# Fundamental solutions for transient heat transfer by conduction and convection in an unbounded, half-space, slab and layered media in the frequency domain

**ABSTRACT** Analytical Green's functions in the frequency domain are presented for the three-dimensional diffusion equation in an unbounded, half-space, slab and layered media. These proposed expressions take into account the conduction and convection phenomena, assuming that the system is subjected to spatially sinusoidal harmonic heat line sources and do not require any type of discretization of the space domain. The application of time and spatial Fourier transforms along the two horizontal directions allows the solution of the three-dimensional time convection-diffusion equation for a heat point source to be obtained as a summation of one-dimensional responses. The problem is recast in the time domain by means of inverse Fourier transforms using complex frequencies in order to avoid aliasing phenomenon. Further, no restriction is placed on the source time dependence, since the static response is obtained by limiting the frequency to zero and the high frequency contribution to the response is small.The proposed functions have been verified against analytical time domain solutions, known for the case of an unbounded medium, and the Boundary Element Method solutions for the case of the half-space, slab and layered media.

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Page 1

Fundamental solutions for transient heat transfer by conduction

and convection in an unbounded, half-space, slab and layered

media in the frequency domain

Nuno Simo ˜es, Anto ´nio Tadeu*

Department of Civil Engineering, University of Coimbra, Pinhal de Marrocos, 3030-290 Coimbra, Portugal

Received 19 October 2004; revised 31 May 2005; accepted 5 June 2005

Available online 26 August 2005

Abstract

Analytical Green’s functions in the frequency domain are presented for the three-dimensional diffusion equation in an unbounded, half-

space, slab and layered media. These proposed expressions take into account the conduction and convection phenomena, assuming that the

system is subjected to spatially sinusoidal harmonic heat line sources and do not require any type of discretization of the space domain. The

application of time and spatial Fourier transforms along the two horizontal directions allows the solution of the three-dimensional time

convection-diffusion equation for a heat point source to be obtained as a summation of one-dimensional responses. The problem is recast in

the time domain by means of inverse Fourier transforms using complex frequencies in order to avoid aliasing phenomenon. Further, no

restriction is placed on the source time dependence, since the static response is obtained by limiting the frequency to zero and the high

frequency contribution to the response is small.

The proposed functions have been verified against analytical time domain solutions, known for the case of an unbounded medium, and the

Boundary Element Method solutions for the case of the half-space, slab and layered media.

q 2005 Elsevier Ltd. All rights reserved.

Keywords: Transient heat transfer; Conduction; Convection; 2.5D Green’s functions; Layered media

1. Introduction

The transient heat diffusion is a fundamental phenom-

enon observed in several applications, such as building

physics and thermal engineering. One of the most important

reference works for transient heat transfer is by Carslaw and

Jaeger [1]. This book contains a set of analytical solutions

and Green’s functions for the diffusion equation, which

gives the response, i.e. temperature field or/and heat fluxes,

of the diffusion equation in the presence of a transient heat

process.

Most of the known techniques to solve transient

convection-diffusion heat problems have been formulated

in the time domain or using Laplace transforms. In the ‘time

marching’ approach, the solution is assessed step by step at

consecutive time intervals after an initially specified state

has been assumed.Using the Laplace transform, a numerical

transform inversion is required to calculate the physical

variables in the real space, after the solution has been

obtained for a sequence of values of the transformed

parameter.

In the ‘time marching’ approach, the result at each time

step is computed directly in the time domain. Chang et al.

[2] and Shaw [3] used a time-dependent fundamental

solution for studying transient heat processes. Later, Wrobel

and Brebbia [4] implemented a Boundary Element Method

BEM formulation for axisymmetric diffusion problems.

Dargush and Banerjee [5] proposed a BEM approach in the

time domain, where planar, three-dimensional and axisym-

metric analyses are all addressed with a time-domain

convolution. Lesnic et al. [6] studied the unsteady diffusion

equation in both one and two dimensions by a time

marching BEM model, taking into account the treatment

of singularities. Davies [7] used a time-domain analysis to

compute the heat flow across a multi-layer wall, considering

surface films to model the radiant and convective exchange.

Engineering Analysis with Boundary Elements 29 (2005) 1130–1142

www.elsevier.com/locate/enganabound

0955-7997/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.enganabound.2005.06.002

*Corresponding author.Tel.: C351 239 797201;fax: C351 239 797190.

E-mail address: tadeu@dec.uc.pt (A. Tadeu).

Page 2

An alternative to the ‘time marching’ approach is to

remove the time dependent derivative, using instead a

transformed variable. The Laplace transform technique

has been extensively reported in the literature, to shift the

solution from the time domain to a transformed domain,

for solving diffusion problems. However, an inverse

transform is then required to reconstitute the solution in

the time domain, which is associated with a loss of

accuracy and may lead to the amplification of small

truncation errors. Several researchers have tried to

overcome this major drawback: algorithms for Laplace

inversion have been proposed by Stehfest [8] and Papoulis

[9]. Rizzo and Shippy [10] used a numerical approach that

incorporates a Laplace transform to create a time-

independent boundary integration in a transform domain.

Other authors have proposed different solutions for

analyzing the diffusion-type problem by means of Laplace

transforms, such as those presented by Cheng et al. [11],

Zhu and Satravaha ([12,13]).

The search for Green’s functions has been extensively

researched, given their interest as benchmark solutions

and in the development of numerical methods, such as

the BEM (e.g. Ochiai [14]) and the Method of the

Fundamental Solutions (MFS) (e.g. Sˇarler [15]). Feng

[16] used a method based on modified Green’s functions

to compute the unsteady heat transfer of a homogeneous

or a composite solid body. Haji-Sheikh et al. [17] present

different types of Green’s functions that are solutions of

the heat conduction diffusion equation in multi-dimen-

sional, multi-layer bodies for different boundary con-

ditions, calculating eigenvalues.

This work presents Green’s functions for calculating the

transient heat transfer wave field in the presence of an

unbounded, half-space, slab and multi-layer formations,

with the occurrence of conduction and convection phenom-

ena. The problem is formulated in the frequency domain

using time Fourier transforms. The proposed technique

allows the use of any type of heat source, and deals with the

static response.

The problem of multi-layer heat transfer has been

broadly studied. The O¨zisik’s book [18] includes a review

of one-dimensional composite media, referring to orthog-

onal expansions, Green’s functions and Laplace transform

techniques. Monte [19] analysed the transient heat

conduction of multi-layer composite slabs, applying the

method of separation of variables to the heat conduction

partial differentialequation.

regression method was developed by Wang and Chen

[20] to compute the heat flow for a one-dimensional

multi-layer model.

The present work defines the transient heat transfer in a

multi-layer system subject to a point, a linear or a plane

source in the presence of both convection and conduction.

This work extends previous work carried out by the

authors to define the response of layered solid media

subjected to a spatially sinusoidal harmonic heat

A frequency-domain

conduction line source, e.g. Tadeu et al. [21], where

only the conduction phenomenon was addressed. The

proposed fundamental solutions relate the heat field

variables (fluxes or temperatures) at some position in

the domain caused by a heat source placed elsewhere in

the media, in the presence of both conduction and

convection phenomena.

As in the previous work, the technique requires the

knowledge of the Green’s function for the unbounded

media, which are developed by first applying a time

Fourier transform to the time diffusion equation for a

heat point source and then a spatial Fourier transform to

the resulting Helmholtz equation, along the z direction, in

the frequency domain. So these functions are written first

as a superposition of cylindrical heat waves along one

horizontal direction (z) and then as a superposition of

heat plane sources.

The Green’s functions for a layered formation are

formulated as the sum of the heat source terms equal to

those in the full-space and the surface terms required to

satisfy the boundary conditions at the interfaces, i.e.

continuity of temperatures and normal fluxes between

layers, and null normal fluxes or null temperatures at the

outer surface. The total heat field is achieved by adding the

heat source terms, equal to those in the unbounded space, to

the sets of surface terms arising within each layer and at

each interface.

The scope of this paper is to present first the three-

dimensional formulation, explaining the mathematical

manipulation to obtain the Green’s functions for a heat

line source as a continuous superposition of heat plane

sources in the frequency domain. The procedure to retrieve

the time domain solutions is also given. Thismethodology is

verified by comparing the results obtained with the exact

time solutions for one, two and three-dimensional point heat

sources placed in an unbounded medium.

This paper then goes on to describe the formulation of a

sinusoidal line heat load applied to a half-space, a slab, a

slab over a half-space medium and a layered formation. The

continuity of temperature and heat fluxes need to be

achieved between two neighbouring layers, while null

temperatures or null heat fluxes may be prescribed along an

external interface of the boundary. The full set of

expressions is corroborated by comparing its solutions

with those provided by the Boundary Element Method,

which requires the discretization of all layer interfaces.

2. Three-dimensional problem formulation and Green’s

functions in an unbounded medium

The transient heat transfer by conduction and convection

in the domain with constant velocities along the x, y and z

directions is expressed by the equation

N. Simo ˜es, A. Tadeu / Engineering Analysis with Boundary Elements 29 (2005) 1130–11421131

Page 3

v2

vx2Cv2

K1

vy2Cv2

?

vT

vt;

vz2

v

vxCVy

??

T

K

Vx

v

vyCVz

v

vz

?

T

Z1

K

(1)

in which Vx, Vyand Vxare the velocity components in the

direction x, y and z, respectively, t is time, T(t,x,y,z) is

temperature, KZk/(rc) is the thermal diffusivity, k is the

thermal conductivity, r is the density and c is the specific

heat. The application of a Fourier transformation in the time

domain to the Eq. (1) gives the equation

v2

vx2Cv2

vy2Cv2

!2!

vz2

??

K1

K

Vx

v

vxCVy

v

vyCVz

v

vz

???

C

ffiffiffiffiffiffiffiffi ffi

K

K iu

r

^Tðu;x;y;zÞ Z0;

(2)

where iZ

the Helmholtz equation by the insertion of a convective

term. For a heat point source, applied at (0,0,0) in an

unbounded medium, of the form p(u, x, y, z, t)Zd(x) d(y)

d(z)i(ut), where d(x), d(y) and d(z) are Dirac-delta functions,

the fundamental solution of Eq. (2) can be expressed as

ffiffiffiffiffiffi

K 1

p

and u is the frequency. Eq. (2) differs from

^Tfðu;x;y;zÞ

ZeðVxxCVyyCVzzÞ=2K

2k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

As the geometry of the problem remains constant along

the z direction, the full three-dimensional problem can be

expressed as a summation of simpler two-dimensional

solutions. This requires the application of a Fourier

transformation along that direction, writing this as a

summation of two-dimensional solutions with different

spatial wavenumbers kz (Tadeu and Kausel [22]). The

application of a spatial Fourier transformation to

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

along the z direction, leads to this fundamental solution

x2Cy2Cz2

p

eK i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

K ðV2

xCV2

yCV2

zÞ=4K2K iu=K

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2Cy2Cz2

p

:

(3)

eK i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2Cy2Cz2

p

K ðV2

xCV2

yCV2

zÞ=4K2K iu=Kx2Cy2Cz2

p

;

(4)

~Tfðu;x;y;kzÞZKieðVxxCVyyCVzzÞ=2K

4k

s

!H0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4K2

KV2

xCV2

yCV2

z

Kiu

KKðkzÞ2

r0

0

@

1

A;

(5)

where H0() are Hankel functions of the second kind and

order 0, and r0Z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2Cy2

p

.

The full three-dimensional solution is then achieved by

applying an inverse Fourier transform along the kxdomain

to the expression ðK i=2ÞH0>

K ðiu=KÞKðkzÞ2r0

can be expressed as a discrete summation if we assume

the existence of virtual sources, equally spaced at Lz, along

z, which enables the solution to be obtained by solving a

limited number of two-dimensional problems,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

K ðV2

xCV2

yCV2

zÞ=4K2

q

?

?

. This inverse Fourier transformation

^Tðu;x;y;zÞZ2p

L

eðVxxCVyyCVzzÞ=2K

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4K2

@

2kx2Cy2Cz2

p

KV2

X

Kiu

M

mZKM

H0

!

xCV2

yCV2

z

KKðkzmÞ2

s

r0

01

AeKikzmz;

(6)

with kzmbeing the axial wavenumber given by KzmZ(2p/Lx)m.

The distance Lz is chosen so as to prevent spatial

contamination from the virtual sources, i.e. it must be

sufficiently large (Bouchon and Aki [23]). This technique is

an adaptation and extension of other mathematical and

numerical formulations used to solve problems such as wave

propagation (Tadeu et al. [24] and Godinho et al. [25]).

Note that Eq. (5) with VzZ0 becomes the fundamental

solution of the differential equation obtained from Eq. (2)

after the application of a spatial Fourier transformation

along the z direction, namely

v2

vx2Cv2

vy2

??

K1

K

Vx

v

vxCVy

v

vy

?

!2!

??

C

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

K

K iu

KðkzÞ2

r

~Tðu;x;y;kzÞ Z0;

(7)

when VzZ0.

Eq. (5), which results when a spatially sinusoidal

harmonic heat line source is applied at the point (0,0)

along the z direction, subject to convection velocities

Vx, Vyand Vz, can be further manipulated and written as a

continuous superposition of heat plane phenomena,

~Tfðu;x;y;kzÞ ZK ieðVxxCVyyCVzzÞ=2K

4pk

!

ð

C N

K N

eK injyj

n

!

eK ikxðxK x0Þdk0

x

(8)

where

n Z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2K

KV2

xCV2

yCV2

z

Kiu

KKðkzÞ2Kk2

x

s

and Im(n)%0, and the integration is performed with respect

to the horizontal wavenumber (kx) along the x direction.

Assuming the existence of an infinite number of virtual

sources, we can discretize these continuous integrals.

The integral in the above equation can be transformed into

N. Simo ˜es, A. Tadeu / Engineering Analysis with Boundary Elements 29 (2005) 1130–11421132

Page 4

a summation if an infinite number of such sources are

distributed along the x direction, spaced at equal intervals

Lx. The above equation can then be written as

~Tfðu;x;y;kzÞ ZK i eðVxxCVyyCVzzÞ=2K

4k

E0

X

EdZeK ikxnðxÞ,

nZC N

nZ K N

E

nn

??

Ed;

(9)

where

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

%0, and kxnZ(2p/Lx)n,which can in turn be approximated

by a finite sum of equations (N). Notice that kzZ0

corresponds to the two-dimensional case.

E0ZKi/2kLx,EZeK innjyj,

nnZ

K ðV2

xCV2

yCV2

zÞ=4K2Kðiu=KÞKðkzÞ2Kk2

xn

q

and Im(nn)

2.1. Responses in the time domain

The heat in the spatial-temporal domain is calculated by

applying a numerical inverse fast Fourier transform in kz, kx

and in the frequency domain. The computations are

performed using complex frequencies with a small

imaginary part of the form ucZu-ih (with hZ0.7Du, and

Du being the frequency step) to prevent interference from

aliasing phenomena. In the time domain, this effect is

removed by rescaling the response with an exponential

window of the form eht. The time variation of the source can

be arbitrary. The time Fourier transformation of the source

heat field defines the frequency domain to be computed. The

response may need to be computed from 0.0 Hz up to very

high frequencies. However, as the heat responses decay very

rapidly with increasing frequency, we may limit the upper

frequency for which the solution is required. The frequency

0.0 Hz corresponds to the static response that can be

computed when the frequency is zero. The use of complex

frequencies allows the solution to be obtained because,

when ucZu-ih (for 0.0 Hz), the arguments of the Hankel

function of the equations are not zero.

The technique proposed in this paper uses Fourier

transformations, which can be written as discrete sum-

mations over wavenumbers and frequencies. The math-

ematical formulation entails the use of sources equally

spaced in the z-axis and x-axis by spatial separations

LzZ2p/Dkzand LxZ2p/Dkx, and also by temporal intervals

TZ2p/Du, with Dkzand Dkxbeing the wavenumber steps.

Note that the use of complex frequencies diminishes the

contribution from the periodic (fictitious) sources to the

response at the time window T.

2.2. Verification of the solution

The formulation described above was implemented and

used to compute the heat field in an unbounded medium. In

order to verify this formulation, the solution is compared

with the analytical response in the time domain.

The exact solution of the convective diffusion, expressed

by Eq. (1), in an unbounded medium subjected to a unit heat

sourceiswellknownanditallowsthecomputationoftheheat

field given by both conduction and convection phenomena in

thepresenceofthree,twoorone-dimensionalproblems.When

the heat source is applied at the point (0,0,0) at time tZt0, the

temperature at (x,y,z) is given by the expression

Tðt;x;y;zÞ ZeðK ðK tVxCxÞ2K ðK tVyCyÞ2K ðK tVzCzÞ2Þ=4Kt

rcð4pKtÞd=2

;

if tOt0;

(10)

where tZt-t0,dZ3,dZ2anddZ1wheninthepresenceofa

three, two and one-dimensional problems, respectively

(Carslaw and Jaeger [1] and Hagentoft [26]).

In the verification procedure, a homogeneous unboun-

ded medium, with thermal properties that allowed

kZ1.4 W mK18CK1, cZ880.0 J KgK18CK1and rZ

2300 Kg mK3, was excited at tZ277.8 h by a unit heat

source placed at xZ0.0 m, yZ0.0 m, zZ0.0 m. The

convection velocities applied in the x, y and z direction

were equal to 1!10K6m sK1.

Theresponseswerecalculatedalongalineof40receivers

placed from (xZK1.5,yZ0.35,zZ0.0) to (xZ1.5,yZ0.35,

zZ0.0), for a plane (dZ1), cylindrical (dZ2) and spherical

(dZ3) unit heat source.

The calculations were first performed in the frequency

range [0, 1024!10K7]Hz with an increment of DuZ10K7

Hz, which defines a time window of TZ2777.8 h. The

solution for the two-dimensional case (cylindrical unit heat

source) was found with Eq. (5), while the results for a plane

unit heat source propagating along the y axis was obtained

ascribing kzZ0 and kxnZ0 to Eq. (9), multiplied by Lx.

Complex frequencies of the form ucZuKi0.7Du have

been used to avoid the aliasing phenomenon. The spatial

period has been set as LxZLzZ2

In Fig. 1, the solid line represents the exact time solution

given by Eq. (10) while the marks show the response

obtained using the proposed Green’s functions. There is

good agreement between these two solutions. Notice that, as

we have assigned a convection velocity in the x direction,

the temperature response along the line of receivers is not

symmetrical. In addition, lower temperatures were regis-

tered at the three-dimensional case, since the energy emitted

by the heat source is dissipated in the three directions.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k=ðrcDfÞ

p

.

3. Green’s functions in a half-space

In this section, a semi-infinite medium bounded by a

surface with null heat fluxes or null temperatures is

considered. The required Green’s functions for a half-

space can be expressed as the sum of the surface terms and

the source terms. The surface terms need to satisfy the

boundary condition of the surface (null heat fluxes or null

temperatures), while the source terms are equal to those

presented for the infinite unbounded medium. The surface

terms for a heat source located at (x0,y0)can be expressed by

N. Simo ˜es, A. Tadeu / Engineering Analysis with Boundary Elements 29 (2005) 1130–11421133

Page 5

~T1ðu;x;y;kzÞ

ZE0eððVxðxK x0ÞCVyðyK y0ÞÞ=2KÞX

where EaZeK inny. Anis the unknown coefficient to be com-

puted, so that the heat field produced simultaneously by the

source and surface terms should produce~T1ðu;x;y;kzÞZ0

or ðv~T1ðu;x;y;kzÞ=vyÞZ0 at yZ0.

The computation of the unknown coefficient is obtained

for each value of n. These coefficients are given below for

the two cases of null heat fluxes and null temperatures at the

surface yZ0.

Null normal flux at yZ0,

nZC N

nZ K N

Ea

nn

An

??

Ed;

(11)

AnZeK inny0;

Null temperature at yZ0,

AnZK eK inny0:

(12)

Replacing these coefficients in Eq. (11), we may compute

the heat terms associated with the surface.

Null normal flux at yZ0,

~T1ðu;x;y;kzÞ ZE0eððVxðxK x0ÞCVyðyK y0ÞÞ=2KÞX

nZC N

nZ K N

Eaf

nn

??

Ed;

Null temperature at yZ0,

~T1ðu;x;y;kzÞ

ZE0eððVxðxK x0ÞCVyðyK y0ÞÞ=2KÞX

nZC N

nZ K N

Eat

nn

??

Ed

(13)

where EafZeK innðyCy0Þand EatZK eK innðyCy0Þ.

The final fundamental solutions for a half-space are

given by adding both terms: the source and the surface

terms, which leads to

Null normal flux at yZ0,

~Tðu;x;y;kzÞ

ZE0eððVxðxK x0ÞCVyðyK y0ÞÞ=2KÞX

nZC N

nZ K N

ECEaf

nn

??

Ed;

(14)

Null temperature at yZ0,

~Tðu;x;y;kzÞ

ZE0eððVxðxK x0ÞCVyðyK y0ÞÞ=2KÞX

nZC N

nZ K N

EKEat

nn

??

Ed:

(15)

0

–1.5

0.2

0.4

0.6

0.8(a)

–1.0–0.50 0.51.01.5

350 h

450 h

550 h

650 h

350 h

450 h

550 h

650 h

350 h

450 h

550 h

650 h

X (m)

–1.5–1.0 –0.50 0.51.01.5

X (m)

–1.5–1.0 –0.50 0.51.01.5

X (m)

Temperature (ºC)

0

0.2

0.1

0.3

0.4

0.5(b)

Temperature (ºC)

0

0.2

0.1

0.3(c)

Temperature (ºC)

Fig. 1. Temperature along a line of 40 receivers, at times 350, 450, 550 and 650 h: (a) for a plane (dZ1) unit heat source; (b) for a cylindrical (dZ2) unit heat

source and (c) for a spherical (dZ3) unit heat source.

N. Simo ˜es, A. Tadeu / Engineering Analysis with Boundary Elements 29 (2005) 1130–11421134

Page 6

3.1. Verification of the solution

A BEM model was used to compute the temperature field

when a heat source is placed in a semi-infinite medium

bounded by a surface with null temperatures or null normal

heat fluxes. This technique entails high computational costs,

since it needs the discretization of the boundary. In order to

simulate the half-space, the boundary is modelled through a

large number of elements distributed along as much of the

surface as necessary. The limited discretization of the

interfaces was achieved by introducing an imaginary part to

the frequencies, ucZuKih (with hZ0.7Du), which intro-

duces damping.

The verification of the solution is illustrated for a

homogeneous half-space medium with thermal material

properties that allow kZ1.4 W mK18CK1, cZ880.0 J KgK1

8CK1andrZ2300 Kg mK3.Theconvectionvelocity,inthey

direction,appliedtothehalf-spacemediumis5!10K7m sK1.

It is assumed that the origin of convection coincides with the

source position. This structure is excited at (xZ0.0 m,yZ

1.0 m) by a line heat source with spatial sinusoidal variation

(kzZ0.4 rad mK1). Fig. 2 gives the results obtained at the

receiver(xZ0.2 m,yZ0.5 m)inthefrequencyrange[0,032!

10K7] Hz with a frequency increment of 1!10K7Hz. In this

plot, the solid lines represent the results provided by the

proposed solutions while the markers correspond to the

solution computed using the Boundary Element technique.

These results show that the responses are similar.

4. Green’s functions in a slab formation

For a slab structure with thickness h, the Green’s

functions can be achieved taking into account the boundary

conditions prescribed at each surface, i.e. null heat fluxes or

null temperatures. They can be expressed by adding the

surface and source terms, which are equal to those in the

full-space.

Three scenarios can be considered: the prescription of

null heat fluxes at the top and bottom interfaces (Case I); or

null temperatures in both surface boundaries (Case II); or

even the consideration of different conditions at each

surface (Case III). At the top and bottom interfaces, surface

terms can be generated and expressed in a form similar to

that of the source term.

Top surface medium

~T1ðu;x;y;kzÞ ZE0eððVxðxK x0ÞCVyðyK y0ÞÞ=2KÞX

nZC N

nZ K N

Ea

nn

At

n

??

Ed;

Bottom surface medium

~T2ðu;x;y;kzÞ

ZE0eððVxðxK x0ÞCVyðyK y0ÞÞ=2KÞX

where EbZeK innjyK hj. At

coefficients to be determined by imposing the appropriate

boundary conditions, so that the field originated simul-

taneously by the source and the surface terms guarantees

null heat fluxes or null temperatures at yZ0 and at yZh.

This problem formulation leads to a system of two

equations in the two unknown constants for each value

of n.

nZC N

nZ K N

Eb

nn

Ab

n

??

Ed;

(16)

n and Ab

n are as yet unknown

Case I. -null heat fluxes at the top and bottom surfaces.

Vy

2KKinn

0

@

Vy

2KKinn

1

A

Vy

2KCinn

0

@

1

AeK innh

1

A

0

@

1

AeK innh

Vy

2KCinn

Vy

2KCinn

0

@

2

6666664

3

7777775

Atn

Ab

n

"#

Z

K

0

@

1

AeK inny0

1

K

Vy

2KKinn

0

@AeK innjhK y0j

2

6666664

3

7777775

:

(17)

–0.030

–0.015

0

0.015

0.030

0.045

0.060(a) (b)

0 1x10–6

2x10–6

3x10–6

real part

imaginary part

Frequency (Hz)

01x10–6

2x10–6

3x10–6

Frequency (Hz)

Amplitude (°C)

–0.03

–0.02

–0.01

0

0.01

0.02

Amplitude (°C)

real part

imaginary part

Fig. 2. Real and imaginary parts of the response for a half-space formation (KzZ0.4 rad mK1): (a) Null normal flux at yZ0; and (b) Null temperature at yZ0.

N. Simo ˜es, A. Tadeu / Engineering Analysis with Boundary Elements 29 (2005) 1130–11421135

Page 7

Case II. -null temperatures at the top and bottom surfaces.

1

K eK innh

eK innh

1

"#

Atn

Ab

n

"#

Z

K eK inny0

K eK innjhK y0j

"#

(18)

Case III. -null heat fluxes at the top surface and null

temperatures at the bottom surface.

Vy

2KKinn

0

@

1

A

Vy

2KCinn

0

@

1

AeK innh

eK innh

1

2

664

3

775

At

n

Ab

n

"#

Z

K

Vy

2KCinn

0

@

1

AeK inny0

K eK innjhK y0j

2

664

3

775:

(19)

Once this system of equations has been solved, the

amplitude of the surface terms has been fully defined, and

the heat in the slab can thus be found. The final expressions

for the Green’s functions are then derived from the sum of

the source terms and the surface terms originated in the two

slab surfaces, which leads to the following expressions,

~Tðu;x;y;kzÞ ZK i

4keððVxðxK x0ÞCVyðyK y0ÞÞ=2KÞH0ðKtr0Þ

nZC N

CE0eððVxðxK x0ÞCVyðyK y0ÞÞ=2KÞX

where

nZ K N

Ea

nn

At

nCEb

nn

Ab

n

??

Ed;

(20)

KtZ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2K

KV2

xCV2

yCV2

z

K iu

K

KðkzÞ2

s

:

4.1. Verification of the solution

The formulation described above related to the slab

formation was used to compute the responses at a receiver

placedinaslab3.0 mthick,subjectedtoaspatiallyharmonic

varying line load in the z direction. The results for the three

different cases of boundary conditions were then compared

with those achieved by using a Boundary Element Method.

In this verification procedure, the medium properties and

convectionvelocityremainthesameasthoseassumedforthe

half-space (kZ1.4 W mK18CK1, cZ880.0 J KgK18CK1

and rZ2300 Kg mK3). The slab is heated by a harmonic

point heat source applied at (xZ0.0 m, yZ1.0 m).

Theresponseisperformedinthefrequencyrange[0,32!

10K7]HzwithafrequencyincrementofDuZ10K7Hz.The

imaginarypartofthefrequencyhasbeensettohZ0.7Du.To

validate the results, the response is computed for a single

value of kz(kzZ0.4 rad mK1). Fig. 3 shows the real and

imaginary parts of the responses at the receiver (xZ0.2 m,

yZ0.5 m). The solid lines represent the discrete analytical

responses, while the marked points correspond to the

Boundary Element Method. The results confirm that the

solutions to the three cases are in very close agreement.

5. Green’s functions in a layered formation

The solutions for more complex structures, such as a

layer over a half-space, a layer bounded by two semi-infinite

media and a multi-layer can be established imposing the

required boundaryconditions at the interfaces and at the free

surface.

–0.050

–0.025

0

0.025

0.050 (a)

(b)

(c)

Amplitude (°C)

Amplitude (°C)

Amplitude (°C)

–0.02

–0.01

0

0.01

0.02

–0.050

–0.025

0

0.025

0.050

0 1x10–6

2x10–6

3x10–6

Frequency (Hz)

0 1x10–6

2x10–6

3x10–6

Frequency (Hz)

0 1x10–6

2x10–6

3x10–6

Frequency (Hz)

real part

imaginary part

real part

imaginary part

real part

imaginary part

Fig. 3. Real and imaginary parts of the responses for a slab formation, when

a heat source is applied at the point (xZ0.0 m, yZ0.0 m): (a) Case I (null

heat fluxes at the top and bottom surfaces); (b) Case II (null temperatures at

the top and bottom surfaces); and (c) Case III (null temperatures at the top

surface and null heat fluxes at the bottom surface).

N. Simo ˜es, A. Tadeu / Engineering Analysis with Boundary Elements 29 (2005) 1130–11421136

Page 8

5.1. Layer over a half-space

Assuming the presence of a layer, h1thick, over a half-

space, we may prescribe null temperature or null heat fluxes

at the free surface (top), while at the interface we need to

satisfy the continuity of temperature and normal heat fluxes.

The solution is again expressed as the sum of the source

terms (the incident field) equal to those in the full-space and

the surface terms. At the interfaces 1 and 2, surface terms

are generated, which can be expressed in a form analogous

to that of the source term.

Layer interface 1

~T11ðu;x;y;kzÞ ZE01eðVy1ðyK y0Þ=2K1ÞX

nZC N

nZ K N

E11

nn1

At

n1

??

Ed:

(21)

Layer interface 2

~T12ðu;x;y;kzÞ ZE01eðVy1ðyK y0Þ=2K1ÞX

nZC N

nZ K N

E12

nn1

Ab

n1

??

Ed:

(22)

Half-space (interface 2)

~T21ðu;x;y;kzÞ ZE02eðVy2ðyK y0Þ=2K2ÞX

nZC N

nZ K N

E21

nn2

At

n2

??

Ed:

(23)

where E0jZðK i=2kjLxÞ, E11ZeK inn1yE12ZeK inn1jyK h1j, E21Z

eK inn2jyK h1j

nnjZ

K ðVyj=2KjÞ2Kðiu=KjÞKk2

Im(nnj)%0 and h1is the layer thickness (jZ1 stands for

the layer (medium 1) while jZ2 indicates the half-space

(medium 2)). Meanwhile, KjZkj/rjcj) is the thermal

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

zKk2

xn

q

with

diffusivity in the medium j (kj, rjand cjare the thermal

conductivity, the density and the specific heat of the

material in the medium j, respectively).

The coefficients At

unknown. They are defined in order to ensure the

appropriate boundary conditions: the field produced

simultaneously by the source and surface terms allows

the continuity of heat fluxes and temperatures at yZh1,

and null heat fluxes (Case I) or null temperatures (Case

II) at yZ0.

Imposing the three stated boundary conditions for each

value of n, a system of three equations in the three unknown

coefficients is defined.

n1, Ab

n1

and At

n2

are as yet

Case I. -null heat fluxes at yZ0.

where

b1ZKVy1

2K1

Cinn1

?

?

?

?

eK inn1y0;

b2ZKVy1

2K1Kinn1

eK inn1jh1K y0j;

b3ZK eK inn1jh1K y0j

when the source is in the layer ðy0!h1Þ;

while

b1Z0;

b2Z

Vy2

2K2

Cinn2

??nn1

nn2

eðVy2ðhK y0Þ=2K2Þ

eðVy1ðhK y0Þ=2K1ÞeK inn2jh1K y0j;

b3Zk1nn1

k2nn2

eðVy2ðhK y0Þ=2K2Þ

eðVy1ðhK y0Þ=2K1ÞeK inn2jh1K y0j;

when the source is in the half-space (y0Oh1).

Case II. -null temperatures at yZ0.

Vy1

2K1

Kinn1

0

@

Vy1

2K1

1

A

Vy1

2K1

Cinn1

0

@

1

AeK inn1h1

Cinn1

0

Kinn1

0

@

1

AeK inn1h1

Vy1

2K1

0

@

1

A

Knn1

nn2

Vy2

2K2

Kinn2

0

@

Kk1nn1

k2nn2

1

AeðVy2ðh1K y0Þ=2K2Þ

eðVy2ðh1K y0Þ=2K2Þ

eðVy1ðh1K y0Þ=2K1Þ

eðVy1ðh1K y0Þ=2K1Þ

eK inn1h1

1

2

666666666664

3

777777777775

At

n1

Ab

n1

At

n2

2

664

3

775Z

b1

b2

b3

2

64

3

75;

(24)

1eK inn1h1

0

Vy1

2K1Kinn1

0

@

1

AeK inn1h1

Vy1

2K1

Cinn1

0

@

1

A Knn1

nn2

Vy2

2K2Kinn2

0

@

Kk1nn1

k2nn2

1

AeðVy2ðh1K y0Þ=2K2Þ

eðVy2ðh1K y0Þ=2K2Þ

eðVy1ðh1K y0Þ=2K1Þ

eðVy1ðh1K y0Þ=2K1Þ

eK inn1h1

1

2

6666666664

3

7777777775

At

n1

Ab

n1

At

n2

2

664

3

775Z

b1

b2

b3

2

64

3

75:

(25)

N. Simo ˜es, A. Tadeu / Engineering Analysis with Boundary Elements 29 (2005) 1130–11421137

Page 9

where

b1ZK eK inn1y0;

b2ZKVy1

2K1

Kinn1

??

eK inn1jh1K y0j;

b3ZK eK inn1jh1K y0j;

when the source is in the layer ðy0!h1Þ;

while

b1Z0;

b2Z

Vy2

2K2

Cinn2

??nn1

nn2

eðVy2ðhK y0Þ=2K2Þ

eðVy1ðhK y0Þ=2K1ÞeK inn2jh1K y0j;

b3Zk1nn1

k2nn2

eðVy2ðhK y0Þ=2K2Þ

eðVy1ðhK y0Þ=2K1ÞeK inn2jh1K y0j;

when the source is in the half–space ðy0Oh1Þ:

The heat field produced within the two media results

from the contribution of both the surface terms generated at

the various interfaces and the source term.

y0!h1(source in the medium 1)

~Tðu;x;y;kzÞ ZK i

4k1

eðVy1ðyK y0Þ=2K1ÞH0ðKt1r0Þ

nZC N

E11

nn1

~Tðu;x;y;kzÞ ZE02eðVy2ðyK y0Þ=2K2Þ

E21

nn2

CE01eðVy1ðyK y0Þ=2K1ÞX

if y!h1;

nZC N

?

nZ K N

At

n1CE12

nn1

Ab

n1

??

Ed;

!

X

nZ K N

At

n2

?

Ed;

if y!h1:

(26)

y0Oh1(source in the medium 2)

~Tðu;x;y;kzÞZE01eðVy1ðyKy0Þ=2K1ÞX

if y!h1;~Tðu;x;y;kzÞZKi

nZCN

nZKN

E11

nn1

At

n1CE12

nn1

Ab

n1

??

Ed;

4k2

?

eðVy2ðyKy0Þ=2K2ÞH0ðKt2r0Þ

CE02eðVy2ðyKy0Þ=2K2ÞX

nZCN

nZK N

E21

nn2

At

n2

?

Ed;if yOh1:

(27)

with

KtjZ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2Kj

K

Vyj

??2

Kiu

KjKðkzÞ2

s

ðjZ1;2Þ:

5.2. Layer bounded by two semi-infinite media

For the case of the layer placed between two semi-

infinite media, the solution needs additionally to account

for the continuity of temperature and heat fluxes at the

interface 1, since the heat propagation also occurs

through the top semi-infinite space (medium 0), which

can be expressed by

~T02ðu;x;y;kzÞ

ZE00eðVy0ðyK y0Þ=2K0ÞX

nZC N

nZ K N

E01

nn0

Ab

n0

??

Ed;

(28)

where

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

The surface terms produced in the interfaces 1 and 2, in

the media 1 and 2 (bottom semi-infinite medium) are

expressed in Eqs. (21)–(23).

The coefficients Ab

respecting the continuity of heat fluxes and temperatures at

yZh1 and yZ0. These surface conditions lead to the

following system of four equations when the heat source is

placed within the layer.

E0jZðK i=2kjLxÞ,E00ZeK inn0y, and

nn0Z

K ðVy0=2K0Þ2Kðiu=K0ÞKk2

zKk2

xn

q

.

n0, At

n1, Ab

n1and At

n2are defined by

where

c3jZe

Vyj

P

j

lZ1

hlK y0

??

=2Kj

??

nnj

;

c4jZe

Vyj

P

jK 1

lZ1

hlK y0

??

=2Kj

??

nnj

;

Vy0

2K0

Cinn0

0

@

1

Ac30

c41

K

Vy1

2K1

? inn1

0

@

1

A

K

Vy1

2K1

Cinn1

0

@

1

AeK inn1h1

0

k1c30

k0c41

K 1

K eK inn1h1

0

0

Vy1

2K1

? inn1

0

@

1

AeK inn1h1

Vy1

2K1

Cinn1

0

@

1

A

K

Vy2

2K2

? inn2

2

4

3

5c42

c31

0eK inn1h1

1

Kk1c42

k2c31

2

66666666666666664

3

77777777777777775

Ab

n0

At

n1

Ab

n1

At

n2

2

666664

3

777775

Z

b1

b2

b3

b4

2

66664

3

77775

(29)

N. Simo ˜es, A. Tadeu / Engineering Analysis with Boundary Elements 29 (2005) 1130–11421138

Page 10

b1Z

Vy1

2K1

Cinn1

??

eK inn1y0;

b2ZeK inn1y0;

b3ZKVy1

2K1

Kinn1

??

eK inn1jh1K y0j;

b4ZK eK inn1jh1K y0j;

when the source is in the layer ð0!y0!h1Þ;

while

b1ZKVy0

2K0Kinn0

??c30

c41

eK inn0jy0j;

b2ZKk1c30

k0c41

eK inn0jy0j;

b3Z0;

b4Z0

when the source is in the half–space ðy0!h1Þ:

and

b1Z0;

b2Z0;

b3Z

Vy2

2K2

Cinn2

??c42

c31

eK inn2jh1K y0j;

b4Zk1c42

k2c31

eK inn2jh1K y0j;

when the source is in the half–space ðy0Oh1Þ:

The temperatures for the three media are then computed

by adding the contribution of the source terms to those

associated with the surface terms originated at the various

interfaces. This procedure produces the following

expressions for the temperatures in the three media.

~Tðu;x;y;kzÞZE00eðVy0ðyKy0Þ=2K0ÞX

~Tðu;x;y;kzÞZKi

4k1

nZCN

E11

nn1

nZCN

nZK N

E01

nn0

Ab

n0

??

Ed;ify!0;

eðVy1ðyKy0Þ=2K1ÞH0ðKt1r0ÞCE01eðVy1ðyKy0Þ=2K1Þ

!

X

nZKN

At

n1CE12

nn1

Ab

n1

??

Ed;if0!y!h1;

~Tðu;x;y;kzÞZE02eðVy2ðyKy0Þ=2K2ÞX

nZCN

nZK N

E21

nn2

At

n2

??

Ed;ifyOh1:

(30)

The derivation presented assumed that the spatially

sinusoidal harmonic heat source is placed within the layer.

However, the equations can be easily manipulated to

accommodate another position of the source.

5.3. Multi-layer

The Green’s functions for a multi-layer are established

using the required boundary conditions at all interfaces.

Consider a system built from a set of m plane layers of

infinite extent bounded by two flat, semi-infinite media, as

shown in Fig. 4. The top semi-infinite medium is called

medium 0, while the bottom semi-infinite medium is

assumed to be the medium mC1. The thermal material

properties and thickness of the various layers may differ.

Different vertical convection velocities can be ascribed at

each layer. The convection is computed assuming that the

origin of convection coincides with the conduction source.

The system of equations is achieved considering that the

multi-layer is excited by a spatially sinusoidal heat source

located in the first layer (medium 1). The heat field at some

position in the domain is computed, taking into account both

the surface heat terms generated at each interface and the

contribution of the heat source term.

For the layer j, the heat surface terms on the upper and

lower interfaces can be expressed as

~Tj1ðu;x;y;kzÞ ZE0jeðVyjðyK y0Þ=2KjÞX

nZC N

nZ K N

Ej1

nnj

At

nj

??

Ed;

~Tj2ðu;x;y;kzÞ ZE0jeVyjðyK y0Þ=2KjX

nZC N

nZ K N

Ej2

nnj

Ab

nj

??

Ed;

(31)

where E0jZK i=2kjLx, Ej1Ze

nnjZ

K ðVy=2KjÞ2CðK iu=KjÞKkJ2

and hlis the thickness of the layer l. The heat surface terms

produced at interfaces 1 and mC1, which govern the heat

that propagates through the top and bottom semi-infinite

media, are ,respectively, expressed by

K innjyKP

jK 1

lZ1

hl

????????, Ej2Ze

xn

; with Im(nxj)%0

K innjyKP

j

lZ1

hl

????????,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

zKk2

q

~T02ðu;x;y;kzÞ ZE00eðVy0ðyK y0Þ=2K0ÞX

~TðmC1Þ2ðu;x;y;kzÞ ZE0ðmC1ÞeðVyðmC1ÞðyK y0Þ=2KðmC1ÞÞX

!

nnðmC1Þ

nZC N

nZ K N

E01

nn0

Ab

n0

??

Ed;

nZC N

nZ K N

EðmC1Þ2

At

nðmC1Þ

??

Ed:

(32)

Medium 1

Y

X

h1

Medium m+1

Medium 0

Interface 1

Interface m+1

Medium 2

Medium m

Interface 2

Interface m

hm

Fig. 4. Geometry of the problem for a multi-layer bounded by two semi-

infinite media.

N. Simo ˜es, A. Tadeu / Engineering Analysis with Boundary Elements 29 (2005) 1130–11421139

Page 11

A system of 2(mC1) equations is derived, ensuring the

continuity of temperatures and heat fluxes along the mC1

interfaces between layers. Each equation takes into account

the contribution of the surface terms and the involvement of

the incident field. All the terms are organized according to

the form FaZb

where

c1jZ

Vyj

2Kj

Cinnj

??

?

;

c2jZ

Vyj

2Kj

? innj

??

;

c3jZe

Vyj

P

j

lZ1

hlK y0

?

=2Kj

nnj

;

c4jZe

Vyj

P

jK 1

lZ1

hlK y0

??

=2Kj

nnj

and

c5jZeK innjhj:

The resolution of the system gives the amplitude of the

surface terms in each interface. The temperature field for

each layer formation is obtained by adding these surface

terms to the contribution of the incident field, leading to the

following equations.

Top semi-infinite medium (medium 0)

~Tðu;x;y;kzÞ ZE00eðVy0ðyK y0Þ=2K0ÞX

if y!0;

nZC N

nZ K N

E01

nn0

Ab

n0

??

Ed;

Layer 1 (source position)

~Tðu;x;y;kzÞZKi

nZCN

4k1

eðVy1ðyKy0Þ=2K1ÞH0ðK1r0ÞCE01eðVy1ðyKy0Þ=2K1Þ

!

X

nZK N

E11

nn1

At

n1CE12

nn1

Ab

n1

??

Ed;if 0!y!h1;

Layer j (js1)

~Tðu;x;y;kzÞZE0jeðVyjðyKy0Þ=2KjÞX

if

X

nZCN

nZKN

Ej1

nnj

At

njCEj2

nnj

Ab

nj

??

Ed;

jK1

lZ1

hl!y!

X

j

lZ1

hl;

Bottom semi-infinite medium (medium mC1)

~TðmC1Þ2ðu;x;y;kzÞ

ZE0ðmC1ÞeðVyðmC1ÞðyKy0Þ=2KmC1ÞX

nZCN

nZK N

EðmC1Þ2

nnðmC1Þ

At

nðmC1Þ

??

Ed:

(34)

Notice that when the position of the heat source

is changed, the matrix F remains the same, while

the independent terms of?b are different. However, as the

equations can be easily manipulated to consider another

position for the source, they are not included here.

5.4. Verification of the solution

Next, the results are found for the three scenarios. First, a

flat layer, 3.0 m thick, is assumed to be bounded by one half-

space. Null heat fluxes or null temperatures are prescribed at

the top surface. Then, a flat layer, also 3.0 m thick, bounded

by two semi-infinite media, is used to evaluate the accuracy

of the proposed formulation. The convection velocities

applied to the three media were 5!10K7, 8!10K7and 1!

10K6m sK1for the top medium, intermediate layer and

bottom medium, respectively. The thermal material proper-

tiesusedintheintermediatelayerwerekZ1.4 W mK18CK1,

cZ880.0 J KgK18CK1and rZ2300.0 Kg mK3, while at

the top and bottom media were kZ63.9 W mK18CK1, cZ

434.0 J KgK18CK1and rZ7832.0 Kg mK3.

c10c30

K c21c41

K c11c41c51

.

000

c30

k0

Kc41

k1

Kc41c51

k1

.

000

0c21c31c51

c11c31

.

000

0

c31c51

k1

.

c31

k1

.

.

000

.....

000

.

K c2mc4m

K c1mc4mc5m

0

000

.

Kc4m

km

Kc41c5m

km

0

000

.

c2mc3mc5m

c1mc3m

K c2ðmC1Þc4ðmC1Þ

000

.

c3mc5m

km

c3m

km

Kc4ðmC1Þ

kðmC1Þ

2

66666666666666666666666666666664

3

77777777777777777777777777777775

Ab

n0

At

n1

Ab

.

n1

Atnm

Ab

nm

At

nðmC1Þ

2

66666666666664

3

77777777777775

Z

c11c41eK inn1y0

c41

k1

eK inn1y0

K c21c31eK inn1jh1K y0j

Kc31

k1

eK inn1jh1K y0j

.

0

0

0

0

2

666666666666666666666666666664

3

777777777777777777777777777775

(33)

N. Simo ˜es, A. Tadeu / Engineering Analysis with Boundary Elements 29 (2005) 1130–11421140

Page 12

The calculations have been performed in the frequency

domain from 0 to 32!10K7Hz, with a frequency increment

of DuZ10K7Hz and considering a single value of kzequal

to 0.4 rad mK1. The amplitude of the response for two

receivers placed in two different media was computed for a

heat point source applied at (xZ0.0 m, yZ1.0 m). The

real and imaginary parts of the response at the receiver 1

(xZ0.2 m, yZ3.5 m) and the receiver 2 (xZ0.2 m, yZ

3.5 m) are displayed in Figs. 5 and 6, with the imaginary

part of the frequency set to hZ0.7Du. The solid lines

represent the analytical responses, while the marked points

correspond to the BEM solution.

As can be seen, these two solutions seem to be in very

close agreement, and equally good results were obtained

Receiver 1 Receiver 2

–0.050

–0.025

0

0.025

0.050(a)

(b)

Amplitude (°C)

Amplitude (°C)

Amplitude (°C)

Amplitude (°C)

–0.002

–0.001

0

0.001

0.002

0.003

–0.02

–0.01

0

0.01

0.02

–0.002

–0.001

0

0.001

0.002

01x10–6

2x10–6

3x10–6

Frequency (Hz)

01x10–6

2x10–6

3x10–6

Frequency (Hz)

0 1x10–6

2x10–6

3x10–6

Frequency (Hz)

01x10–6

2x10–6

3x10–6

Frequency (Hz)

real part

imaginary part

real part

imaginary part

real part

imaginary part

real part

imaginary part

Fig. 5. Real and imaginary parts of the responses for a layer over a half-space: (a) Case I-null heat fluxes at yZ0, responses at receivers 1 and 2; and (b) Case II-

null temperatures at yZ0, responses at receivers 1 and 2.

–0.010

–0.005

0

0.005

0.010 (a)(b)

Amplitude (°C)

Amplitude (°C)

–0.002

–0.001

0

0.001

0.002

01x10–6

2x10–6

3x10–6

Frequency (Hz)

01x10–6

2x10–6

3x10–6

Frequency (Hz)

real part

imaginary part

real part

imaginary part

Fig. 6. Real and imaginary parts of the responses for a layer bounded by two semi-infinite media: (a) Receiver 1; and (b) Receiver 2.

N. Simo ˜es, A. Tadeu / Engineering Analysis with Boundary Elements 29 (2005) 1130–11421141

Page 13

from tests in which heat sources and receivers were situated

at different points.

6. Conclusions

This paper has presented the 2.5 Green’s functions for

computing the transient heat transfer by conduction and

convection in an unbounded medium, half-space, slab and

layered media. In this approach the calculations are first

performed in the frequency domain. The results for a

layered formation are obtained adding the heat source term

and the surface terms, required to satisfy the interface

boundary conditions (temperature and heat fluxes con-

tinuity). Notice that null convection velocities can be

prescribed for the different layers, allowing solid layers to

be modelled.

The unbounded medium formulation was corroborated

by comparing its time responses and the exact time

solutions. In turn, the analytical solutions used in the half-

space, slab systems and layered media formulation were

verified using a BEM algorithm. Very good agreement was

found between the solutions.

The 2.5 Green’s functions for a layered medium can be

useful for solving problems such as the heat performance of

a multi-layer construction element. Using these funda-

mental solutions together with the BEM or MFS algorithms

can be helpful in resolving engineering problems, such as

the case of layered formations with buried inclusions.

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