L=Lr over Ce and r,ir,j isindependent of r, it is possible to write that

LlijLn

~ 2rdij, on Ce for 2D problems 54

Llij

Ln~

dij{r,ir,j

r2, on Ce for 3D problems 55

Using polar coordinates for 2D problems and

spherical coordinates for 3D problems, the final

form of IIIij is obtained as follows

IIIij ~4pH j dij, for 2D problems 56

IIIij ~8

3pH j dij, for 3D problems 57

Substituting the obtained expressions for IIIij into

equation (49) and the resulting relation into equa-

tion (45) then, after some mathematical manipula-

tions, the following expression can be obtained for

the derivatives of the displacement components

LuiLj

j

~ C {Uik,jtkzT

ik,juk dC

{k

C

HLlij

Ln{lij

LH

Ln

dC

zk

V{Ve

lijs x

kdV 58

where

k~a 1zn

8p 1{n 59

The thermo-elastic stressstrain relation can befound using equation (17) which can be expressed

as follows

sij~2G

1{2n1{2n eijznekkdij{ 1zn aHdij

60

Equation (60) is valid for 3D problems as well asplane strain problems. The strain for an internal

point j can be expressed as follows

eij j ~ 12

LuiLjj

zLuj

Lji

61

Substituting equation (61) into equation (60) results

in

sij j ~ 2Gn1{2n

LukLjk

dijzGLuiLjj

zLuj

Lji

{Ea

1{2n H j dij 62

Substituting equation (58) into equation (62) then,

after some mathematical manipulations, the follow-

ing stress integral equation in terms of heat source

function is obtained

sij j ~C

Uijktk{Tijkuk

dCz

C

lijLH

Ln{lijH

dC

z1

kV{Ve

lijs x

dV{Ea

1{2n H j

dij

63

where Uijk and T

ijk are the commonly used singular

functions in elastostatic analysis [8, 29] and

lij~Ea

4p 1{n ln1

r{

1z2n

2

dij

1{2n {r,ir,j !

64

lij ~Ea

4p 1{n r 2r,ir,j{dij

1{2n

Lr

Ln

{ r,injzr,jni 65for 2D problems, andlij~

Ea

8p 1{n rdij

1{2n{r,ir,j

66

lij ~Ea

8p 1{n r2 3r,ir,j{dij

1{2n

Lr

Ln

{ r,injzr,jni

67

for 3D problems.





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The domain integral in the stress integral equation

(63) is weakly singular but the domain integral in

equation (45) is strongly singular. The method for

the evaluation of the domain integral in equation

(63) with boundary-only discretization is described

in the next section.

It can be seen from equation (63) that in order to

find the stresses, the temperature and its gradient at

the boundary should be known. Therefore, the heat

conduction problem should be solved before the

evaluation of the stresses.

6 THE EVALUATION OF THE DOMAININTEGRALS WITHOUT INTERNALDISCRETIZATION USING THE CTM

In this section, the methods for the evaluation of the

domain integrals appearing in the developed integral

equations are described. First, it is shown that

domain integrals corresponding to a wide variety of

non-uniform heat sources can be exactly trans-

formed into boundary integrals using the CTM.

Then, it is shown that for cases with very compli-

cated non-uniform heat sources the domain inte-

grals can also be evaluated using the CTM without

internal discretization.

Three domain integrals appeared in the formula-

tions presented in the previous sections. The firstone is the domain integral in the temperature

integral equation (equation (18))

I1~

V{Ve

Hs x dV 68

The second one is the domain integral in the

displacement integral equation (equation (39))

I2~

V{Ve

vis x dV 69

The third one is the domain integral in the stress

integral equation (equation (63))

I3~

V{Ve

lijs x dV 70

The kernels H, vi, and lij for 2D and 3D problemsare given in equations (19), (20), (33), (34), (64), and

(66). Since I1 and I3 are weakly singular and I2 is a

regular integral, V{Ve can be replaced with V in the

corresponding equations; however, these integrals

are expressed and used in the original form to showthe correctness of the proceeding operations.

6.1 Exact transformation of the domain integralsinto boundary integrals for a wide variety ofnon-uniform heat sources

In this section, it is shown that for cases where the

heat source function s x , has a quadratic variationin one Cartesian direction and arbitrary variations in

other directions, all domain integrals can be exactly

transformed into boundary integrals. With no loss of

generality, only 3D cases are considered in this work.

Suppose that the heat source function has a

quadratic variation in the x1-direction and arbitrary

variations in the x2 and x3 directions, as follows

s x ~f0 x2,x3 zf1 x2,x3 x11zf2 x2,x3 x21 71

where f0, f1, and f2 are arbitrary functions. Substitut-

ing this expression in equation (68) and using theCTM, equation (9) with i~1, results in

I1~

C,Ce

f0H

zf1x

11H

zf2x

21H

dx1 !

n1 dC

72

In this equation H is weakly singular; therefore, theboundary integral over Ce vanishes. Since the

functions f0, f1, and f2 are independent of x1, it is

possible to write that

I1~C

f0H dx1zf1

x11H dx1zf2

x21H dx1

n1 dC

73or

I1~

C

f0F0zf1F1zf2F2 n1 dC 74

where

Fc~

xc1H

dx1, c~0, 1, 2

F0, F1, and F2 can simply be evaluated analytically

(see Appendix 3). The boundary integral in equation(74) can be found numerically with little difficulty.

Similarly, it can be shown that the 1D integralsxx1vi dx1 and

xx1l

ij dx1 can also be evaluated

analytically (see appendix 3) and therefore, the

domain integrals in equations (69) and (70) can be

exactly transformed into boundary integrals.

A heat source function with a quadratic variation

in the x2-direction and arbitrary variations in the x1and x3 directions or another function with a

quadratic variation in the x3-direction and arbitrary

variations in the x1 and x2 directions can also betreated with the same procedure. This means that all





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domain integrals corresponding to a heat source

function with the following form, can be exactly

transformed into boundary integrals

s x

~f0 x2,x3

zf1 x2,x3

x11 zf2 x2,x3

x21

zg0 x1,x3 zg1 x1,x3 x12 zg2 x1,x3 x22zh0 x1,x2 zh1 x1,x2 x13 zh2 x1,x2 x23 75

Similarly, for 2D problems with a heat source

function with the following form, the same proce-

dure can be applied

s x ~f0 x2 zf1 x2 x11zf2 x2 x21 zg0 x1 zg1 x1 x12zg2 x1 x22 76

Most heat source functions can be expressed interms of equations (75) and (76). For very compli-

cated non-uniform heat sources the domain inte-

grals must be evaluated by another version of the

CTM, which is described in the next section.

6.2 The evaluation of the domain integrals without internal cells for non-uniform heatsources with very complicated forms

Suppose that the heat source function is too

complicated to be expressed or approximated in

the form of equations (75) and (76). In such cases theCTM is again employed, but the inner line integral,

in equation (9), is evaluated numerically. To evaluate

the domain integral (68), it is regularized as follows

I1~

V{Ve

H j, x s x dV

~s j V{Ve

H j, x dV

z

V{Ve

H j, x s x {s j dV 77

Using the CTM, it is possible to rewrite equation (77)

in the following form

I1~s j C

H

dx1

n1 dC

z

C

x1a

H s x {s j dx1

& 'n1 dC 78

The inner line integral in the first integral in

equation (78) can be evaluated analytically using

the procedure described in the previous section. The

line integral in the second integral in equation (78) isindependent of domain geometry and can be

evaluated numerically. It should be noted that the

boundary integrals over Ce in equation (78) are equal

to zero and have been removed. The constant a, the

lower bound of the inner 1D integral in equation

(78), is arbitrary. It is suggested to set

a~x1 minzx1 max

279

where x1 min and x1 max are, respectively, the mini-

mum and maximum values of x1 over the boundary

of the problem.

The domain integral in equation (69) is regular

and can be evaluated by the same method (CTM)

with no need for any regularization. It can be

expressed as follows

I2~V{Ve vis

x dV

~

C

x1a

vis x dx1 !

n1 dC 80

The domain integral in equation (70) is weakly

singular and can be expressed as follows

I3~

V{Ve

lijs x dV

~s j C

lij dx1

n1 dC

z

C

x1

a

lij s(x){s j dx1& '

n1 dC 81

The inner line integrals in equations (78), (80), and

(81) are written with respect to the x1-direction.

Alternatively, these equations can be written with

respect to either the x2 or x3 directions. The

boundary integrals in equations (78), (80), and (81)

are evaluated using the Gaussian quadrature

method, while the inner line integrals are evaluated

using the composite Gaussian quadrature method

with an arbitrary number of integration intervals and

integration points.

When the heat source function is defined at

discrete data points, the domain integrals can be

evaluated by the same method without internal cells.

In such cases an appropriate mesh-free interpolation

method must be employed for evaluation of the heat

source function at integration points. In this man-

ner, the radial point interpolation method [31],

moving least-squares method [32], or one of several

other methods [3336] can be used. An important

advantage of the present method is that there is no

need for the existence of any particular solutions forthe interpolating functions.





12/24

7 EXAMPLES

Three examples including different forms of heat

sources are analysed to demonstrate the validity and

efficiency of the present methods. In all examples,

regular integrals over the boundary elements are

evaluated using the four-point Gaussian quadrature

method and singular integrals over the boundary

elements are evaluated analytically.

7.1 Example 1: a 2D domain subjected todifferent forms of heat sources

In this example, a rectangular domain with the

structural and thermal boundary conditions shown

in Fig. 4 is considered. The problem is analysed

under a plane strain condition with E~200GPa,n~0:3, a~11:7|10{6 uC21, k~60 W=m uC. The sides

of the rectangle are a~0:15m and b~0:3 m.

In the first case, the heat source function is con-

sidered to be as follows

s~kH02p

b

2sin

2py

b

82

where H0~50 uC. It can be shown that the exact

temperature distribution for this problem can be

expressed as follows

H~H0 sin2py

b

Displacement, stress, and strain fields can also be

found analytically as follows

ux~0, uy~1zn

1{n

aH0b

2p1{ cos

2py

b

!

exx~0, eyy~1zn

1{n aH0 sin2py

b

, exy~0

sxx~{E

1{naH0 sin

2py

b

, syy~0, sxy~0

The results obtained by the proposed BEM are

compared with those obtained using the finite

element method (FEM) and also exact solutions.

The FEM analysis of the problem was carried out

with a developed code. For the FEM analysis, four-

node quadrilateral elements were used. In eachelement the heat source function was interpolated

by the FEM shape functions. For the BEM analysis of

the problem, linear boundary elements were used

for the discretization of the boundary. The domain

integrals corresponding to the heat source function

were exactly transformed into boundary integrals.

Two different meshes, mesh I and mesh II, as

shown in Figs 5 and 6, were considered in the

analyses. The results obtained for the temperature,

vertical displacement, and horizontal stress along

Fig. 4 A 2D thermo-elastic problem

Fig. 5 Mesh I for the FEM and BEM discretization ofthe domain in example 1: (a) FEM mesh and (b)BEM mesh

Fig. 6 Mesh II for the FEM and BEM discretization of

the domain in example 1: (a) FEM mesh and (b)BEM mesh





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the line AB, shown in Fig. 4, with mesh I are shown

in Fig. 7. The results obtained with mesh II are

shown in Fig. 8. It can be observed that the results

obtained by mesh II (finer mesh) are better than

those obtained by mesh I. Also, it can be seen that

the results obtained by the BEM are better than

those obtained by the FEM with the same order of

discretization.

In the second case, a more complicated heatsource function with the following form was con-

Fig. 7 Results obtained by the FEM and BEM for theheat source function given by equation (82)using mesh I, in comparison with the exact

solution, example 1: (a) temperature, (b) dis-placement, and (c) stress

Fig. 8 Results obtained by the FEM and BEM for theheat source function given by equation (82)using mesh II, in comparison with the exactsolution, example 1: (a) temperature, (b) dis-placement, and (c) stress





14/24

sidered

s~4|105 1z3xz10x2

sin 2py

b

2 !83

In this case, no exact solution exists and the results

obtained by the proposed BEM are compared with

the accurate FEM solutions. The FEM solutions were

obtained using a fine mesh with 4608 four-node

quadrilateral elements. Since the heat source func-

tion has a quadratic variation in the x-direction, the

corresponding domain integrals are transformed

exactly into boundary integrals in the BEM analysis.

The results obtained for the temperature, vertical

displacement, and horizontal stress along the line AB

are shown in Fig. 9. It can be seen that the BEM

results with the coarse mesh (only 12 linearelements) are in good agreement with the accurate

FEM solutions.

In the third case, a very complicated heat source

function with the following form was considered

s~8|105 cosp

2

x

a

h isin 2p

y

b

2 !84

Since it does not have a quadratic variation in either

the x or y directions, it must be treated with the

method described in section 6.2. The resultsobtained for this case in comparison with the

accurate FEM solutions (4608 elements) are shown

in Fig. 10. It can be seen that the BEM results with

only 24 linear elements are in excellent agreement

with the accurate FEM solutions.

In the fourth case, it was assumed that the heat

source was distributed over a portion of the main

domain. As shown in Fig. 11, a heat source with

uniform intensity of 2 MW

m3 was considered

over a circular region with radius of 0.04m and

centred at (0.085, 0.065). The problem boundary

was discretized using 24 linear elements and the

boundary of the circular region was discretized

using 16 linear elements. It should be noted that

the nodes and elements on the circle are only used

for the evaluation of the domain integrals and they

do not introduce any additional unknowns into the

problem. This point is an important advantage of

the BEM in comparison with the FEM. For the

FEM modelling of the problem, the circular region

and the other portion of the domain must be

discretized with a compatible mesh and a large

number of elements and nodes must be employed.The results obtained for this case in comparison

with the accurate FEM solutions (4929 elements)

are shown in Fig. 12. It can be observed that the

BEM results are again in a very good agreement

with the accurate FEM solutions.

Fig. 9 Results obtained by the BEM (12 elements) forthe heat source function given by equation (83),in comparison with the accurate FEM solution(4608 elements), example 1: (a) temperature,(b) displacement, and (c) stress





15/24

7.2 Example 2: a 2D multiply connected domainsubjected to a non-uniform heat source

In this example, a 2D multiply connected domain

with the structural and thermal boundary conditions

shown in Fig. 13 is considered. It was analysed under

a plane stress condition with E~200GPa, n~0:3,

a~11:7|10{6 uC21, k~60 W=m uC. The considered

heat source function had the following form

s~106 xz3y 85

For the BEM analysis of the problem, the outerboundary (rectangle) was discretized using 80 linear

boundary elements and the inner boundary (circle)

was discretized using 36 linear boundary elements.

Since this problem has no exact solution, the results

obtained by the proposed BEM are compared with

the accurate FEM solutions. The FEM solutions were

obtained using a fine mesh with 13 512 four-node

quadrilateral elements. All domain integrals were

transformed exactly into boundary integrals in the

BEM analysis. The results obtained for the tempera-

ture, horizontal displacement, and vertical stress

along the line y~0:05 are shown in Fig. 14. It can beseen that the BEM results are in a very good

agreement with the accurate FEM solutions.

7.3 Example 3: a 3D domain subjected todifferent forms of heat sources

In this example, a cube with edge length of

a~10 mm, as shown in Fig. 15, is considered. The

used material properties were E~210 GPa, n~0:3,

a~11|10{6 uC21 . The temperature at the surface

y~

0 was taken to be zero and the other surfaces ofthe boundary were insulated. The surface y~0 was

Fig. 11 The BEM discretization of the domain inexample 1 for a heat source distributed overthe circular region with radius of 0.04 m andcentred at (0.085, 0.065)

Fig. 10 Results obtained by the BEM (24 elements) forthe heat source function given by equation(84), in comparison with the accurate FEMsolution (4608 elements), example 1: (a)temperature, (b) displacement, and (c) stress





16/24

assumed to be traction free and the displacement of

the other surfaces in the normal direction was taken

to be zero.

In the first case, the heat source function wasconsidered to be

s~s0 exp {my 86

with s0=k~1 and m~0:2. The boundary of the

problem was discretized using 600 constant bound-

ary elements as shown in Fig. 15. A similar problem

was analysed by Ochiai [13] with the same number

of boundary elements and 64 internal points. There

is no need to consider any internal points to analyse

the problem by the proposed method.

The exact solution of this problem for the

temperature, displacement in the y direction, and

stresses in x and z directions are, respectively

H~s0

km21{ exp {my { s0 exp {ma

kmy

uy~s0a 1zn km2 1{n y{az

m exp {ma 2

a2{y2 &

z1

mexp {my { exp {ma

'

sxx~szz~{s0aE

km2 1{n 1{exp {my {m exp {ma y

For the BEM analysis of the problem, all domain

integrals were exactly transformed into boundaryintegrals. The results obtained for H, uy, and szz

Fig. 12 Results obtained by the BEM (24 elements) forthe heat source over a portion of the domain(Fig. 11), in comparison with the accurateFEM solution (4929 elements), example 1: (a)temperature, (b) displacement, and (c) stress

Fig. 13 A 2D thermo-elastic problem with multiply-connected domain





17/24

along the line x~z~a=2 are shown in Fig. 16. It can

be seen that the BEM results are in very good

agreement with the exact solutions.

In the second case, a more complicated heat source

function with the following form was considered

s~s0 1z0:1xz0:2x2

exp {myz 87

where s0=k~1 and m~0:2. In this case, no exact

solution exists and the results obtained by the

proposed BEM are compared with the accurate FEM

solutions. The FEM solutions were obtained using a

fine mesh with 64 000 eight-node elements. Since the

heat source function has a quadratic variation in thex

direction, the corresponding domain integrals are

transformed exactly into boundary integrals in the

BEM analysis. The obtained results for H, uy, and szzalong the line x~z~a=2 are shown in Fig. 17.

In the third case, the following heat source func-

tion was considered

s~s0 exp {mffiffiffiffiffiffiffiffi

xyzp 88

where s0=k~1 and m~0:2. Since this function cannot

be expressed with a quadratic variation in anydirection, it must be treated with the method

described in section 6.2. The results obtained for

this case in comparison with the accurate FEM

solutions (64 000 elements) are shown in Fig. 18. It

can be observed that the BEM results are very good.

8 CONCLUSIONS

A BEM formulation for the analysis of steady state

thermo-elastic problems involving arbitrary non-

uniform heat sources was presented. Both 2D and

Fig. 14 Results obtained by the BEM (116 elements) incomparison with the accurate FEM solution(13 512 elements), example 2: (a) temperature,(b) displacement, and (c) stress

Fig. 15 A 3D domain with heat source, discretized byboundary elements





18/24

3D thermo-elastic problems with various kinds of

heat sources can be analysed by the proposed BEM

with boundary-only discretization and there is no

need to define any internal points. It was seen that

the results obtained by the proposed BEM are

accurate and better than those obtained by the

FEM with the same order of discretization. It can

also be claimed that the present formulation is very

efficient from a modelling viewpoint.

Fig. 16 Results obtained by the BEM for the heatsource function given by equation (86), incomparison with the exact solution, example3: (a) temperature, (b) displacement, and (c)stress

Fig. 17 Results obtained by the BEM for the heatsource function given by equation (87), incomparison with the accurate FEM solution,example 3: (a) temperature, (b) displacement,and (c) stress





19/24

ACKNOWLEDGEMENT

The second author would like to acknowledge thefinancial support from the Vice Chancellor of

Research at Shiraz University under grant 88-GR-ENG-54.

F Authors 2010

REFERENCES

1 Burczynski, T. and Dlugosz, A. Evolutionaryoptimization in thermoelastic problems using theboundary element method. Comput. Mech., 2002,28, 317324.

2 Rizzo, F. J. and Shippy, D. J. An advanced boundaryintegral equation method for three-dimensional ther-moelasticity. Int. J. Numer. Methods Engng, 1977, 11,7531768.

3 Karami, G. and Kuhn, G. Implementation of ther-moelastic forces in boundary element analysis ofelastic contact and fracture mechanics problems.Engng Analysis Boundary Elements, 1992, 10, 313322.

4 Cheng, A. H. D., Chen, C. S., Golberg, M. A., andRashed, Y. F. BEM for thermoelasticity andelasticity with body force a revisit. Engng AnalysisBoundary Elements, 2001, 25, 377387.

5 Gao, X. W. Boundary element analysis in thermo-elasticity with and without internal cells. Int. J. Numer. Methods Engng, 2003, 57, 975990.

6 Hematiyan, M. R. Exact transformation of domainintegrals into boundary integrals in BEM formula-tion of 2D and 3D thermoelasticity. In Advances inboundary element techniques VIII(Eds V. Minutoloand M. H. Aliabadi), 2007, pp. 263268 (EC Ltd,Eastleigh, Hampshire, UK).

7 Fazeli, R. and Hematiyan, M. R. An efficientmethod for boundary element analysis of thermo-

elastic problems without domain discretization. InAdvances in boundary element techniques IX(Eds R. Abascal and M. H. Aliabadi), 2008, pp. 4148 (ECLtd, Eastleigh, Hampshire, UK).

8 Aliabadi, M. H. The boundary element method, vol.2,2002 (John Wiley, Chichester, West Sussex, UK).

9 Shiah, Y. C. and Tan, C. L. Exact boundary integraltransformation of the thermoelastic domain inte-gral in BEM for general 2D anisotropic elasticity.Comput. Mech., 1999, 23, 8796.

10 Shiah, Y. C. and Tan, C. L. Determination ofinterior point stresses in two dimensional BEMthermoelastic analysis of anisotropic bodies. Int. J.

Solids Structs., 2003, 37, 809829.11 Shiah, Y. C. and Lin, Y. J. Multiple reciprocity

boundary element analysis of two dimensionalanisotropic thermoelasticity involving an internalarbitrary non-uniform volume heat source. Int. J.Solids Structs., 2003, 40, 65936612.

12 Shiah, Y. C. and Huang, J. H. Boundary elementmethod interior stress/strain analysis for two-dimensional static thermoelastic involving nonuni-form volume heat sources. J. Thermal Stresses,2005, 23, 363390.

13 Ochiai, Y. Three-dimensional steady thermal stressanalysis by triple-reciprocity boundary element

method. Int. J. Numer. Methods Engng, 2005, 63,17411756.

Fig. 18 Results obtained by the BEM for the heatsource function given by equation (88), incomparison with the accurate FEM solution,example 3: (a) temperature, (b) displacement,and (c) stress





20/24

14 Ochiai, Y. Steady heat conduction analysis inorthotropic bodies by triple-reciprocity BEM. Com-put. Modelling Engng Sci., 2001, 2, 435445.

15 Ochiai, Y. Two dimensional unsteady heat con-duction analysis with heat generation by triple-

reciprocity BEM. Int. J. Numer. Methods Engng,2001, 51, 143157.16 Chandra, A. and Mukherjee, S. Boundary element

methods in manufacturing, 1997 (Oxford UniversityPress, New York, USA).

17 Cruse, T. A., Snow, D. W., and Wilson, R. B.Numerical solution in axisymmetric elasticity.Comput. Struct., 1977, 7, 445451.

18 Danson, D. J. A boundary element formulation forproblems in linear isotropic elasticity with body for-ces. In Boundary element methods(Ed. C. A. Brebbia),1981, pp. 105122 (Springer, Berlin, Germany).

19 Pape, D. A. and Banerjee, P. K. Treatment of bodyforces in 2D elastostatic BEM using particular inte-

grals. Trans. ASME, J. Appl. Mech., 1987, 54, 866871.20 Partridge, P. W., Brebbia, C. A., and Wrobel, L. C.

The dual reciprocity boundary element method, 1992(Computational Mechanics Publications, South-ampton, UK).

21 Nowak, A. J. and Brebbia, C. A. The multiple-reciprocity method, a new approach for transform-ing BEM domain integrals to the boundary. Engng Analysis Boundary Elements, 1989, 6, 164167.

22 Gao, X. W. The radial integration method forevaluation of domain integrals with boundary-onlydiscretization. Engng Analysis Boundary Elements,2002, 26, 905916.

23 Lai, W. M., Rubin, D., and Kremple, E. Introduc-tion to continuum mechanics, third edition, 1999(Butterworth-Heinemann, USA).

24 Hematiyan, M. R. Exact transformation of a widevariety of domain integrals into boundary integralsin boundary element method. Communs. Numer.Methods Engng, 2008, 24, 14971521.

25 Hematiyan, M. R. A general method for evaluationof 2D and 3D domain integrals without domaindiscretization and its application in BEM. Comput.Mech., 2007, 39, 509520.

26 Hematiyan, M. R., Mohammadi, M., Fazeli, R.,and Khosravifard, A. Cartesian transformationmethod for evaluation of regular and singular

domain integrals in boundary element methodand other mesh reduction methods. In Advances inboundary element techniques IX (Eds R. Abascaland M. H. Aliabadi), 2008, pp. 93100 (EC Ltd,Eastleigh, Hampshire).

27 Khosravifard, A. and Hematiyan, M. R. A newmethod for meshless integration in 2D and 3DGalerkin meshfree methods. Engng Analysis Bound-ary Elements, 2010, 34, 3040.

28 Brebbia, C. A., Telles, J. C. F., and Wrobel, L. C.Boundary element techniques theory and applica-tions in engineering, 1984 (Springer-Verlag, Berlin,Germany).

29 Paris, F. and Canas, J. Boundary element method,1997 (Oxford University Press, New York, USA).

30 Noda, N., Hetnarski, R. B., and Tanigawa, Y.Thermal stresses, second edition, 2003 (Taylor andFrancis, New York, USA).

31 Liu, G. R. and Gu, Y. T. An introduction to meshfreemethods and their programming, 2005 (Springer,

Dordrecht, The Netherlands).32 Lancaster, P. and Salkauskas, K. Surface generated

by moving least squares methods. Math. Comput.,1981, 37, 141158.

33 Shepard, D. A two dimensional function for irregu-larly spaced data. In Proceedings of theACM NationalConference, New York, USA, 1968, pp. 517524.

34 Micchelli, C. A. Interpolation of scattered data.Constructive Approximation, 1986, 2, 1222.

35 Welch, W. and Witkin, A. Variational surface model-ing. Comput. Graphics, 1992, 26, 157166.

36 Ochiai, Y. and Seliya, T. Generation of free-formsurface in CAD for dies. Adv. Engng Software, 1995,

22, 113118.

APPENDIX 1

In this appendix, the Laplacian ofvi is evaluated.

2D problems

vi~{L

Lxir2 ln r

~{ri 2 ln rz1 89

vi,k~{dik 2 ln rz1 {ri 2r,kr

90

+2vi~vi,kk~{dik

2r,kr

{dik

2r,kr

{2rir,kkr{r,kr,k

r2

91

Noting that r,kr,k~1 and r,kk~1=r, the following

result is obtained

+2vi~{4r,i=r 92

3D problems

vi~L

Lxir ~r,i 93

+2vi~+2r,i~ +2r

,i 94





21/24

Noting that +2r~2=r, the following result is obtained

+2vi~

{2r,ir2

95

APPENDIX 2

In this appendix, the expression for Uik,kj is found.

2D problems


r,ir

96

Using equation (92) the following equation isobtained

Uik,k~1{2n

16pG 1{n +2vi 97

Substitutingvi~{ r2 ln r

,i

from equation (89) into

equation (97) leads to

Uik,k~{ 1{2n

16pG 1{n +2 r2 ln r

,i

h i98

and therefore

Uik,kj~{ 1{2n

16pG 1{n +2 r2 ln r

,ij

h i99

3D problems


r,ir2

100

Using equation (95), the following equation isobtained

Uik,k~1{2n

16pG 1{n +2vi 101

Substitutingvi~r,i from equation (93) into equation

(101) the following result is obtained

Uik,k~1{2n

16pG 1{n +2 r,i 102

and therefore

Uik,kj~1{2n

16pG 1{n +2 r,ij 103

APPENDIX 3

For the exact transformation of the domain integrals

into boundary integrals by the CTM, the integralsxc1H

dx1,

xc1vi dx1, and

xc1lij dx1 (c~0, 1, 2) must

be evaluated analytically. All possible integrals in 2D

and 3D thermo-elastic problems are given as follows.

2D problem

I1~

ln

1

r

dx1~{r1 ln r{r2 tan{1 r1

r2

zr1

I2~

x1 ln

1

r

dx1~{

1

2r2z2j1r1

ln r

{j1r2 tan{1 r1

r2

z

1

4r2zj1r1

I3~

x21 ln

1

r

dx1~{

1

3r31 zj1r

2zj21r1

ln r

z1

3r3

2{j2

1r2 tan{1 r1

r2

{1

3r1r

22 z

1

9r31 z

1

2j1r

2zj21r1

I4~

r1 ln

1

r

dx1~{

1

2r2 ln rz

1

4r2

I5~

x1r1 ln

1

r

dx1~{

1

3r31 z

1

2j1r

2

ln r

z1

3

r32 tan{1 r1

r2 z

1

9

r31

{1

3r1r

22 z

1

4j1r

2

I6~

x21r1 ln

1

r

dx1

~{1

4r41 {r

42 z

8

3j1r

31 z2j

21r

2

ln r

z2

3j1r

32 tan

{1 r1r2

z

1

16r41

{ 18

r21 r22 z 29j1r31 { 23

j1r1r22 z 14j21r2





22/24

I7~

r2 ln

1

r

dx1~r2I1

I8~

x1r2 ln1r

dx1~r2I2

I9~

x21r2 ln

1

r

dx1~r2I3

I10~

r1 dx1~

1

2r21

I11~

x1r1 dx1~ 13

r31 z 12j1r21

I12~

x21r1 dx1~

1

4r41 z

2

3j1r

31 z

1

2j21r

21

I13~

r2 dx1~x1r2

I14~

x1r2 dx1~1

2x2

1r2

I15~

x21r2 dx1~

1

3x31r2

I16~

r,1r,2 dx1~

r1r2r2

dx1~r2 ln r

I17~ x1r,1r,2 dx1~ x1r1r2

r2

dx1

~r2 j1 ln r{r2 tan{1 r1

r2

zr1

!

I18~

x21r,1r,2 dx1~

x21

r1r2r2

dx1

~r2 j21{r

22

ln r{2j1r2 tan

{1 r1r2

z

1

2r21z2j1r1

!

I19~

r,1r,1 dx1~r21

r2 dx1~{r2 tan{1 r1

r2

zr1

I20~

x1r,1r,1 dx1~

x1

r21r2

dx1

~{r22 ln r{j1r2 tan{1 r1

r2 z

1

2

r21 zj1r1

I21~

x21r,1r,1 dx1~

x21

r21r2

dx1

~{2j1r22 ln r{ j

21r2{r

32

tan{1

r1r2

z1

3r31 zj1r

21 zj

21r1{r1r

22

I22~

r,2r,2 dx1~

r2

2r2

dx1~r2 tan{1 r1r2

I23~

x1r,2r,2 dx1~

x1

r22r2

dx1

~r22 ln rzj1r2 tan{1 r1

r2

I24~ x21r,2r,2 dx1~ x21r22

r2dx1

~2j1r22 ln rz j

21r2{r

32

tan{1

r1r2

zr1r

22

3D problem

V1~

1

rdx1~ ln r1zr

V2~

x1

1

rdx1~j1 ln r1zr zr

V3~

x21

1

rdx1~

1

22j21{r

2zr21

ln r1zr

zr1

2r1z2j1

V4~

r,1 dx1~r1

r dx1~r





23/24

V5~

x1r,1 dx1~

x1

r1r

dx1

~{1

2

r2{r21 ln r1zr z

1

2

r r1z2j1

V6~

x21r,1 dx1~

x21

r1r

dx1

~{j1 r2

{r21

ln r1zr

z1

3r 3r21 {2r

2z3j1r1z3j

21

V7~ r,2 dx1~ r2r

dx1~r2V1

V8~

x1r,2 dx1~

x1

r2r

dx1~r2V2

V9~

x21r,2 dx1~

x21

r2

rdx1~r2V3

V10~

r,3 dx1~

r3r

dx1~r3V1

V11~

x1r,3 dx1~

x1

r3r

dx1~r3V2

V12~

x21r,3 dx1~

x21

r3r

dx1~r3V3

V13~

r,1r,1

rdx1~

r21r3

dx1~ ln r1zr {r,1

V14~

x1

r,1r,1r

dx1~

x1

r21r3

dx1

~j1 ln r1zr z2r{r1r,1{j1r,1

V15~

x21

r,1r,1r

dx1~

x21

r21r3

dx1

~1

23r21 {3r

2z2j21

ln r1zr

z

1

2 r,1 3r2{2r21 {2j21

z2j1 2r{r1r,1

V16~

r,1r2

dx1~

r1r3

dx1~{1

r

V17~

x1

r,1r2

dx1~

x1

r1r3

dx1~ ln r1zr {x1r

V18~

x21

r,1r2

dx1~

x21

r1r3

dx1

~2j1 ln r1zr z2r{x21

r

V19

~ r,1r,2r

dx1

~ r1r2r3

dx1

~r2

V16

V20~

x1

r,1r,2r

dx1~

x1

r1r2r3

dx1~r2V17

V21~

x21

r,1r,2

rdx1~

x21

r1r2

r3dx1~r2V18

V22~r,1r,3

r dx1~

r1r3r3 dx

1~r3V16

V23~

x1

r,1r,3

rdx1~

x1

r1r3

r3dx1~r3V17

V24~

x21

r,1r,3r

dx1~

x21

r1r3r3

dx1~r3V18

V25~ 1r3

dx1~r,1

r2{r21

V26~

x1

1

r3dx1~

j1r,1

r2{r21{

1

r

V27~

x21

1

r3dx1~ ln r1zr {r,1z j

21r,1

r2{r21{2

j1

r

V28~ r,2r,2

r dx1~ r

22

r3 dx1~

r

2

2 V25





24/24

V29~

x1

r,2r,2r

dx1~

x1

r22r3

dx1~r22 V26

V30~

x21r,2r,2

rdx1~

x21

r22r3

dx1~r22 V27

V31~

r,3r,3

rdx1~

r23r3

dx1~r23 V25

V32~

x1

r,3r,3r

dx1~

x1

r23r3

dx1~r23 V26

V33~

x21

r,3r,3r

dx1~

x21

r23r3

dx1~r23 V27

V34~

r,2r,3r

dx1~

r2r3r3

dx1~r2r3V25

V35~

x1

r,2r,3

rdx1~

x1

r2r3

r3dx1~r2r3V26

V36~

x21

r,2r,3r

dx1~

x21

r2r3r3

dx1~r2r3V27



boundary element analysis of thermo-elastic problems with non-uniform heat sources

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