This article was downloaded by: [University of California, San Francisco]On: 04 November 2014, At: 20:27Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Numerical Heat Transfer, Part B:Fundamentals: An International Journalof Computation and MethodologyPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/unhb20

Estimation of Surface Conditions forNonlinear Inverse Heat ConductionProblems Using the Hybrid InverseSchemeHan-Taw Chen a & Xin-Yi Wu aa Department of Mechanical Engineering , National Cheng KungUniversity , Tainan City, Taiwan, Republic of ChinaPublished online: 24 Feb 2007.

To cite this article: Han-Taw Chen & Xin-Yi Wu (2007) Estimation of Surface Conditions for NonlinearInverse Heat Conduction Problems Using the Hybrid Inverse Scheme, Numerical Heat Transfer, PartB: Fundamentals: An International Journal of Computation and Methodology, 51:2, 159-178, DOI:10.1080/10407790600878734

To link to this article: http://dx.doi.org/10.1080/10407790600878734

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoever orhowsoever caused arising directly or indirectly in connection with, in relation to or arisingout of the use of the Content.

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &

Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a, S

an F

ranc

isco

] at

20:

27 0

4 N

ovem

ber

2014

ESTIMATION OF SURFACE CONDITIONS FORNONLINEAR INVERSE HEAT CONDUCTIONPROBLEMS USING THE HYBRID INVERSE SCHEME

Han-Taw Chen and Xin-Yi WuDepartment of Mechanical Engineering, National Cheng Kung University,Tainan City, Taiwan, Republic of China

A hybrid numerical method involving the Laplace transform technique and finite-difference

method in conjunction with the least-squares method and actual experimental temperature

data inside the test material is proposed to estimate the unknown surface conditions of

inverse heat conduction problems with the temperature-dependent thermal conductivity

and heat capacity. The nonlinear terms in the differential equations are linearized using

the Taylor series approximation. In this study, the functional form of the surface conditions

is unknown a priori and is assumed to be a function of time before performing the inverse

calculation. In addition, the whole time domain is divided into several analysis subtime inter-

vals and then the unknown estimates on each subtime interval can be predicted. In order to

show the accuracy and validity of the present inverse scheme, a comparison among the

present estimates, direct solution, and actual experimental temperature data is made.

The effects of the measurement errors, initial guesses, and measurement location on the esti-

mated results are also investigated. The results show that good estimation of the surface

conditions can be obtained from the present inverse scheme in conjunction with knowledge

of temperature recordings inside the test material.

INTRODUCTION

It is usually assumed in both theoretical and industrial applications that theboundary conditions must be accurately given. However, in many heat transfersituations, it is difficult to measure or produce the approximate boundary conditionsfor a real problem. In general, the estimation of the unknown surface conditionsfrom the measured temperature histories at the accessible interior locations of theconducting material can be regarded as the inverse heat conduction problem (IHCP).The IHCP is often regarded as an ill-posed problem because the solution does notsatisfy the general requirements of existence, uniqueness, and stability due to smallchanges in the input data. The main difficulty of the IHCP is that its solution is verysensitive to changes in the input data resulting from measurement errors [1–3]. Todate, various methods [1–20], such as the regularization, least-squares, sequential,

Received 30 April 2006; accepted 16 June 2006.

The authors gratefully acknowledge the financial support provided by the National Science Council

of the Republic of China under Grant NSC 90-2212-E-145-001.

Address correspondence to Han-Taw Chen, Department of Mechanical Engineering, National

Cheng Kung University, Tainan City, Taiwan 701, Republic of China. E-mail: htchen@mail.ncku.edu.tw

159

Numerical Heat Transfer, Part B, 51: 159–178, 2007

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7790 print=1521-0626 online

DOI: 10.1080/10407790600878734

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a, S

an F

ranc

isco

] at

20:

27 0

4 N

ovem

ber

2014

conjugate gradient, function specification, Kalman filter, group-preserving, andhybrid inverse methods, have been developed for solving the IHCP. Most of theprevious works were confined to problems with constant thermal properties.However, the material properties can be a function of temperature for most realisticproblems that are encountered in engineering applications. Under this circumstance,the conduction problem will become nonlinear. It can be found from [5–11] that theestimation of the unknown surface conditions for the nonlinear IHCP is moredifficult than those for the linear IHCP.

Most numerical schemes for the IHCP may be sensitive to measurement noise.This sensitivity often depends on the time step. In general, the smaller the time stepis, the more ill-posed the problem becomes. In order to overcome this drawback,Chen and Chang [12] first introduced a hybrid scheme of the Laplace transformand finite-difference methods to estimate the unknown surface temperature inone-dimensional IHCPs using measured nodal temperatures inside the material with-out measurement errors. However, the measurement location had to be placed nearthe active boundary in order to obtain a more accurate estimation. Later, Chen et al.[13] and Chen and Wu [14–16] applied a similar scheme in conjunction with asequential-in-time concept and the least-squares method to estimate the unknownsurface conditions from temperature measurements with measurement errors. Itcan be observed from the works of Chen et al. [13] and Chen and Wu [14–16] that

NOMENCLATURE

[A] global conduction matrix

Cj undetermined coefficients

C0 coefficient corresponding to heat

capacity per unit volume shown

in Eq. (12b)

d�j new correction of Cj

F1 unknown function

½F � forcing matrix

k thermal conductivity, W=m K

k0 coefficient corresponding to thermal

conductivity shown in Eq. (12a)

‘ distance between two neighboring

nodes

L length of the test material

M number of discrete measurement times

Mt number of discrete times

n total number of nodes

p number of subtime intervals

s Laplace transform parameter

t time

tf final time

tr discrete measurement time

t0 initial measurement time

T temperature, KeTT temperature in the transform domain

½eTT � global matrix of the nodal

temperatures in the transform domain

T0 initial temperature, K

x spatial coordinate

xm location of thermocouple

a0 referenced thermal diffusivity, m2=s

b coefficient corresponding to thermal

conductivity shown in Eq. (12a)

c coefficient corresponding to heat

capacity per unit volume shown

in Eq. (12b)

d relative error between the current value

and the value of its previous iteration

e prescribed accuracy

g dimensionless spatial coordinate

ð¼x=LÞrexa standard deviation of the mean with

respect to exact data

rmea standard deviation of the mean with

respect to temperature measurements

s dimensionless time ð¼ t=tf Þx�r random error of the measurement

Subscripts

cal calculated value

exa exact value

exp experimental data

mea measured value

160 H.-T. CHEN AND X.-Y. WU

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a, S

an F

ranc

isco

] at

20:

27 0

4 N

ovem

ber

2014

the estimated surface temperatures or heat fluxes are in good agreement with theexact solution of the direct problem for various cases. The main advantages of thishybrid inverse method are that the estimates at any specific time can be obtainedwithout step-by-step computation and are independent of the time step in perform-ing the inverse calculation, due to the application of the Laplace transform.

It is worth mentioning that a few investigators predicted the unknown surfaceconditions from actual experimental data. Moreover, these investigators are oftenfree to choose their required type of boundary condition in order to show the accu-racy of their inverse schemes, such as the Dirichlet, Neumann, and mixed boundaryconditions. Ji et al. [18] applied a recursive least-squares algorithm to estimate theunknown surface heat flux of the one-dimensional IHCP from actual experimentaldata. Ji and Jang [19] provided experimental data to verify the stability of theKalman filter technique for IHCPs. In these studies [18, 19], Neumann boundaryconditions were chosen. Li [20] applied the implicit finite-difference method in con-junction with the linear least-squares errors method and experimental data to predictthe unknown surface temperature with Dirichlet boundary conditions. It can beobserved that Li’s predicated results did not agree well with experimental tempera-ture data for short times. However, it can be found from [14] that the estimates ofthe unknown surface temperature and heat flux obtained from the hybrid inversescheme agreed well with the direct solutions and actual experimental temperaturedata for various types of boundary condition.

The application of the least-squares method to nonlinear IHCPs can yield amore accurate result as the number of unknown parameters increases. However, thisincrease also causes instabilities. Thus, Beck et al. [5] applied the sequential methodto avoid the instabilities by subdividing the time domain and then solved the non-linear IHCP over each subdomain using the least-squares method. Their method[5] made assumptions about the behavior of experimental data at future time steps.Dorai and Tortorelli [6] used Newton’s method to estimate the unknown boundaryconditions of the linear and nonlinear IHCP. For the linear IHCP, the estimationcan be obtained in one iteration. However, the inverse solution sometimes divergeddue to a small radius of convergence for the nonlinear IHCP. The conjugate gradientmethod (CGM) was employed to solve the nonlinear IHCP by Alifanov and Artyu-khin [7]. Wang et al. [8] applied the CGM to investigate the effects of the surfacetemperature and conditions on the surface absorptivity in the laser surface heatingprocess, where the heat capacity and thermal conductivity were dependent on tem-perature. It can be observed from this work [8] that the thermocouple was placednear the heated surface in order to obtain a more accurate estimate. Wang et al.[8] stated that the CGM is essentially an iterative regularization scheme in whichan initial guess for the iterations must be chosen appropriately. However, the initialguess is difficult to determine. In addition, good convergence is difficult or imposs-ible to obtain if the initial guess is not chosen properly. Thus, in order to increase theaccuracy of the estimates, the result obtained from the least-squares method wasused as an initial guess for the CGM in their study [8]. Khachfe and Jarny [9] usedthe finite-element technique together with the CGM to solve the two-dimensionalnonlinear IHCP. Yang [10] applied a sequential method in conjunction with thefinite-element method, the concept of future time, and a modified Newton-Raphsonmethod to estimate the unknown boundary conditions of the nonlinear IHCP.

NONLINEAR INVERSE HEAT CONDUCTION 161

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a, S

an F

ranc

isco

] at

20:

27 0

4 N

ovem

ber

2014

Daouas and Radhouani [11] applied the smoothing technique associated with theKalman filter to solve a nonlinear one-dimensional IHCP. Chen and Lin [21, 22]have applied the hybrid method of the Laplace transform and finite-difference meth-ods to solve nonlinear transient thermal problems. It can be observed from these twoworks that this hybrid method is efficient and accurate for the nonlinear transientthermal problems. Thus this study further applies this hybrid scheme in conjunctionwith a sequential-in-time concept, the Taylor series approximation, least-squaresmethod, and actual experimental temperature data inside the test material to esti-mate the unknown surface conditions for the nonlinear IHCP. In order to showthe accuracy of the present inverse scheme for the nonlinear IHCP, a comparisonof the estimated surface temperature among the present estimates, direct solution,and actual experimental temperature data is made.

MATHEMATICAL FORMULATION

The IHCP investigated here involves the estimation of the unknown surfacetemperature and heat flux from transient temperature measurements inside the testcylindrical bar. The test cylindrical bar is enclosed by a highly insulated material andis heated at one end. The ratio of the length to the diameter is large enough to ensurethat the assumption of a one-dimensional model can be used. The physical geometryof the present problem is shown in Figure 1. For the direct heat conduction problem,the temperature field can be determined provided the surface conditions at x ¼ 0 andx ¼ L are given. However, one of the surface conditions for the present IHCP isunknown. It cannot be estimated unless additional information about the tempera-ture history in the test material is given. In this study, a thermocouple of type T iswelded at x ¼ xm to record the temperature history Tmeaðxm; trÞ during the heatingprocess, where tr is the discrete measurement time. The thermal conductivity kðTÞand heat capacity per unit volume qcpðTÞ are assumed to be functions of tempera-ture. Thus the unknown surface temperature and heat flux can be obtained simul-taneously by solving the one-dimensional nonlinear IHCP in conjunction withtransient temperature measurements inside the test cylindrical bar. The initialtemperature of this test material can vary with the spatial coordinate in the presentstudy. The governing differential equation, boundary conditions, and initialcondition are expressed as

qqx

kðTÞ qT

qx

� �¼ qcpðTÞ

qT

qtfor 0 < x < L; 0 < t � tf ð1Þ

Figure 1. Schematic diagram of the inverse problem.

162 H.-T. CHEN AND X.-Y. WU

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a, S

an F

ranc

isco

] at

20:

27 0

4 N

ovem

ber

2014

Tð0; tÞ ¼ F0ðtÞ ¼ ? ð2Þ

TðL; tÞ ¼ FLðtÞ ð3Þ

and

Tðx; 0Þ ¼ T0ðxÞ ð4Þ

where T is the temperature, T0 is the initial temperature, t and x denote the tem-poral- and spatial-domain variables, respectively, and tf is the final time.

For convenience of the numerical analysis, the following dimensionless para-meters are introduced:

g ¼ x

Land s ¼ t

tfð5Þ

The substitution of Eq. (5) into Eqs. (1)–(4) gives the following equations:

qqg

kðTÞ qT

qg

� �¼ L2

tfqcpðTÞ

qT

qsfor 0 < g < 1; 0 < s � 1 ð6Þ

Tð0; sÞ ¼ F1ðsÞ ¼ ? ð7Þ

Tð1; sÞ ¼ FnðsÞ ð8Þ

and

Tðg; 0Þ ¼ TinðgÞ ð9Þ

where F1ðsÞ is the unknown function. It can be estimated provided that the know-ledge of temperature history at g ¼ gm is given.

NUMERICAL ANALYSIS

In industrial applications, the measured temperature profiles often exhibit ran-dom oscillations due to measurement errors [8, 18–20]. On the other hand, due toexperimental uncertainties, more realistic measurements should add simulated smallrandom errors to the solution obtained from the related direct problem, as shown in[8, 13–16]. Thus, in order to simulate the experimental temperature Texp, the exactsolution Texa can be modified by adding small random errors. The simulated tem-perature Texp used in the present inverse analysis can be expressed as

Texpðgm; srÞ ¼ Texaðgm; srÞð1� x�r Þ for r ¼ 0; 1; . . . ;M � 1 ð10Þ

where M denotes the number of the discrete measurement times, x�r is a randomnumber generated by QuickBASIC 4.50 and is assumed to be within 2.5% in thepresent study. It can be found from [14, 16, 23] that a smooth curve is frequently

NONLINEAR INVERSE HEAT CONDUCTION 163

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a, S

an F

ranc

isco

] at

20:

27 0

4 N

ovem

ber

2014

selected to present experimental measured data. Thus a curve-fitted profile generatedthrough a least-squares process is used to fit experimental temperature data. Later,the temperature measurement Tmea is obtained from this curve-fitted profile in thepresent study. The standard deviations of the mean of the experimental temperatureTexp with respect to the exact solution Texa and the temperature measurement Tmea

are defined, respectively, as [23]

rexa ¼1

M

XM�1

r¼0

Texpðgm; srÞ � Texaðgm; srÞ� �2( )1=2

ð11aÞ

and

rmea ¼1

M

XM�1

r¼0

Texpðgm; srÞ � Tmeaðgm; srÞ� �2( )1=2

ð11bÞ

Assume that the thermal conductivity and heat capacity within the temperaturerange considered can be expressed as

kðTÞ ¼ k0ð1þ bTÞ ð12aÞ

and

qcpðTÞ ¼ C0ð1þ cTÞ ð12bÞ

where k0, C0, b, and c are coefficients corresponding to the thermal conductivity andheat capacity per unit volume.

The application of central-difference approximation to Eq. (6) in conjunctionwith Eq. (12) can yield the following discretized form:

Ti�1� 2Ti þTiþ1

‘2þ b

2

T2i�1� 2T2

i þT2iþ1

‘2¼ 1

Fo

qTi

qsþ c

2

qT2i

qs

� �for i ¼ 2;3; . . . ;n� 1

ð13Þ

where Fo is defined as Fo ¼ a0tf =L2, a0 ¼ k0=C0 is the referenced thermal diffusiv-ity, ‘ ¼ 1=ðn� 1Þ denotes the distance between two neighboring nodes and is uni-form, and n is the nodal number.

The application of the Laplace transform technique is restricted to linear sys-tems, so the nonlinear terms in Eq. (13) must be linearized. In this study, the Taylorseries approximation is applied to linearize the nonlinear terms. Thus the nonlinearterm T2

j can be linearized into [21, 22]

T2j ¼ 2TjTj � T

2

j for j ¼ i � 1; i; i þ 1 ð14Þ

where the overbar denote the initial guess or the previously iterated solution.

164 H.-T. CHEN AND X.-Y. WU

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a, S

an F

ranc

isco

] at

20:

27 0

4 N

ovem

ber

2014

Substituting Eq. (14) into Eq. (13) yields the linear algebraic equation

ð1þ bTi�1ÞTi�1 � 2ð1þ bTiÞTi þ ð1þ bTiþ1ÞTiþ1

¼ b2

T2

i�1 � 2T2

i þ T2

iþ1

� �þ ‘2

Fo1þ cTi

qTi

qsfor i ¼ 2; 3; . . . ; n� 1 ð15Þ

The Laplace transform technique is employed to remove the time-dependentterms in Eqs. (7), (8), and (15). Thus the Laplace transform of Eqs. (7), (8), and(15) with respect to s in conjunction with the initial condition (9) gives

ð1þ bTi�1ÞeTTi�1 � 2 1þ bTi

þ ‘2

Fo1þ cTi

s

� �eTTi þ 1þ bTiþ1

eTTiþ1

¼ b2s

T2

i�1 � 2T2

i þ T2

iþ1

� �� ‘2

Fo1þ cTi

TinðgiÞ for i ¼ 2; 3; . . . ; n� 1 ð16Þ

eTTð0; sÞ ¼ eTT1 ¼ eFF1ðsÞ ð17aÞ

and

eTTð1; sÞ ¼ eTTn ¼ eFFnðsÞ ð17bÞ

where s is the Laplace transform parameter. eTTi is defined as

eTTiðsÞ ¼Z 1

0

e�stTiðsÞ ds ð18Þ

Rearrangement of Eqs. (16) and (17) yields the following matrix equation:

½A�½eTT � ¼ F½ � ð19Þ

where [A] is a matrix with the Laplace transform parameter s, ½eTT � is a matrix repre-senting the unknown nodal temperatures in the transform domain, and F½ � is amatrix representing the forcing term. [A] and F½ � can be determined provided thatthe initial guesses of Ti, i ¼ 2, 3, . . . , n� 1, are given in advance. The direct Gaussianelimination algorithm and the numerical inversion of the Laplace transform [25] areapplied to solve Eq. (19) in order to determine the nodal temperatures at a specifictime. The computational procedure is performed repeatedly until the relative errorsbetween the current nodal temperatures and the values of its previous iteration areall less than the tolerance value d. In other words, an iterative solution is said tobe convergent if

TðmÞi � T

ðmþ1Þi

��� ���TðmÞi

< d for i ¼ 2; 3; . . . ; n� 1 ð20Þ

where the superscript m denotes the mth iteration. One of the advantages of thepresent method is that the unknown estimation does not always need to proceed withstep-by-step computation from the initial time s ¼ 0, due to the application of theLaplace transform.

NONLINEAR INVERSE HEAT CONDUCTION 165

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a, S

an F

ranc

isco

] at

20:

27 0

4 N

ovem

ber

2014

The functional form of the unknown surface temperature F1ðsÞ in the presentstudy is assumed to be a function of s before performing the inverse calculation.However, it is not easy to obtain an approximate polynomial function which cancompletely fit F1ðsÞ for the whole time domain considered. Under this circumstance,a sequential-in-time procedure is introduced to estimate F1ðsÞ. In other words, thewhole time domain 0 < s � 1 is divided into p subtime intervals. The unknown sur-face temperature F1ðsÞ on each subtime interval can be approximated by an(N� 1)th-degree polynomial guess function of s and is expressed as

F1ðsÞ ¼XN

j¼1

Cjsj�1 ð21Þ

where Cj , j ¼ 1, 2, . . . , N, are the unknown coefficients and can be estimated usingthe least-squares method in conjunction with the temperature measurements on eachsubtime interval.

The discrete time sr during the inverse calculation can be written as sr ¼ r Dse,r ¼ 0, 1, . . . , Mt � 1, where Mt denotes the number of the discrete times and the timestep Dse is defined as Dse ¼ 1=ðMt � 1Þ. It can be obtained from the definitions of N,p, and Mt that the Mt value equals ‘‘p(N� 1)þ 1.’’

The least-squares minimization technique is applied to minimize the sum of thesquares of the deviations between the calculated and measured temperatures at anapproximate sensor location gm. The error in the estimates EðC1;C2; . . . ;CNÞ on eachanalysis interval, si � sr � siþN�1, i ¼ 0, N� 1, . . . , Mt �N, can be expressed as

EðC1;C2; . . . ;CNÞ ¼XiþN�1

r¼i

Tcalðgm; srÞ � Tmeaðgm; srÞ½ �2 ð22Þ

where Tcalðgm; srÞ denotes the calculated temperature. Tmeaðgm; srÞ is the tempera-ture measurement obtained from the curve-fitted profile of the experimental data.The estimated values of Cj, j ¼ 1, 2, . . . , N, are determined provided that the valueof EðC1;C2; . . . ;CNÞ is minimum. The detailed computational procedures for esti-mating the unknown coefficients Cj, j ¼ 1, 2, . . . , N, can be found in [13–16]. In orderto avoid repetition, they are not shown in this article.

In order to yield the minimum value of E with respect to Cj , differentiating Ecorresponding to the new correction d�j is performed. Thus the correction equationscorresponding to Cj can be expressed as

XN

j¼1

XN

i¼1

xjix

ki d�j ¼ �

XN

j¼1

xkj ej for k ¼ 1; 2; . . . ;N ð23Þ

where xkj and ej are defined as xk

j ¼ qTcal=qCj and ej ¼ Tcalðgm; srÞ � Tmeaðgm; srÞfor j ¼ r� i þ 1 and i � r � i þN � 1. The new correction d�j , j ¼ 1, 2, . . . , N, areobtained from Eq. (23). Later, Cj ¼ Cj þ d�j can be determined and is regarded asthe new guess of Cj. The above numerical procedures are repeated until the valuesof Tcalðgm; srÞ � Tmeaðgm; srÞ=Tmeaðgm; srÞj j, r ¼ 0, 1, . . . , Mt � 1, are all less thana prescribed accuracy e. In the present study, e ¼ 1� 10�6 is taken through all thecases.

166 H.-T. CHEN AND X.-Y. WU

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a, S

an F

ranc

isco

] at

20:

27 0

4 N

ovem

ber

2014

Once the unknown surface temperature is determined, the temperature distri-bution can be obtained using the direct method. The unknown surface heat fluxqð0; sÞ can be determined from the following expression [1, 16]:

qð0; sÞ ¼ � kðTÞL

qT

qgð0; sÞ ¼ � kðT1Þ

L

4T2 � 3T1 � T3

2‘ð24Þ

where T1, T2, and T3 respectively denote the nodal temperatures at x ¼ 0, ‘, and 2‘,as shown in Figure 1. It is worth mentioning that the present inverse scheme can alsobe applied to predict the unknown surface temperature and heat flux using the Neu-mann boundary condition. However, its inverse computational procedures for theIHCP with the temperature-dependent thermal conductivity are more complicatedthan those using the Dirichlet boundary condition, as shown in Eq. (7). Moreover,the temperature distribution can be calculated provided the unknown surface tem-perature is obtained using the direct method. Consequently, the unknown surfaceheat flux can also be determined using the temperature distribution in conjunctionwith Eq. (24). For these reasons, the inverse computational procedure for theNeumann boundary condition is not shown in this study.

RESULTS AND DISCUSSION

In order to demonstrate the accuracy and efficiency of the present inversescheme in estimating the unknown surface conditions from the knowledge of thetemperature measurements at the selected location of the test material, a numericalexample and an experimental example are illustrated. At the same time, a compari-son among the present estimates, direct solution, and actual experimental tempera-ture data is made. In addition, the effects of the initial guesses, measurementlocation, measurement errors, and number of divided subtime intervals on thepresent estimates are also investigated. AISI-304 stainless steel is used as the materialof the test cylindrical bar. Its thermal conductivity and heat capacity per unit volumewithin the temperature range considered can be expressed by Eqs. (12a) and (12b)with k0 ¼ 9, C0 ¼ 363, b ¼ 0;00222, and c ¼ 0:00105 [24]. All the computations ofthe illustrated examples are performed on a PC.

Numerical Example

The inverse problem considered here is concerned with the estimation of theunknown surface temperature and heat flux at g ¼ 0 from the transient temperaturemeasurements taken at g ¼ gm, which is a function of time obtained form the relateddirect problem with the following boundary conditions and initial condition:

Tð0; sÞ ¼ 310es=4 ð25aÞ

Tð1; sÞ ¼ 300es=25 ð25bÞand

Tðg; 0Þ ¼ TinðgÞ ¼ 310� 14gþ 4g2 ð26Þ

NONLINEAR INVERSE HEAT CONDUCTION 167

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a, S

an F

ranc

isco

] at

20:

27 0

4 N

ovem

ber

2014

The Laplace transforms of the boundary conditions at g ¼ 0 and g ¼ 1 aregiven as

eTT1 ¼310

s� 0:25ð27aÞ

and

eTTn ¼300

s� 0:04ð27bÞ

Rearrangement of Eqs. (16) and (27) yields the matrix equation shown in Eq. (19).The temperature histories for various g values are shown in Figure 2. The tempera-ture histories at g ¼ 1=3 or g ¼ 2=3 can be chosen as additional temperatureinformation in order to predict the unknown surface conditions at g ¼ 0. Asecond-degree polynomial function (N ¼ 3) is selected to predict the unknownsurface temperature F1ðsÞ on each subtime interval. Thus the values of C1, C2,and C3 in Eq. (21) will be predicted. The Laplace transform of Eq. (21) is given as

eFF 1 ¼X3

j¼1

CjCðjÞ

sjð28Þ

where CðjÞ is the gamma function.

Figure 2. Temperature histories at g ¼ 0, 1=3, 2=3, and 1.

168 H.-T. CHEN AND X.-Y. WU

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a, S

an F

ranc

isco

] at

20:

27 0

4 N

ovem

ber

2014

The unknown coefficients C1, C2, and C3 used to begin the iterations are takenas unity. Once the unknown surface temperature Tð0; sÞ is determined, the tempera-ture distribution in the test material can be obtained using the direct method. Conse-quently, the unknown surface heat flux qð0; sÞ can also be obtained from Eq. (24). Inorder to investigate the number of the subtime intervals on the present estimates,Table 1 displays comparisons of the unknown surface temperature Tð0; sÞ and heatflux qð0; sÞ between the present estimates and exact solution for x�r ¼ 0, gm ¼ 1=3,n ¼ 16, and various p values at various dimensionless times. It can be observed fromTable 1 that the present estimates are convergent and agree well with the exact sol-ution for various p values. Thus p ¼ 2 is taken for all the calculations of this example.

In order to investigate the effects of the initial guesses C1, C2, and C3 on thepresent estimates, Table 2 shows comparisons of the unknown surface temperatureTð0; sÞ and heat flux qð0; sÞ for x�r ¼ 0, gm ¼ 1=3, n ¼ 16, p ¼ 2, and various valuesof the initial guesses. It is found from Table 2 that the differences of the presentestimates for C1 ¼ C2 ¼ C3 ¼ 0:1, C1 ¼ C2 ¼ C3 ¼ 1, and C1 ¼ C2 ¼ C3 ¼ 10 arevery small. The effect of the measurement location on the present estimates is also

Table 1. Present estimates of Tð0; sÞ and qð0; sÞ for x�r ¼ 0, n ¼ 16, gm ¼ 1=3, and various p values

Testð0; sÞ qestð0; sÞ

s T exað0; sÞ p ¼ 1 p ¼ 2 p ¼ 3 qexað0; sÞ p ¼ 1 p ¼ 2 p ¼ 3

0.1 317.85 317.84 317.85 317.85 4:1243� 103 4:1188� 103 4:1233� 103 4:1241� 103

0.2 325.89 325.88 325.89 325.89 5:3461� 103 5:3419� 103 5:3463� 103 5:3463� 103

0.3 334.14 334.13 334.15 334.14 6:3525� 103 6:3521� 103 6:3536� 103 6:3518� 103

0.4 342.60 342.60 342.61 342.60 7:2939� 103 7:2972� 103 7:2941� 103 7:2935� 103

0.5 351.28 351.29 351.27 351.28 8:2296� 103 8:2353� 103 8:2262� 103 8:2300� 103

0.6 360.17 360.19 360.17 360.17 9:1851� 103 9:1909� 103 9:1857� 103 9:1840� 103

0.7 369.29 369.31 369.29 369.28 1:0172� 104 1:0175� 104 1:0174� 104 1:0171� 104

0.8 378.63 378.64 378.64 378.64 1:1196� 104 1:1192� 104 1:1196� 104 1:1196� 104

0.9 388.22 388.19 388.21 388.22 1:2260� 104 1:2245� 104 1:2256� 104 1:2260� 104

1.0 398.05 397.96 398.01 398.04 1:3368� 104 1:3335� 104 1:3354� 104 1:3362� 104

Table 2. Effects of initial guesses on the present estimates for p ¼ 2, x�r ¼ 0, n ¼ 16, and gm ¼ 1=3

T estð0; sÞ qestð0; sÞ

sC1 ¼ C2

¼ C3 ¼ 0:1

C1 ¼ C2

¼ C3 ¼ 1

C1 ¼ C2

¼ C3 ¼ 10

C1 ¼ C2

¼ C3 ¼ 0:1

C1 ¼ C2

¼ C3 ¼ 1

C1 ¼ C2

¼ C3 ¼ 10

0.1 317.8457 317.8457 317.8457 4123.2713 4123.2713 4123.2713

0.2 325.8929 325.8929 325.8928 5346.2913 5346.2913 5346.2913

0.3 334.1461 334.1461 334.1461 6353.6093 6353.6093 6353.6093

0.4 342.6054 342.6054 342.6054 7294.1333 7294.1333 7294.1333

0.5 351.2707 351.2707 351.2707 8226.1933 8226.1933 8226.1933

0.6 360.1667 360.1667 360.1666 9185.6733 9185.6667 9185.6600

0.7 369.2898 369.2898 369.2897 10173.6667 10173.6600 10173.6533

0.8 378.6386 378.6385 378.6384 11196.1200 11196.1200 11196.1067

0.9 388.2130 388.2130 388.2129 12255.5933 12255.5867 12255.5733

1.0 398.0132 398.0131 398.0130 13353.6800 13353.6733 13353.6533

NONLINEAR INVERSE HEAT CONDUCTION 169

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a, S

an F

ranc

isco

] at

20:

27 0

4 N

ovem

ber

2014

investigated. Figure 3 shows a comparison of the unknown surface temperatureTð0; sÞ between the present estimates and exact solution for x�r ¼ 0, n ¼ 16, p ¼ 2,and various gm values. It can be observed from this figure that the deviationsbetween the present estimates and exact solution are small even though the measure-ment location is far from the position of the unknown boundary condition. In mostof the previous works, the measurement location is generally positioned in the neigh-borhood of the estimated unknown boundary condition in order to obtain a moreaccurate estimate. These comparative results imply that the effects of the initialguesses and measurement location on the present estimates are not very significantfor the present hybrid inverse method.

In order to show the effect of measurement errors on the present estimates,simulated experimental temperatures can be generated form Eq. (10). Later, anapproximate polynomial function generated through a least-squares process is usedto fit these experimental temperatures, as shown in Figure 4. The standard deviationof the mean of the temperature measurements with respect to the experimental tem-peratures for x�r ¼ 2:5% is rmea ¼ 0:055 K. Comparisons of the unknown surfacetemperature Tð0; sÞ and heat flux qð0; sÞ between the present estimates and the exactsolution for gm ¼ 1=3, p ¼ 2, n ¼ 16, and various x�r values are shown in Figures 5and 6, respectively. It can be observed from Figures 5 and 6 that the present esti-mates of both Tð0; sÞ and qð0; sÞ perform stably and deviate slightly from the exactsolution and direct solution, respectively. The maximum iterations of this numericalexample are 4 times on a subtime interval. The computational time is about 15 s.

Figure 3. Comparison of Tð0; sÞ between the present estimates and exact solution for x�r ¼ 0, p ¼ 2,

n ¼ 16, and various gm values.

170 H.-T. CHEN AND X.-Y. WU

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a, S

an F

ranc

isco

] at

20:

27 0

4 N

ovem

ber

2014

Figure 4. Simulated experimental temperatures at gm ¼ 1=3 and its curve-fitted values for x�r ¼ 2:5%.

Figure 5. Comparison of Tð0; sÞ between the present estimates and exact solution for gm ¼ 1=3, p ¼ 2,

n ¼ 16, and various x�r values.

NONLINEAR INVERSE HEAT CONDUCTION 171

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a, S

an F

ranc

isco

] at

20:

27 0

4 N

ovem

ber

2014

Experimental Example

Schematic diagrams of the present experimental arrangement and the test cyl-indrical bar are shown in Figures 7 and 8. AISI-304 stainless steel is used as thematerial of the test cylindrical bar. The test cylindrical bar, 500 mm in length and12 mm in diameter, is enclosed by a highly adiabatic material and heated at one

Figure 6. Comparison of qð0; sÞ between the present estimates and exact solution for gm ¼ 1=3, p ¼ 2,

n ¼ 16, and various x�r values.

Figure 7. Schematic diagram of the experimental arrangement.

172 H.-T. CHEN AND X.-Y. WU

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a, S

an F

ranc

isco

] at

20:

27 0

4 N

ovem

ber

2014

end with an electrical heater. Four thermocouples of type T are mounted at 200, 250,300, and 350 mm from the heating end of the test cylindrical bar with cyanoacrylateadhesive (Salton Enterprise Corporation, D-3) as shown in Figure 8. These thermo-couples are connected to a PC-based data acquisition system (National InstrumentsCorporation, SCXI system) for online downloading of temperature data from theexperiment. The limit of error for the thermocouple is 0.4%. The temperature mea-surements at 200 and 350 mm from the heating end are regarded as the boundaryconditions of the present experimental problem. Thus the length of the test cylindri-cal bar is 150 mm. The measured temperature histories for these four locations areshown in Figure 9. It can be observed form this figure that the experimental tempera-ture data exhibit slight random oscillations. The polynomial functions generatedthrough a least-squares process are also used to fit these experimental temperaturedata for the whole time domain, 0 � s � 1. The functional forms of these curve-fitted temperature measurements with respect to s can be expressed as follows:

Tðg ¼ 0; sÞ ¼322:1644996þ 328:054803s� 1466:114207s2 þ 4011:185446s3

�6149:232013s4 þ 4012:723782s5 for s < 0:4345:5013334þ 38:50390872sþ 33:35034934s2 � 171:9438394s3

þ183:1523453s4 � 64:85812025s5 for s � 0:4

8>><>>:9>>=>>;

ð29aÞ

T g ¼ 1

3; s

� �¼ 307:2644224þ 206:9819441s� 544:7901713s2 þ 795:1598014s3

� 603:6588482s4 þ 183:7697758s5 ð29bÞ

T g ¼ 2

3; s

� �¼ 303:4520202þ 117:9251814s� 192:8294568s2 þ 135:1223982s3

� 16:25534214s4 � 15:22385796s5 ð29cÞ

Figure 8. Schematic illustration of arrangement of test cylindrical bar.

NONLINEAR INVERSE HEAT CONDUCTION 173

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a, S

an F

ranc

isco

] at

20:

27 0

4 N

ovem

ber

2014

and

Tðg ¼ 1; sÞ ¼ 302:3925946þ 56:79664454sþ 10:79650568s2 � 202:087052s3

þ 256:5783034s4 � 100:8625637s5 ð29dÞ

The standard deviations of the mean of these temperature measurementswith respect to the experimental temperature data are rg¼0

mea ¼ 0:0024 K, rg¼1=3mea ¼

0:0014 K, rg¼2=3mea ¼ 0:0015 K, and rg¼1

mea ¼ 0:0016 K, respectively. In order to showthe accuracy and validity of the present inverse method further, it is assumed thatthe temperature history at g ¼ 0 will be estimated in this example. The temperaturemeasurements at g ¼ 1=3 or g ¼ 2=3 are selected as additional temperature infor-mation to perform the inverse calculations. A second-degree polynomial function(N ¼ 3) is still selected to predict the history of the unknown surface temperatureat g ¼ 0 on each subtime interval. The unknown coefficients C1, C2, and C3 usedto begin the iterations are still taken as unity. The estimated results show no obviousdeviations for p ¼ 4 and p ¼ 5. Thus p ¼ 4 is taken for the following calculations inthe present experimental example. Figure 10 shows the comparison of the unknownsurface temperature Tð0; sÞ between the present estimates and actual experimental

Figure 9. Experimental temperature data and its curve-fitted values at various g values.

174 H.-T. CHEN AND X.-Y. WU

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a, S

an F

ranc

isco

] at

20:

27 0

4 N

ovem

ber

2014

data for two different measurement locations, gm ¼ 1=3 and gm ¼ 2=3. It can beobserved form this figure that the present estimates of Tð0; sÞ for gm ¼ 1=3 are moreaccurate than those for gm ¼ 2=3. However, the present estimates for gm ¼ 1=3 andgm ¼ 2=3 both deviate slightly from experimental temperature data. The relativeerrors between the present estimates and actual experimental temperature are within1.6% for the whole time domain considered. This result shows further that thepresent inverse method is accurate and reliable. Once the unknown surface tempera-ture Tð0; sÞ is determined, the temperature distribution in the test material can beobtained using the direct method. Consequently, the unknown surface heat fluxqð0; sÞ can also be obtained from Eq. (24), as shown in Figure 11. It can be foundfrom this figure that the relative errors of the unknown surface heat flux qð0; sÞ ofthe present estimates for gm ¼ 1=3 and gm ¼ 2=3 are within 12.6%. Obviously,the present estimate of the unknown surface temperature Tð0; sÞ is more accuratethan that of the unknown surface heat flux qð0; sÞ. These deviations can result fromvarious experimental uncertainties, such as the heat loss from the test cylindrical barto the environment through the adiabatic material in the radius direction, the cali-bration and fluctuation in the thermocouple readings, and the composition of the

Figure 10. Comparison of Tð0; sÞ between the present estimates and actual experimental temperature data

for n ¼ 16, p ¼ 4, and various gm values.

NONLINEAR INVERSE HEAT CONDUCTION 175

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a, S

an F

ranc

isco

] at

20:

27 0

4 N

ovem

ber

2014

test material in the present experiment. The maximum iterations of the presentexperimental example are 4 times on a subtime interval, and the computational timeis about 22 s.

CONCLUSIONS

The present study proposes a hybrid inverse scheme involving the Laplacetransform and finite-difference methods in conjunction with a sequential-in-timeconcept, the Taylor series approximation, least-squares method, and actual experi-mental temperature data inside the test material to estimate the unknown surfacetemperature and heat flux for the nonlinear IHCP. Due to the application of theLaplace transform, the unknown surface temperature and heat flux can be estimatedsimultaneously at a specific time without step-by-step computations from t ¼ 0. It isfound from the above results that the present inverse method can give a goodestimation of the unknown surface conditions and is not very sensitive to themeasurement locations and initial guesses of the unknown coefficients. In addition,the present estimates also exhibit stable behavior for the cases with measurementerrors and deviate slightly from the exact solution for the present problem. In the

Figure 11. Estimated results of qð0; sÞ obtained from present inverse scheme for n ¼ 16, p ¼ 4, and various

gm values.

176 H.-T. CHEN AND X.-Y. WU

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a, S

an F

ranc

isco

] at

20:

27 0

4 N

ovem

ber

2014

experimental example, the present estimate of the unknown surface temperaturedeviates slightly from actual experimental temperature data for various measurementlocations. This implies that the present inverse method has good accuracy andreliability.

REFERENCES

1. M. N. Ozisik, Heat Conduction, 2nd ed., Wiley, New York, 1993.2. K. Kurpisz and A. J. Nowak, Inverse Thermal Problems, Computational Mechanics Pub-

lications, Southampton, UK, 1995.3. E. Hensel, Inverse Theory and Applications for Engineers, Prentice-Hall, Englewood Cliffs,

NJ, 1991.4. H. Y. Li and M. N. Ozisik, Identification of the Temperature Profile in an Absorbing,

Emitting, and Isotropically Scattering Medium by Inverse Analysis, ASME J. Heat Trans-fer, vol. 114, pp. 1060–1063, 1992.

5. J. V. Beck, B. Litkouhi, and C. R. St. Chair, Efficient Sequential Solution of the Non-linear Inverse Heat Conduction Problem, Numer. Heat Transfer, vol. 5, pp. 275–286,1982.

6. G. A. Dorai and D. A. Tortorelli, Transient Inverse Heat Conduction Problem Solutionvia Newton’s Method, Int. J. Heat Mass Transfer, vol. 40, pp. 4115–4127, 1997.

7. O. M. Alifanov and E. A. Artyukhin, Regularized Numerical Solution of NonlinearInverse Heat Conduction Problem, J. Eng. Phys., vol. 29, pp. 934–938, 1975.

8. J. T. Wang, C. I. Weng, and J. G. Chang, The Influence of Temperature and SurfaceConditions on Surface Absorptivity in Laser Surface Treatment, J. Appl. Phys., vol. 87,pp. 3245–3253, 2000.

9. R. Abou Khachfe and Y. Jarny, Numerical Solution of 2-D Nonlinear Inverse HeatConduction Problems Using Finite-Element Techniques, Numer. Heat Transfer B,vol. 37, pp. 45–67, 2000.

10. C. Y. Yang, Estimation of Boundary Conditions in Nonlinear Inverse Heat ConductionProblems, J. Thermophys. Heat Transfer, vol. 17, pp. 389–395, 2003.

11. N. Daouas and M. S. Radhouani, A New Approach of the Kalman Filter Using FutureTemperature Measurements for Nonlinear Inverse Heat Conduction Problems, Numer.Heat Transfer B, vol. 45, pp. 565–585, 2004.

12. H. T. Chen and S. M. Chang, Application of the Hybrid Method to Inverse Heat Conduc-tion Problems, Int. J. Heat Mass Transfer, vol. 33, pp. 621–628, 1990.

13. H. T. Chen, S. Y. Lin, and L. C. Fang, Estimation of Surface Temperature in Two-Dimensional Inverse Heat Conduction Problems, Int. J. Heat Mass Transfer, vol. 44,pp. 1455–1463, 2001.

14. H. T. Chen and X. Y. Wu, Application of the Hybrid Method to the Estimation of Sur-face Conditions from Experimental Data, Trends Heat, Mass, Momentum Transfer, vol. 8,pp. 103–114, 2002.

15. H. T. Chen, X. Y. Wu, and Y. S. Hsiao, Estimation of Surface Condition from the Theoryof Dynamic Thermal Stresses, Int. J. Thermal Sci., vol. 43, pp. 95–104, 2004.

16. H. T. Chen and X. Y. Wu, Estimation of Surface Absorptivity in Laser Surface HeatingProcess with Experimental Data, J. Phys. D: Appl. Phys., vol. 39, pp. 1141–1148, 2006.

17. C. W. Chang, C. S. Liu, and J. R. Chang, A Group Preserving Scheme for Inverse HeatConduction Problems, CMES, vol. 10, pp. 13–38, 2005.

18. C. C. Ji, P. C. Tuan, and H. Y. Jang, A Recursive Least-Squares Algorithm for On-Line1-D Inverse Heat Conduction Estimation, Int. J. Heat Mass Transfer, vol. 40, pp. 2081–2096, 1997.

NONLINEAR INVERSE HEAT CONDUCTION 177

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a, S

an F

ranc

isco

] at

20:

27 0

4 N

ovem

ber

2014

19. C. C. Ji and H. Y. Jang, Experimental Inverse Investigation in Inverse Heat ConductionProblem, Numer. Heat Transfer A, vol. 34, pp. 75–91, 1998.

20. H. Y. Li, Theoretical and Experimental Studies of the Inverse Heat Conduction Problems,Report of National Council Science, NSC 87-2212-E-211-007, 1998.

21. H. T. Chen and J. Y. Lin, Hybrid Laplace Transform Technique for Nonlinear TransientThermal Problems, Int. J. Heat Mass Transfer, vol. 34, pp. 1301–1308, 1991.

22. H. T. Chen and J. Y. Lin, Application of the Laplace Transform to Nonlinear TransientProblems, Appl. Math. Modelling, vol. 15, pp. 144–151, 1991.

23. J. P. Holman, Experimental Methods for Engineers, 7th ed., pp. 48–143, McGraw-Hill,New York, 2001.

24. V. S. Arpaci, S. H. Kao, and A. Selamet, Introduction to Heat Transfer, Prentice-Hall,Upper Saddle River, NJ, 1999.

25. G. Honig and U. Hirdes, A Method for the Numerical Inversion of Laplace Transforms,J. Comput. Appl. Math., vol. 10, pp. 113–132, 1984.

178 H.-T. CHEN AND X.-Y. WU

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a, S

an F

ranc

isco

] at

20:

27 0

4 N

ovem

ber

2014