3D Transient Heat Transfer by Conduction and … · 3D Transient Heat Transfer by Conduction and...

Copyright c© 2005 Tech Science Press CMES, vol.9, no.3, pp.221-233, 2005

3D Transient Heat Transfer by Conduction and Convection across a 2D Mediumusing a Boundary Element Model

N. Simoes 1,2 and A. Tadeu 2

Abstract: The use of the Boundary Element Method(BEM) to formulate the 3D transient heat trans-fer through cylindrical structures with irregular cross-sections, bounded by a homogeneous elastic medium,is described in this paper. In this formulation, both theconduction and the convection phenomena are modeled.This system can be subjected to heat emitted by eitherpoint or line sources located somewhere in the media.The solution is first obtained in the frequency domainfor a wide range of frequencies and axial wavenumbers.Time domain responses are later calculated by means of(fast) inverse Fourier transforms into space-time. The ap-propriate fundamental solution (Green’s functions) em-ployed in this BEM model takes the convection phe-nomenon into account. The model is implemented andvalidated by comparing it with analytical solutions for afilled cylindrical circular ring core placed in an infinitemedium and subjected to a heat line source.

keyword: Transient heat transfer, conduction, convec-tion, Boundary Element Method, Green’s functions.

1 Introduction

Most heat transfer problems involve heterogeneous el-ements and unsteady exchanges of energy between dif-ferent media. Formulations for studying those systemsshould therefore contemplate transient heat phenomena.Several numerical approaches have been developed tostudy heat transfer. These include the Finite Elements[e.g. Bathe (1976)], the Finite Differences, the BoundaryElements Method [Brebbia et al. (1984), Pina and Fer-nandez (1984)] and Meshless techniques [Lin and Atluri(2000), Atluri and Shen (2002), Sladek et al. (2004)].Of these techniques, the BEM only requires the dis-cretization of the material boundaries, while it also takesinto account the radiation conditions in the far field,

1 [email protected] Department of Civil Engineering, University of Coimbra, Pinhalde Marrocos, 3030-290 Coimbra, Portugal

which makes the description of the region quite compact.The Finite Elements and the Finite Differences Methods,however, need the full discretization of the domain beingstudied. The BEM is therefore an efficient method sincefewer equations are needed. Furthermore, it is probablythe method best suited to analyzing problems involvinginfinite or semi-infinite domains, since it automaticallysatisfies the far field conditions. Boundary Element mod-els require the prior knowledge of Green’s functions: thereference works by Carslaw and Jaeger (1959), Ozisik(1993) and Beck (1992) contain compilations of certainGreen’s functions and their applicability.

Most of the known schemes devised to solve transientdiffusion heat problems have either been formulated inthe time domain (“time-marching” approach) or else theyuse Laplace transforms. The BEM has been employed inboth techniques. In the “time-marching” approach, theBEM can be used to compute the solution directly in thetime domain, step by step, at successive time increments.Chang et al. (1973) described the first time-domain di-rect boundary integral method to study planar transientheat conduction. Models involving time-dependent so-lutions were also proposed by: Shaw (1974), to dealwith the heat diffusion in 3D bodies; Wrobel and Breb-bia (1981), to study axisymmetric diffusion problems;Dargush and Banerjee (1991), to model planar, three-dimensional and axisymmetric analyses and Lesnic et al.(1995) to study unsteady diffusion equation in both oneand two dimensions by a time marching BEM model,taking into account the treatment of singularities.

A disadvantage of the “time-marching” techniques is thattheir solutions may be unstable. An alternative is to applya Laplace transform to move the solution from the timedomain to a transformed variable. However, this processrequires an inverse transform to find the solution in thetime domain. Rizzo and Shippy (1970) used a Laplacetransform associated with a boundary integral represen-tation for transient heat conduction analysis. Since then,several authors have applied Laplace transforms to vari-

222 Copyright c© 2005 Tech Science Press CMES, vol.9, no.3, pp.221-233, 2005

ous diffusion problems, including Cheng et al. (1992),Zhu et al. (1994) and Sutradhar et al. (2002). Thelast author employed a Laplace transform BEM approachin a 3D transient heat conduction problem, consideringthat the thermal conductivity and the specific heat couldchange exponentially in one coordinate.

One drawback of using Laplace transforms is the lossof accuracy in the inversion process, which amplifiessmall truncation errors. Stehfest (1970) has proposed amore stable algorithm to overcome this problem. A timeFourier transform scheme has been suggested by the au-thors [Tadeu et al. (2004)] to deal with the time variablein the transient heat conduction equation. This approachallows the calculation to be made in the frequency do-main.

In the work described here, the time Fourier Transformis again used to compute the transient heat transfer whencylindrical inclusions of infinite length are located insidea homogeneous elastic medium. However, the convec-tion phenomenon is also considered in this formulation.

The rest of this paper is organized as follows: first, thefundamental equations of the three-dimensional heat dif-fusion problems are described; then, the main integralsrequired to solve the BEM are indicated, including thenecessary Green’s functions; next, the frequency heat re-sponses through a cylindrical circular ring core are ob-tained using the BEM model and compared with analyt-ical solutions; the BEM model is then used to simulatethe heat propagation generated by a heat point sourceplaced in the vicinity of an inclusion with an irregularcross-section, which encloses a circular cylinder.

2 Three-dimensional problem formulation

The transient heat transfer in a homogeneous, isotropicbody, involving the conduction and convection phenom-ena, can be expressed by the diffusion equation below,when constant velocities, Vx, Vy and Vz, are assumedalong the domain in the x, y and z directions respectively,

∇2T − 1K

(Vx

∂T∂x

+Vy∂T∂y

+Vz∂T∂z

)=

1K

∂T∂t

(1)

in which ∇2 =(

∂2

∂x2 + ∂2

∂y2 + ∂2

∂z2

), t is time, T (t, x, y, z)is

temperature, K = k/(ρ c)is the thermal diffusivity, kis

the thermal conductivity, ρ is the density and c is the spe-cific heat. A Fourier transformation in the time domain

applied to eq. (1) gives the equation below, expressed inthe frequency domain⎛⎝∇2 − 1

K

(Vx

∂∂x

+Vy∂∂y

+Vz∂∂z

)+

(√−iω

K

)2⎞⎠

T (ω,x,y, z) = 0 (2)

where i =√−1 and ω is the frequency. Eq. (2) differs

from the Helmholtz equation by the presence of a con-vective term. For a heat point source applied at (0,0, 0)in an unbounded medium, of the form p(ω,x,y, z, t) =δ(x)δ(y)δ(z)ei(ωt), where δ(y) and δ(z) are Dirac-deltafunctions, the fundamental solution of eq. (2) can be ex-pressed as

Tf (ω,x,y, z)

=e

Vxx+Vyy+Vzz2K

2k√

x2 +y2 + z2e−i√−V2

x +V 2y +V 2

z4K2 − iω

K

√x2+y2+z2

(3)

When the geometry of the problem does not changealong one direction (z) the full 3D problem can be ex-pressed as a summation of simpler 2D solutions. This re-quires the application of a Fourier transformation alongthat direction, which can be expressed as a summa-tion of 2D solutions with different spatial wavenumberskz[Tadeu & Kausel (2000)].

The application of a spatial Fourier transformation to

e−i

√−V2

x +V2y +V2

z4K2 − iω

K

√x2+y2+z2

√x2+y2+z2

along the z direction, leads to

Tf (ω,x,y,kz) =−i e

Vxx+Vyy+Vzz2K

4k

H0

⎛⎝√−V 2

x +V 2y +V 2

z

4K2 − iωK

− (kz)2 r0

⎞⎠

(4)

where H0 are second-kind Hankel functions of the order0, and r0 =

√x2 +y2.

This response is related to a spatially varying heat linesource of the type p(ω,x,y,kz, t) = δ(x)δ(y)ei(ωt−kzz)

(see 1).

The full three-dimensional solution can be syn-thesized by applying an inverse Fourier trans-form along the kz domain to the expression

3D Transient Heat Transfer 223

x

yz

(0,0,0)

Figure 1 : Spatially harmonic varying line load

−i2 H0

(√−V 2

x +V 2y +V 2

z

4K2 − iωK − (kz)

2 r0

). If we as-

sume the existence of virtual sources, equally spacedsufficiently far apart, this inverse Fourier transformationcan be formulated as a discrete summation, whichenables the solution to be obtained by solving a limitednumber of two-dimensional problems,

T (ω,x,y, z) =2πL

eVxx+Vyy+Vzz

2K

2k√

x2 +y2 + z2

M

∑m=−M

H0

⎛⎝√

−V 2x +V 2

y +V 2z

4K2 − iωK

− (kzm)2 r0

⎞⎠e−ikzmz

(5)

where kzm is the axial wavenumber given bykzm = 2πLz

m.The distance Lz must be large enough to prevent spatialcontamination from the virtual sources [Bouchon & Aki(1977)]. An analogous approach has been used by Tadeuet al (2002) and Godinho et al (2001) to solve problemsof wave propagation.

The fundamental solution of the differential equation ob-tained from eq. (2) after the application of a spatialFourier transformation along the z direction is eq. (4)with Vz = 0.⎛⎝∇2 − 1

K

(Vx

∂∂x

+Vy∂∂y

)+

(√−iω

K− (kz)

2

)2⎞⎠

T (ω,x,y,kz) = 0 (6)

with ∇2 =(

∂2

∂x2 + ∂2

∂y2

).

The frequency responses may need to be computed from0.0Hz up to very high frequencies. However, since theheat responses decay very fast as the frequency increases,it allows us to limit the upper frequency where the solu-tion is required. The use of complex frequencies leads

to arguments for the Hankel function different from zero(ωc =−iη for 0.0Hz), when the frequency is zero, whichallows the calculation of the static response.

The heat responses in the spatial-temporal domain arethen obtained by means of an inverse fast Fourier trans-form in kz and in the frequency domain. In order to pre-vent the aliasing phenomena, complex frequencies witha small imaginary part of the form ωc = ω − iη (withη = 0.7∆ω, and ∆ω being the frequency step) are usedin the computation procedure. The constant η cannot bemade arbitrarily large, since this leads to severe loss ofnumerical precision in the evaluation of the exponentialwindows [see Kausel and Roesset (1992)]. The time evo-lution of the heat source amplitude can be easily changed.

3 Boundary Element Method formulation

The fundamental BEM equations can be found in Wro-bel (1981), where they are described in detail. The BEMcan be used to solve eq. (6) for each value of kz, corre-sponding to individual 2D problems.

The boundary integral equations for a homogeneousisotropic medium layer that is embedded in an infinitemedium and contains a cylindrical body (bounded by asurface S), when this system is subjected to an incidentheat field given by Tinc, are expressed as follows,

along the exterior domain

p T (ext)(x0,y0,kz,ω)

=Z

S

q(ext)(x,y,ηn,kz,ω)G(ext)(x,y,x0,y0,kz,ω)ds

−Z

S

H(ext)(x,y,ηn,x0,y0,kz,ω)T (ext)(x,y,kz,ω)ds

−Z

S

G(ext)(x,y,x0,y0,kz,ω)T (ext)(x,y,kz,ω)V (ext)n ds

+ Tinc(x0,y0,kz,ω) (7)

along the interior domain

p T (int)(x0,y0,kz,ω)

=Z

S

q(int)(x,y,ηn,kz,ω)G(int)(x,y,x0,y0,kz,ω)ds

−Z

S

H(int)(x,y,ηn,x0,y0,kz,ω)T (int)(x,y,kz,ω)ds

−Z

S

G(int)(x,y,x0,y0,kz,ω)T (int)(x,y,kz,ω)V (int)n ds (8)


These boundary integral equations incorporate a term re-lated to the convection phenomenon: Vn = Vxnx +Vyny.In eqs. (7) and (8), the interior and exterior domains areidentified by the superscripts int and ext respectively, νn

is the unit outward normal along the boundary, G and Hare respectively the fundamental solutions (Green’s func-tions) for the temperature (T ) and heat flux (q), at (x,y)due to a virtual point heat load applied at (x0,y0), p isa constant defined by the shape of the boundary, with avalue of 1/2 if (x0,y0) ∈ S when the boundary is smooth.Note that this formulation assumes initial conditions ofnull temperatures and null heat fluxes throughout the do-main. Other initial conditions would require the evalua-tion of the surface integrals.

If the boundary is discretized into N straight boundaryelements, with one nodal point in the middle of each ele-ment, eqs. (7) and (8) take the form,

along the exterior domain

N∑

l=1q(ext) klG(ext) kl −

N∑

l=1T (ext) klH(ext) kl−

N∑

l=1G(ext) kl T (ext) klV (ext)

n + T kincl

= cklT(ext) k

l

(9)

along the interior domain

N

∑l=1

q(int) klG(int) kl −N

∑l=1

T (int) klH(int) kl

−N

∑l=1

G(int) kl T (int) klV (int)n = cklT

(int) kl (10)

where q(ext) kll and T (ext) kl

l are the nodal heat fluxes and

temperatures in the exterior domain, q(int) kll and T (int) kl

lare the nodal heat fluxes and temperatures in the interiordomain,

H(ext)kll =

Z

Cl

H(ext) (ω,xl,yl,ηl,xk,yk,kz) dCl

H(int)kll =

Z

Cl

H(int) (ω,xl,yl,ηl,xk,yk,kz) dCl

G(ext)kll =

Z

Cl

G(ext) (ω,xl,yl,xk,yk,kz) dCl

G(int)kll =

Z

Cl

G(int) (ω,xl,yl,xk,yk,kz) dCl

where ηl is the unit outward normal for the lth

boundary segment Cl. In equations (9) and (10),H(ext) (ω,xl,yl,ηl,xk,yk,kz) and G(ext) (ω,xl,yl,xk,yk,kz)are respectively the Green’s functions for the heat fluxesand temperature components in the exterior mediumof the inclusion, while H(int) (ω,xl,yl,ηl,xk,yk,kz) andG(int) (ω,xl,yl,xk,yk,kz) are respectively the Green’sfunctions for the heat fluxes and temperature componentsin the interior medium of the inclusion, at point(xl,yl),caused by a concentrated heat load acting at the sourcepoint (xk,yk). If the loaded element coincides with the el-ement being integrated, the factor ckl takes the value1/2.

The two-and-a-half dimensional Green’s functions fortemperature and heat fluxes in Cartesian co-ordinates arethose for an unbounded medium,

G(x,y,x0,y0,kz,ω) =−i4k

eVr02K H0 (ktr r)

H (ω,xl,yl,ηl,xk,yk,kz) =−i4

eVr02K[

V2K

(∂r0

∂ηl

)H0 (ktr r)−ktrH1 (ktr r)

(∂r∂ηl

)](11)

where V is the radial convection velocity, ktr =√−V 2

4K2 + −iωK − (kz)

2, r =√

(xl −xk)2 +(yl −yk)

2, r0 =√x2 +y2 is the distance to the convection source posi-

tion, and Hn are Hankel functions of the second kind andorder n.

If the element to be integrated is not the loaded element,the integrations in equations (9) and (10) are evaluatedusing a Gaussian quadrature scheme, while for the loadedelement, the existing singular integrands in the sourceterms of the Green’s functions are calculated in closedform [Tadeu (1999)].

The final system of equations is assembled assuring thecontinuity of temperatures and heat fluxes along theboundary of the inclusion. The unknown nodal tempera-tures and heat fluxes are obtained by solving this systemof equations, allowing the heat field along the domain tobe defined.

The final integral equations are manipulated and com-bined so as to impose the continuity of temperatures andheat fluxes along the boundary of the inclusion, and asystem of equations is assembled. The solution of thissystem of equations gives the nodal temperatures andheat fluxes, which allow the reflected heat field to be de-fined.


A numerical inverse fast Fourier transform in kz and fre-quency domain is then used to compute the temperaturefield in the spatial-temporal domain.

4 Bem validation

The BEM algorithm was implemented and validated byapplying it to a filled cylindrical circular ring core (seefigure 2), subjected to a harmonic heat line source appliedat point O (x0,y0), for which the solution is known inclosed form and described in Appendix A. The continuityof normal heat fluxes and temperatures at both interfacesalong the material interfaces (a = 0.5m and b = 1.0m)are assumed. The thermal properties and the convectionvelocities allowed at each of the three media are listed infigure 1. The reflected heat responses were calculated inthe frequency domain from 0Hz to 64×10−7 Hz, with afrequency increment of ∆ω = 1×10−7 Hz and consider-ing a single value of the parameter kz equal to 0.4rad / m.The simulated system is heated by a harmonic line sourcelocated in the inner medium of the circular ring core(x = −0.3m,y = 0.0m).

Figure 2 : Circular ring core geometry

Fig. 3 shows the real and imaginary parts of the re-sponses at receivers placed in each medium: Rec. 1 (x =0.2m,y = 0.2m), in medium 3; Rec. 2 (x = 0.2m,y =0.7m), in medium 2; and Rec. 3 (x = 0.2m,y = 1.2m),in medium 1. The solid lines represent the analyticalresponses, while the marked points correspond to theBEM solution, computed using 100 constant boundaryelements. The round and the triangle marks designate thereal and imaginary parts of the responses, respectively.

The results obtained by the two formulations are in veryclose agreement. Very good results were also obtainedfor heat sources and receivers placed at different posi-tions.

5 Applications

The BEM formulation has been used to study the heatpropagation in a system in which a hollow square(medium 2) is buried in an unbounded water medium(medium 1), and the square, 0.5 m length, contains asolid cylinder, 0.15 m of radius, which is filled with oil(medium 3), as illustrated in Figure 4. The inclusionshave been modeled using a total of 500 straight bound-ary elements.

Two different cases have been considered ascribing dif-ferent properties to the medium 2. The hollow square(containing the cylindrical ring core) (medium 2) is de-fined as being made of steel (Case 1) or concrete (Case2), for which the thermal properties are listed in the figure2. The thermal conductivity (k1 = 0.606W.m - 1.◦C−1),

the density (ρ1 = 998.0Kg.m - 3) and the specific heat

(c1 = 4181.0 J.Kg - 1.◦C−1) of the host medium (wa-ter) are kept constant in all the analyses and it is as-sumed that they do not vary with temperature. Thethermal properties of the oil (medium 3) are assumedto remain constant with temperature variations, allow-ing k3 = 0.145W.m - 1.◦C−1, ρ3 = 887.1 Kg.m - 3 andc3 = 1888.5 J.Kg - 1.◦C−1.

This system is subjected to a heat point source, placedin the host medium at (x0 = −0.15 m, y0 = −0.1 m,z0 = 0.0 m), which starts emitting energy at t ≈ 0.76 h.Its power is increased linearly from 0.0 W to 1000.0 W,reaching maximum power at t ≈ 3.46 h. This peak ismaintained for a period of t ≈ 2.72 h. The power is thenreduced linearly to 0.0 W, which occurs at t ≈ 8.89 h.The heat field is computed for several grids of receiversin the frequency range of

[0.0, 128.0e−5Hz

], with a fre-

quency increment of 0.5e−5 Hz, giving a time period of55.56 h. The spatial intervals between virtual sources aregiven by the higher value of L, depending on the mate-

rial’s thermal properties, L = (2√

ki/(ρi ci∆ f )+2).

Next, two situations are presented: the modeling of boththe conduction and convection phenomena, or the con-duction phenomenon alone. The convection velocity (inthe y direction) affecting the water medium may be 2.0×


Table 1 : Material thermal properties

Medium 1 Medium 2 Medium 3

Thermal conductivity(

W.m - 1.oC−1)

k1 = 0.12 k2 = 0.72 k0 = 0.12

Density(

Kg.m - 3)

ρ1 = 1380.0 ρ2 = 780.0 ρ0 = 1380 .0

Specific heat(

J.Kg - 1.oC−1)

c1 = 510.0 c2 = 1860.0 c0 = 510.0

Convection velocity (m / s) V1 = 1×10−6 V2 = 2×10−6 V3 = 1×10−6

Table 2 : Thermal properties of medium 2

Medium 2 Thermal conductivity,k [W.m - 1.oC−1]

Specific heat, c [J.Kg - 1.oC−1] Density, ρ [Kg.m - 3]

Steel 63.9 434.0 7832.0Concrete 1.4 880.0 2300.0

10−6 m / s, while for the oil a velocity of 1×10−6 m / s isallowed.

Table 3 : Coordinates of receivers Rec. 1 to Rec. 5,placed at z = 0.0m and z = 0.5m

Receiver x (m) y (m)Rec. 1 −0.1835 0.2563Rec. 2 −0.1835 0.5601Rec. 3 0.0063 0.2563Rec. 4 0.1835 0.4335Rec. 5 0.1835 0.5601

Figures 5 and 6 illustrate the temperature evolutionrecorded at receivers Rec. 1 to Rec. 5, located atz = 0.0m and z = 0.5m, respectively. The receivers areplaced inside the inclusions and also in the host medium,as shown in Figure 4 and listed in Table 3.

Since null initial temperatures and heat fluxes are the ini-tial conditions prescribed for the full domain, all the re-ceivers exhibit null temperatures in the initial momentsof the time responses. After the heat source has startedemitting energy, the receiver closest to the heat source(Rec. 1), on plane z = 0.0m, is the first to register achange of temperature. This same receiver also recordsa higher maximum temperature during the time domainunder study than the rest of the receivers, for both cases.For Case 1, receiver Rec. 1 reaches almost 47.0 ◦C, whilea maximum of 130.0 ◦C is registered when Case 2 is com-

puted. At z = 0.5m, when Case 1 is modeled, receiverRec. 1 records a higher maximum temperature (23.2 ◦C)than the same receiver does in Case 2 (15.9 ◦C). For Case1, the amplitude decreases 23.8 ◦C between the planesz = 0.0m and z = 0.5m, while the difference is 114.1 ◦Cfor the Case 2. We may conclude that in the presence ofthe hollow concrete inclusion, the energy does not prop-agate so easily in the z direction, as it does when the in-clusion is made of steel. So, the amplitude differencesbetween the two cases, found at z = 0.0m, result fromheat accumulation in the source plane (z = 0.0m), whenthe inclusion medium presents a significantly lower ther-mal diffusivity.

At receiver Rec. 5, which is placed further away fromthe heat source, the temperature increase smoothly andits maximum is reached at later times (when the sourcepower is no longer emitting energy), which means thatthe energy is still propagating to the zones with lowertemperatures in order to achieve energy equilibrium.

The distances from the source to the receivers Rec. 3 andRec. 4, located at z = 0.0m, are 0.389m and 0.629m,respectively. Receiver Rec. 3 exhibits a far higher maxi-mum temperature than Rec. 4 in Case 2 (concrete mate-rial). When steel is used, however, these receivers regis-ter similar maximum temperatures. We may also observethat although receivers Rec. 2 and Rec. 4, located atz = 0.0m, are almost the same distance from the source,Rec. 4 registers a higher maximum temperature soonerthan Rec. 2, in Case 1.


-0.10

-0.05

0

0.05

0.10

0 2x10-6 4x10-6 6x10-6

Frequency (Hz)

Am

plit

ude

a)

-0.2

0

0.2

0.4

0 2x10-6 4x10-6 6x10-6

Frequency (Hz)

Am

plit

ud

e

b)

-0.2

0

0.2

0.4

0 2x10-6 4x10-6 6x10-6

Frequency (Hz)

Am

plit

ud

e

c)

Figure 3 : Real and imaginary parts of the heat responseswhen a heat source is placed in medium 3 (x = −0.3m,y = 0.0m): a) Receiver Rec. 1; b) Receiver Rec. 2; c)Receiver Rec. 3

x

y

O (-0.15,-0.1,0.0)

Rec. 1

1 Medium

2 3

Rec. 2

Rec. 3

Rec. 4

Rec. 5

0.15 m

(0.25,0.0) (-0.25,0.0)

Figure 4 : Geometry of the problem and position of aset of receivers (Receivers Rec. 1 to Rec. 5) placed atz = 0.0m and z = 0.5m

The factors involved in these dissimilar responses are re-lated to the thermal diffusivity differences between thethree media and also to the receivers’ distance from theheat source. When no convection phenomenon is mod-eled, lower temperatures are recorded by all the receivers,as shown in figure 5. The presence, direction and ampli-tude of the convection velocity may have a considerableinfluence on the heat responses.

In figure 6 a set of snapshots, obtained at t = 10 hand t = 20 h, displays the temperature field for bothcases, the hollow square steel and concrete structures.The results are presented using two-dimensional contourfigures (transversal and longitudinal grid of receivers).These last illustrations are obtained from a transversalgrid of receivers (80×80) placed at z = 0.0 m and from alongitudinal grid of receivers, distributed in the x z plane,crossing the hollow inclusion at y = 0.25 m.

At z = 0.0m and time t = 10 h it is interesting to note thatround isothermals can be found within the oil domainfor Case 1 (see figure 6a)), which means that the heatpropagates faster around the steel structure than throughthe oil fluid. The temperature around the full circularfluid medium is similar and the heat propagation processfor this fluid medium is from the boundary to the centre.However, when the hollow square is made of concrete itcan be observed that the temperature distribution mostlydepends on the distance between each point and the heatsource, since the thermal diffusivity difference betweenmedia 2 and 3 is less than for the steel material. The re-ceivers placed in the concrete medium, behind the cylin-drical circular fluid medium (opposite to the source side),


Case 1 Case 2

0 10 20 30 40 500

15

30

45

60

Time (h)

Tem

pera

ture

(

篶

)

Rec. 1Rec. 2Rec. 3Rec. 4Rec. 5

0 10 20 30 40 50

0

50

100

150

200

Time (h)

Tem

pera

ture

(

篶

)


a)

0 10 20 30 40 500

10

20

30

Time (h)

Tem

pera

ture

(

篶

)


0 10 20 30 40 50

0

10

20

30

40

Time (h)

Tem

pera

ture

(

篶

)


b)

Figure 5 : Heat curves registered for Cases 1 and 2 when both conduction and convection phenomena are assumed:a) Receivers Rec. 1 to Rec. 5, placed at z = 0.0m; b) Receivers Rec. 1 to Rec. 5, placed at z = 0.5m

Case 1 Case 2

0 10 20 30 40 500

15

30

45

60

Time (h)

Tem

pera

ture

(

篶

)


0 10 20 30 40 50

0

50

100

150

200

Time (h)

Tem

pera

ture

(

篶

)


Figure 6 : Heat curves registered for Cases 1 and 2 when the conduction phenomenon is assumed alone: ReceiversRec. 1 to Rec. 5, placed at z = 0.0m


0.0 mz = 0.25 my =

-0.5 -0.25 0 0.25 0.5-0.25

0

0.25

0.5

0.75

0.05

1

50

0.25

5

1

1000

0.52.5

10

25

5

100

300

y(m

)

x(m) -0.5 -0.25 0 0.25 0.50

0.25

0.5

0.75

1

2.5

1

1

5

0.25

0.2

5

0.5

10

10

2.5

0.5

2 .5

0.5

0.05

10 .0 50.2

5

25

5

x(m)

z(m

)

a)

-0.5 -0.25 0 0.25 0.5-0.25

0

0.25

0.5

0.755

100

300

25

0.5

400200

50

10

2.51

y(m

)

x(m) -0.5 -0.25 0 0.25 0.50

0.25

0.5

0.75

1

2.5

5

100.

5

1

25

5

51

10

2. 5

0. 25

0.5

25

10

x(m)

z(m

)

b)

-0.5 -0.25 0 0.25 0.5-0.25

0

0.25

0.5

0.75

1000

5

2.5100

0.5

50500

25

1

0.25

10

0.05

200

300

y(m

)

x(m) -0.5 -0.25 0 0.25 0.50

0.25

0.5

0.75

1

0.25

1

25

50.5

0.05

10

x(m)

z(m

)

c)

-0.5 -0.25 0 0.25 0.5-0.25

0

0.25

0.5

0.75 0.05

200

10

400

50

0.5

500

300

100

25

2.5

0.25

5

1

y(m

)

x(m) -0.5 -0.25 0 0.25 0.50

0.25

0.5

0.75

1

0.25

100

5

1

50

10

2.5

0.50.05

25

x(m)

z(m

)

d)

Figure 7 : Two-dimensional snapshots of the temperature fields (◦C) when both conduction and convection phe-nomena are modelled: a) Case 1, at t = 10h; b) Case 1, at t = 20h; c) Case 2, at t = 10h; d) Case 2, at t = 20h


register lower temperatures along the transversal grid ofreceivers than the other regions that are the same distancefrom the source, which means that the heat flows throughthis inclusion more slowly. As time elapses, the tempera-ture differences between the region behind the inclusionand in its vicinity decrease (t = 20 h in figure 6d)), con-firming the equilibrium evolution. It is interesting to notealso that the temperature rises in the zones a long awayfrom the source, since the energy is still flowing awaythrough the domain, while the temperature closer to theheat source decreases sharply.

The results computed at a grid of receivers distributedalong the xz plane at z = 0.025 m are useful to better un-derstand the heat diffusion process in the z direction. Thereceivers are spaced 0.025m along the z direction: thegrid is formed by 80×41 receivers.

Figures 7 c) and d) illustrate the high heat diffusivity[faster heat propagation] through the steel material: thepresence of isothermals that are parallel with the x direc-tion reflects this behaviour. The temperature distributioninside the oil reveals a long delay in the energy progressin the z direction. It is clear that the presence of the cir-cular heterogeneity introduces an abrupt disruption of theheat propagation along the transversal domain.

6 Final Remarks

A BEM model has been implemented and used to com-pute three-dimensional transient heat transfer along asquare hollow inclusion of infinite length, itself contain-ing an inclusion filled with oil, located inside a homoge-neous elastic medium and heated by a spherical heat loadplaced inside the host medium. The phenomena stud-ied here include heat conduction and convection. Theincorporation of convection diffusion changes the heatresponses in keeping with the amplitude and velocitiesassumed for each material.

A Fourier transform applied in the time domain is thetechnique used to deal with the time variable of the diffu-sion equation. Notice that, if the geometry does not varyalong one direction, the solution becomes simpler, sincethe full 3D problem can be computed as a summation of2D solutions with different spatial kz wavenumbers. Theapplication of an inverse Fourier transform along the kz

domain can be expressed as a discrete summation if asequence of sources, equally-spaced, is considered. Anumerical inverse fast Fourier transform in kz and in the

frequency domain is then used to compute the tempera-ture field in the spatial-temporal domain.

References

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Brebbia, C. A.; Telles, J. C.; Wrobel, L. C. (1984):Boundary Elements Techniques: Theory and Applica-tions in Engineering. Berlin-New York: Springer-Verlag.

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Lin, H.; Atluri, S.N. (2000): Meshless Local Petrov-Galerkin (MLPG) Method for Convection-DiffusionProblems. CMES: Computer Modeling in Engineering& Sciences, vol. 1(2), pp. 45-60.

Atluri, S. N.; Shen, S. (2002): The Meshless LocalPetrov-Galerkin (MLPG) Method: A Simple \& Less-costly Alternative to the Finite Element and BoundaryElement Methods. CMES: Computer Modeling in Engi-neering & Sciences, vol. 3(1), pp. 11-52.

Sladek, J.; Sladek, V., Atluri, S.N. (2004): Mesh-less Local Petrov-Galerkin Method for Heat ConductionProblem in an Anisotropic Medium. CMES: ComputerModeling in Engineering & Sciences, vol. 6(3), pp. 309-318.

Carslaw, H. S.; Jaeger, J. C. (1959): Conduction of heatin solids. Second edition, New York: Oxford UniversityPress.

Ozisik, M. N. (1993): Heat Conduction. Second edition,New York: John Wiley & Sons.

Beck, J. V.; Cole, K. D.; Haji-Sheikh, A.; Litkouhi,B. (1992): Heat Conduction using Green’s Functions.Washington, DC: Hemisphere Publishing Corporation.

Chang, Y. P.; Kang, C. S.; Chen D. J. (1973): The Useof Fundamental Green Functions for Solution of Prob-lems of Heat Conduction in Anisotropic Media. Int. J.Heat and Mass Transfer, 16, pp. 1905-1918.

Shaw R. P. (1974): An Integral Equation Approach toDiffusion. Int. J. Heat and Mass Transfer, vol. 17, pp.693-699.

Wrobel, L. C.; Brebbia, C. A. (1981): A Formulation ofthe Boundary Element Method for Axisymmetric Tran-sient Heat Conduction. Int. J. Heat and Mass Transfer,Vol. 24, pp. 843-850.


Dargush, G. F.; Banerjee P. K. (1991): Application ofthe Boundary Element Method to Transient Heat Con-duction. Int. J. Numerical Methods in Engineering, vol.31, 1231-1247.

Lesnic, D.; Elliot L.; Ingham DB. (1995): Treatmentof Singularities in Time-dependent Problems using theBoundary Element Method. Engineering Analysis withBoundary Elements-EABE, vol. 16, pp. 65-70.

Rizzo, F. J.; Shippy, D. J. (1970): A Method of Solu-tion for Certain Problems of Transient Heat Conduction.AIAA Journal, Vol. 8, pp. 2004-2009.

Cheng, A.; Abousleiman, Y.; Badmus, T. (1992): ALaplace Transform BEM for Axisymmetric DiffusionUtilizing Pre-tabulated Green’s Function. EngineeringAnalysis with Boundary Elements-EABE, vol. 9, pp. 39-46.

Zhu, S.P.; Satravaha, P.; Lu, X. (1994): Solving theLinear Diffusion Equations with the Dual ReciprocityMethods in Laplace Space. Engineering Analysis withBoundary Elements-EABE, vol. 13, pp. 1-10.

Sutradhar, A.; Paulino, G. H.; Gray, L. J. (2002):Transient Heat Conduction in Homogeneous and Non-Homogeneous Materials by the Laplace TransformGalerkin Boundary Element Method. Engineering Anal-ysis with Boundary Elements-EABE, vol. 26 (2), pp. 119-132.

Stehfest, H. (1970): Algorithm 368: Numerical Inver-sion of Laplace Transform. Communications of the As-soc. Comp. Machinery, vol. 13(1), pp. 47-49.

Tadeu, A.; Antonio, J.; Simoes, N. (2004): 2.5DGreen’s Functions in the Frequency Domain for HeatConduction Problems in Unbounded, Half-space, Slaband Layered Media. Computer Modeling in Engineering& Sciences - CMES, vol. 6(1), pp. 43-58.

Tadeu, A; Kausel E. (2000): Green’s Functions forTwo-and-a-half Dimensional Elastodynamic Problems.J. Engng. Mechanics – ASCE, vol. 126(10), pp. 1093-1097.

Bouchon, M.; Aki, K. (1977): Discrete Wave-numberRepresentation of Seismic-source Wave Field. Bulletinof the Seismological Society of America, vol. 67, pp.259-277.

Tadeu, A.; Godinho, L.; Santos P. (2002): Wave Mo-tion Between Two Fluid Filled Boreholes in an ElasticMedium, Engineering Analysis with Boundary Elements-

EABE, vol. 26(2), pp. 101-117.

Godinho, L.; Antonio, J.; Tadeu, A. (2001): 3D SoundScattering by Rigid Barriers in the Vicinity of Tall Build-ings. Journal of Applied Acoustics, vol. 62(11), pp.1229-1248.

Kausel, E.; Roesset, J. M. (1992): Frequency DomainAnalysis of Undamped Systems. Journal of EngineeringMechanics, ASCE, vol. 118(4), pp. 721-734.

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Tadeu, A.; Santos, P.; Kausel, E. (1999): Closed-formIntegration of Singular Terms for Constant, Linear andQuadratic Boundary Elements - Part II: SV-P Wave Prop-agation, Engineering Analysis with Boundary Elements-EABE, vol. 23(9), pp. 757-768.

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Appendix A: Analytical Solution of the 3D Tran-sient Heat Transfer through a Cylin-drical Circular Ring Core

Consider a ring defined by the internal and external radii,a and b respectively, bounded by an exterior and interiormedium, as illustrated in figure 8. This ring is heated bya harmonic source with radial convection, placed in theexterior solid medium (with a thermal conductivity k1, adensity ρ1, a specific heat c1 and a convection velocityV1). The heat generated by this source propagates andhits the outer surface of the ring. After striking the outersurface of the cylindrical ring, part of the incident en-ergy is reflected back into the exterior solid medium, andthe remaining energy is transmitted into the ring material(with a thermal conductivity k2, a density ρ2, a specificheat c2 and a convection velocity V2), in the form of prop-agating energy. This energy continues to propagate anduntil it strikes the inner surface of the ring. There, a simi-lar phenomenon may occur, with part of the energy beingtransmitted to the inner medium (with a thermal conduc-tivity k3, a density ρ3, a specific heat c3 and a convectionvelocity V3) and the rest being reflected back to the ringmedium. This process will be repeated until all the en-ergy is dissipated.


Figure 8 : Circular ring geometry and representation ofthe surface terms

Incident heat field (or free-field)

The three-dimensional incident field for a point pressuresource placed at (x0,0,0), can be expressed as

Tinc(ω, r, r′, z) =Ae

V1r2K1

2k1

e−i√− V2

14K2

1− iω

K1

√r′2+z2

√r′2 + z2

with r′ =√

(x−x0)2 +y2 (12)

where the subscript inc denotes the incident field, A is theheat amplitude, K1 = k1

ρ1 c1and r′ defines the distance be-

tween the source and the receiver. When a Fourier trans-formation is applied along the z direction, the incidentfield can be expressed as a summation of 2D sources,with different spatial wavenumbers,

Tinc(ω, r, r′) =2πL

M

∑m=−M

Tinc(ω, r, r′,kzm)e−ikzmz (13)

with Tinc (ω, r,kzm) = −iA4k1

eV1r2K1 H0 (kα1r′) and kα1 =√

−V 21

4K21

+ −iωK1

− (kzm)2.

Eq. (13) expresses the incident field as heat terms cen-tred at the source point (x0, 0, 0), and not at the axis ofthe cylindrical inclusion, and this constitutes a difficulty.In order to overcome this problem, the incident heat fieldcan be expressed as heat terms centred at the origin. Thisis achieved by applying Graf’s addition theorem [Wat-son (1980)], which results in the expressions below (in

cylindrical coordinates):

Tinc(ω, r,θ,kzm)

= − iA4k1

eV1r2K1

∞

∑n=0

(−1)n εnJn(kα1 r0)Hn(kα1 r)cos(nθ)

when r > r0

Tinc(ω, r,θ,kzm)

= − iA4k1

eV1r2K1

∞

∑n=0

(−1)n εnHn(kα1r0)Jn(kα1 r)cos(nθ),

when r < r0, (14)

in which r0 is the distance from the source to the axis ofthe inclusion, Jn(. . . ) are Bessel functions of order n, and

εn ={

1 i f n = 02 i f n �= 0

.

The scattered heat field in the outer medium

The heat generated in the exterior medium depends onheat coming from the external surface of the cylindricalring, which propagates away from it. The outgoing heatcan be defined using the following equation,

T1(ω, r,θ,kzm) = eV1r2K1

∞

∑n=0

AnHn(kα1r)cos(nθ) (15)

where An are unknown amplitudes.

The heat field in the ring

Two distinct groups of heat fields exist inside the ring,corresponding to the heat generated at the external sur-face and travelling inwards, and to the heat generated atthe internal surface of the pipe, which travels outwards.For the terms generated at the external boundary, the cor-responding standing heat field is given by


∞

∑n=0

BnJn(kα2 r)cos(nθ) (16)

where Bn are unknown amplitudes.

For the heat generated at the internal boundary, there is acorresponding diverging heat field, which can be definedby,


∞

∑n=0

CnHn(kα3r)cos(nθ) (17)

where kα3 =√

−V 23

4K23

+ −iωK3

− (kzm)2 and Cn are unknown

amplitudes.


The heat field in the inner medium

In the inner medium (medium 3), the heat field dependsonly on heat coming from the internal surface of thecylindrical ring core, and thus only inward-propagatingheat is generated. The corresponding heat field is givenby:


∞

∑n=0

DnJn(kα3 r)cos(nθ) (18)

where Dn are unknown amplitudes.

The unknown coefficients An, Bn, Cn and Dn are deter-mined by imposing the required boundary conditions.For the case described here, the boundary conditions arethe continuity of temperatures and normal heat fluxes onthe two interfaces. The four equations defined give riseto a system of four equations with four unknowns, whichyields the unknown coefficients.

Tinc(ω,b,θ,kzm)+ T1(ω,b,θ,kzm) = T2(ω,b,θ,kzm)at r = b

k1∂[Tinc(ω,b,θ,kzm)

]∂r

+k1∂[T1(ω,b,θ,kzm)

]∂r

= k2∂[T2(ω,b,θ,kzm)

]∂r

at r = b

T2(ω,a,θ,kzm)+ T3(ω,a,θ,kzm) = T4(ω,a,θ,kzm)at r = a

k2∂[T2(ω,a,θ,kzm)

]∂r

+k2∂[T3(ω,a,θ,kzm)

]∂r

= k3∂[T4(ω,a,θ,kzm)

]∂r

at r = a (19)

Combining eqs. (A.3) and eqs. from (15) to (A.8) oneobtains a system of equations which is then used to findthe unknown coefficients (An, Bn, Cn, Dn):⎡⎢⎢⎣

a11 a12 a13 a14

a21 a22 a23 a24

a31 a32 a33 a34

a41 a42 a43 a44

⎤⎥⎥⎦⎡⎢⎢⎣

An

Bn

Cn

Dn

⎤⎥⎥⎦

= (−1)n εn

⎡⎢⎢⎣

b1

b2

b3

b4

⎤⎥⎥⎦ (20)

with

a11 = eV1b2K1 Hn(kα1 b);

a12 = −eV2b2K2 Jn(kα2b);

a13 = −eV2b2K2 Hn(kα2b);

a14 = 0;

a21 = 0;

a22 = eV2a2K2 Jn(kα2a);

a23 = eV2a2K2 Hn(kα2 a);

a24 = eV3a2K3 Jn(kα3a);

a31 = k1

{ V1b2K1

Hn(kα1 b)+[nHn(kα1 b)− (kα1b)Hn+1(kα1 b)]

}e

V1b2K1 ;

a32 = −k2

{ V2b2K2

Jn(kα2b)+[nJn(kα2b)− (kα2 b)Jn+1(kα2 b)]

}e

V2b2K2 ;

a33 = −k2

{ V2b2K2

Hn(kα2b)+[nHn(kα2 a)− (kα2 a)Hn+1(kα2b)]

}e

V2b2K2 ;

a34 = 0;

a41 = 0;

a42 = k2

{ V2a2K2

Jn(kα2a)+[nJn(kα2a)− (kα2 a)Jn+1(kα2a)]

}e

V2a2K2 ;

a43 = k2

{ V2a2K2

Hn(kα2 a)+[nHn(kα2 a)− (kα2a)Hn+1(kα2 a)]

}e

V2a2K2 ;

a44 = −k3

{ V3a2K3

Hn(kα3 a)+[nJn(kα3a)− (kα3 a)Jn+1(kα3 a)]

}e

V3a2K3 ;

b1 =iA4k1

eV1b2K1 Hn(kα1 r0)Jn(kα1 b);

b2 = 0;

b3 =iA4

eV1b2K1 Hn(kα1r0)

⎧⎨⎩

V1b2K1

Jn(kα1 b)+[nJn(kα1 b)

−(kα1 b)Jn+1(kα1b)

] ⎫⎬⎭ ;

b4 = 0.

Notice that when the position of the heat source ischanged, the terms ai j of the matrix remain the same,while the independent terms bi are different. However, asthe equations can be easily manipulated to consider an-other position for the source, they are not included here.

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