International Journal of Heat and Mass Transfer 60 (2013) 125–133

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Inverse hyperbolic thermoelastic analysis of a functionally graded hollowcircular cylinder in estimating surface heat flux and thermal stresses

Yu-Ching Yang a, Wen-Lih Chen a,⇑, Huann-Ming Chou a, Jose Luis Leon Salazar b

a Clean Energy Center, Department of Mechanical Engineering, Kun Shan University, Yung-Kang, Tainan 710-03, Taiwan, ROCb Material Engineering School, Costa Rica Institute of Technology, Cartago, Costa Rica

a r t i c l e i n f o

Article history:Received 10 September 2012Received in revised form 10 December 2012Accepted 27 December 2012Available online 28 January 2013

Keywords:Inverse problemHyperbolic heat conductionHollow cylinderFGMsConjugate gradient method

0017-9310/$ - see front matter � 2013 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.12

⇑ Corresponding author. Tel.: +886 6 2050496; fax:E-mail address: wlchen@mail.ksu.edu.tw (W.-L. Ch

a b s t r a c t

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principleis applied to solve the inverse hyperbolic heat conduction problem in estimating the unknown time-dependent inner-wall heat flux of a hollow cylinder from the knowledge of temperature measurementstaken within the medium. The inverse solutions have been justified through the numerical experimentsin two specific cases to determine the unknown heat flux. Temperature data obtained from the directproblem are used to simulate the temperature measurements. The influence of measurement errors uponthe precision of the estimated results is also investigated. Results show that excellent estimation on thetime-dependent heat flux can be obtained for the test cases considered in this study. Once heat flux var-iation is accurately estimated, the evolution of temperature, displacement, and stress distributions can becalculated in great precision.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Functionally graded materials (FGMs), originally proposed byJapanese researchers [1], are a brand of composite materials whichare designed to optimize the stress and thermal resistance andtemperature distribution in components or structures subjectedto extreme thermal or mechanical loading. Unlike traditional lam-inated composite materials where several layers of different mate-rials are stuck together, the transition between different materialsis made gradually in order to reduce the local stress concentrationinduced by abrupt changes in material properties. In recent years,FGMs have even been proposed as a solution for aerospace indus-try where temperature resistant, light-weight structures are re-quired to meet the challenges faced by future high-speed spacevehicles. These novel materials have excellent thermo-mechanicalproperties to withstand high temperature and have been exten-sively applied to important structures, such as nuclear reactors,pressure vessels and pipes, chemical plants, etc. [2–4]. For exam-ple, a thin functionally graded thermal shield can sustain steeptemperature gradients without excessive thermal stresses. Similaradvantages can be realized with functionally graded heat exchan-ger pipes and heat engine components, in which FGMs that contin-uously transit from ceramic to metallic materials would avoid themismatch of the thermal expansion coefficient found in laminatedmaterials.

ll rights reserved..052

+886 6 2050509.en).

Fourier’s law has been traditionally the mainstream theory usedto solve heat conduction problems. Although Fourier’s law bears atheoretical flaw that thermal signal travels at an infinite speed, itsolves most large time and/or length scales engineering heat con-duction problems with satisfactory accuracy. Yet the developmentof laser heating and nanotechnology has created heat conductionproblems within very small time and/or length scales. For example,a carefully controlled incident beam can be used to heat up a verysmall patch of area at a rate up to 180 K/s for a few nanoseconds[5]. In such situations, researchers have reported that the predic-tions by Fourier heat conduction do not agree well with experi-mental observations. Maurer and Thompson [6] observed thesurface temperature of a slab taken immediately after a suddenthermal shock is 300 K higher than that predicted by Fourier’slaw. The disagreement between Fourier prediction and such exper-imental observation is rooted in the unrealistic propagation speedof thermal signal adopted by Fourier’s law. In reality, a thermal sig-nal travels at a finite speed, making a thermal response to behavelike a wave. To better describe this wave-like behavior, instead ofusing Fourier’s law, the Maxwell–Cattaneo equation, which takesfinite thermal signal travelling speed into account, can be used.This approach, however, has led to a more complicated hyperbolicgoverning equation on heat conduction problems. Because FGMsare usually subjected to high temperature, a precise evaluation oftheir thermal characteristics is of great interest for engineering de-sign and manufacture [7]. Therefore, hyperbolic heat conduction inFGMs is an interesting subject to study. However, the complexityof solving hyperbolic heat conduction in FGMs analytically or

Fig. 1. Geometry and coordinate system.

Nomenclature

E elastic modulus (GPa)h convection heat transfer coefficient (W m�2 K�1)J functionalJ0 gradient of functionalk thermal conductivity (W m�1 K�1)p direction of descentq heat flux at the inner surface (W m�2)r space coordinate in r-direction (m)r1 inner radius of the cylinder (m)r2 outer radius of the cylinder (m)T temperature (K)Tr reference temperature (K)T1 outer ambient temperature (K)t time (s)u displacement radial component (m)

Greek symbolsD small variation qualitya thermal diffusivity (m2 s�1)

b step sizec conjugate coefficiente very small valuek variable used in the adjoint problemn transformed time coordinateq density (kg m�3)r standard deviationrr stress radial component (GPa)rz stress axial component (GPa)rh stress tangential component (GPa)s relaxation time (s)x thermal expansion coefficient (1/K)- random variable

SuperscriptsK iterative number⁄ dimensionless quantity

126 Y.-C. Yang et al. / International Journal of Heat and Mass Transfer 60 (2013) 125–133

numerically presents a difficult challenge, and only very few at-tempts to do so have been documented in open literature [8–10].

Quantitative studies of heat transfer processes occurring inmany industrial applications often require accurate knowledge ofboundary conditions, such as heat flux, or thermophysical proper-ties of the materials involved. These important quantities wereconventionally obtained by expensive experimental methodswhich normally involve delicate and sophisticated equipments.In recent years, however, the studies of inverse heat conductionproblem (IHCP) have offered convenient alternatives, which largelyscale down experimental work, to obtain accurate thermophysicalquantities such as heat sources, material’s thermal properties, andboundary temperature or heat flux distributions, in many heat con-duction problems. To date, a variety of analytical and numericaltechniques have been developed for the solution of the inverseheat conduction problems, for example, the conjugate gradientmethod (CGM) [11–13], the genetic algorithm [14], and the linearleast-squares error method [15], etc.

Although there have been a great number of reports dealingwith the inverse solutions of classical Fourier heat conductionproblems, the reports on inverse hyperbolic heat conduction prob-lems have been much fewer in open literature. Huang and Wu [16]studied the inverse hyperbolic problem of a straight fin by an iter-ative regularization method in estimating the unknown base tem-perature based on the boundary temperature measurements. Yang[17] proposed a sequential method for estimating the boundaryconditions in a two-dimensional hyperbolic heat conduction prob-lem. Das et al. [18] estimated the extinction coefficient and theconduction-radiation parameter simultaneously in a non-Fourierconduction and radiation heat transfer problem by the geneticalgorithm in combination with the lattice Boltzmann method andthe finite-volume method. Yet, as far as applying an inverse analy-sis on hyperbolic heat conduction in FGMs is concerned, to the bestof the authors’ knowledge, there has not been such a report in openliterature.

Many engineering components assume the shape of a hollowcylinder. The inverse analysis of the heat transfer process of a hol-low cylinder allows accurate measurements to be taken with muchless experimental effort and sheds light into the heat transfer pro-cess and the reactions of the material’s stresses to thermal loading.Therefore, the focus of the present study is to develop an inversehyperbolic analysis for estimating the unknown time-dependent

heat flux at the inner wall of a hollow cylinder constructed withFGMs from the knowledge of temperature measurements takenwithin the medium. To this end, we present the conjugate gradientmethod and the discrepancy principle [19] to estimate the un-known time-dependent heat flux by using the simulated tempera-ture measurements. The CGM derives from the perturbationprinciples and transforms the inverse problem to the solution ofthree problems, namely, the direct, sensitivity and the adjointproblem, which will be discussed in detail in the following sec-tions. On the other hand, the first-and second-derivatives in allgoverning equations in the three problems are discretized by back-ward differencing to avoid adding the tedious Laplace inversetransform on top of this already complicated inverse process.

2. Analysis

2.1. Direct problem

To illustrate the methodology of developing expressions for theuse in determining the unknown time-dependent heat flux at theinner surface of a functionally graded hollow cylinder in an inverse

Y.-C. Yang et al. / International Journal of Heat and Mass Transfer 60 (2013) 125–133 127

hyperbolic heat conduction problem, a FGM cylinder with inner ra-dius r1, outer radius r2, and infinite length, as shown in Fig. 1, isconsidered. Assume the cylinder and the outer surrounding are ini-tially at temperature T1, and for time t > 0, the cylinder is suddenlysubjected to a heat flux q(t) at its inner wall r = r1. The convectionheat transfer coefficient at the outer wall is h. Then the dimension-less governing equations and the associated boundary and initialconditions for this linear hyperbolic heat conduction problem canbe written as [17,20]:

s�

a�ðr�Þ@2T�ðr�; t�Þ

@t�2þ 1

a�ðr�Þ@T�ðr�; t�Þ

@t�

¼ 1k�ðr�Þ

1r�

@

@r�k�ðr�Þr� @T�ðr�; t�Þ

@r�

� �; ð1Þ

s� @q�ðt�Þ@t�

þ q�ðt�Þ ¼ �k�ðr�Þ @T�ðr�; t�Þ@r�

; at r� ¼ r�1; ð2Þ

�k�ðr�Þ @T�ðr�; t�Þ@r�

¼ h� � T�ðr�; t�Þ; at r� ¼ r�2; ð3Þ

T�ðr�; t�Þ ¼ @T�ðr�; t�Þ@t�

¼ 0; for t� ¼ 0; ð4Þ

q�ðt�Þ ¼ 0; for t� ¼ 0: ð5Þ

The dimensionless variables used in the above formulation are de-fined as follows:

r� ¼ r=r1; r�1¼ r1=r1; r�2¼ r2=r1; t� ¼a0t=r21; T� ¼ ðT�T1Þ=Tr ;

k� ¼ k=k0; a� ¼a=a0; h� ¼hr1=k0; q� ¼ r1q=k0Tr ; s� ¼a0s=r21;

ð6Þ

where Tr is a reference temperature; k0 and a0 are reference valuesof thermal conductivity and thermal diffusivity, respectively. Wefurther assume the thermal conductivity k and thermal diffusivitya change smoothly and continuously through the thickness of theFGM hollow circular cylinder. The direct problem considered hereis concerned with the determination of the medium temperaturewhen the heat flux q⁄(t⁄), thermophysical properties of the cylin-ders, and initial and boundary conditions are known.

2.2. Inverse problem

For the inverse problem, the heat flux function q⁄(t⁄) is regardedas being unknown, while everything else in Eqs. (1)–(5) is known.In addition, temperature readings taken at r = rm are consideredavailable. The objective of the inverse analysis is to estimate theunknown time-dependent heat flux, q⁄(t⁄), merely from the knowl-edge of these temperature readings. Let the measured temperatureat the measurement position r = rm and time t be denoted byY(rm, t). Then this inverse problem can be stated as follows: by uti-lizing the above mentioned measured temperature data Y(rm, t), theunknown q⁄(t⁄) is to be estimated over the specified time domain.

The solution of the present inverse problem is to be obtained insuch a way that the following functional is minimized:

J½q�ðt�Þ� ¼Z t�

f

t�¼0T� r�m; t

�� �� Y� r�m; t

�� �� �2dt�; ð7Þ

where Y� r�m; t�� �¼ ½Yðrm; tÞ � T1�=Tr , and T� r�m; t

�� �is the estimated

(or computed) temperature at the measurement locationr� ¼ r�mð¼ rm=r1Þ. In this study, T� r�m; t

�� �are determined from the

solution of the direct problem given previously by using an esti-mated q⁄K(t⁄) for the exact q⁄(t⁄), here q⁄K(t⁄) denotes the estimatedquantities at the Kth iteration. t�f is the final time of the measure-ment. In addition, in order to develop expressions for the determi-

nation of the unknown q⁄(t⁄), a ‘‘sensitivity problem’’ and an‘‘adjoint problem’’ are constructed as described below.

2.3. Sensitivity problem

The sensitivity problem is obtained from the original directproblem defined by Eqs. (1)–(5) in the following manner: It is as-sumed that when q⁄(t⁄) undergoes a variation Dq⁄(t⁄), T⁄(r⁄, t⁄) isperturbed by T⁄(r⁄, t⁄) + DT⁄(r⁄, t⁄). Then replacing in the directproblem q⁄ by q⁄ + Dq⁄ and T⁄ by T⁄ + DT⁄, subtracting from theresulting expressions the direct problem, and neglecting the sec-ond-order terms, the following sensitivity problem for the sensitiv-ity function DT⁄(r⁄, t⁄) can be obtained:

s�

a�ðr�Þ@2DT�

@t�2þ 1

a�ðr�Þ@DT�

@t�¼ 1

k�ðr�Þ1r�

@

@r�k�ðr�Þr� @DT�

@r�

� �; ð8Þ

s� @Dq�ðt�Þ@t�

þ Dq�ðt�Þ ¼ �k�ðr�Þ @DT�ðr�; t�Þ@r�

; at r� ¼ r�1; ð9Þ

� k�ðr�Þ @DT�ðr�; t�Þ@r�

¼ h� � DT�ðr�; t�Þ; at r� ¼ r�2; ð10Þ

DT�ðr�; t�Þ ¼ @DT�ðr�; t�Þ@t�

¼ 0; for t� ¼ 0; ð11Þ

Dq�ðt�Þ ¼ 0; for t� ¼ 0: ð12Þ

The sensitivity problem of Eqs. (8)–(12) can be solved by the samemethod as the direct problem of Eqs. (1)–(5).

2.4. Adjoint problem and gradient equation

To formulate the adjoint problem, Eq. (1) is multiplied by theLagrange multiplier (or adjoint function) k⁄(r⁄, t⁄), and the resultingexpressions are integrated over the time and correspondent spacedomains. Then the results are added to the right hand side of Eq.(7) to yield the following expression for the functional J[q⁄(t⁄)]:

J½q�ðt�Þ� ¼Z t�

f

t�¼0

Z r�2

r�¼1½T�ðr�; t�Þ � Y�ðr�; t�Þ�2d r� � r�m

� �dr�dt�

þZ t�

f

t�¼0

Z r�2

r�¼1r� � k�ðr�; t�Þ � 1

k�ðr�Þ1r�

@

@r�k�ðr�Þr� @T�

@r�

� ��

� s�

a�ðr�Þ@2T�

@t�2� 1

a�ðr�Þ@T�

@t�

)dr�dt�; ð13Þ

where d(�) is the Dirac function. The variation DJ is derived afterq⁄(t⁄) is perturbed by Dq⁄(t⁄) and T⁄(r⁄, t⁄) is perturbed by DT⁄(r⁄, t⁄)in Eq. (13). Subtracting from the resulting expression the originalEq. (13) and neglecting the second-order terms, we thus find:

DJ½q�ðt�Þ� ¼Z t�

f

t�¼0

Z r�2

r�¼12½T�ðr�; t�Þ � Y�ðr�; t�Þ�DT� � d r� � r�m

� �dr�dt�

þZ t�

f

t�¼0

Z r�2

r�¼1r� � k�ðr�; t�Þ � 1

k�ðr�Þ1r�

@

@r�k�ðr�Þr� @DT�

@r�

� ��

� s�

a�ðr�Þ@2DT�

@t�2� 1

a�ðr�Þ@DT�

@t�

)dr�dt�: ð14Þ

We can integrate the second integral terms in Eq. (14) by parts, uti-lizing the initial and boundary conditions of the sensitivity problem.Then DJ is allowed to go to zero. The vanishing of the integrandscontaining DT⁄ leads to the following adjoint problem for the deter-mination of k⁄(r⁄, t⁄):

� s�

a�ðr�Þ@2k�

@t�2þ 1

a�ðr�Þ@k�

@t�þ 1

r�@

@r�k�ðr�Þr� @

@r�k�

k�ðr�Þ

� ��

þ 2½T�ðr�; t�Þ � Y�ðr�; t�Þ�DT� � dðr� � r�mÞr�

¼ 0; ð15Þ

128 Y.-C. Yang et al. / International Journal of Heat and Mass Transfer 60 (2013) 125–133

k�ðr�Þ @k�

@r�¼ dk�ðr�Þ

dr�k�; at r� ¼ r�1; ð16Þ

k�ðr�Þ @k�

@r�þ h� � dk�ðr�Þ

dr�

� �k� ¼ 0; at r� ¼ r�2; ð17Þ

k� ¼ @k�

@t�0; for t� ¼ t�f : ð18Þ

The adjoint problem is different from the standard initial valueproblem in that the final condition at time t� ¼ t�f is specified in-stead of the customary initial condition at time t⁄ = 0. However, thisproblem can be transformed to an initial value problem by thetransformation of the time variable as n� ¼ t�f � t�. Then the adjointproblem can be solved by the same method as the direct problem.

Finally the following integral term is left:

DJ¼Z t�

f

t�¼0k� r�1;t

�� ��s�k� r�1;t

�� �dðt� �0Þ�s�

@k� r�1;t�� �

@t�

� �Dq�ðt�Þ

k� r�1� �

dt�:

ð19Þ

From the definition used in Ref. [11], we have:

DJ ¼Z t�

f

t�¼0J0ðt�ÞDq�ðt�Þdt�; ð20Þ

where J0(t⁄) is the gradient of the functional J(q). A comparison ofEqs. (19) and (20) leads to the following form:

J0ðt�Þ¼ k� r�1;t�� ��s�k� r�1;t

�� �dðt� �0Þ�s�

@k� r�1;t�� �

@t�

� �k� r�1� �

: ð21Þ

2.5. Conjugate gradient method for minimization

The following iteration process based on the conjugate gradientmethod is now used for the estimation of q⁄(t⁄) by minimizing theabove functional J[q⁄(t⁄)]:

q�Kþ1ðt�Þ ¼ q�Kðt�Þ � bK pKðt�Þ; K ¼ 0;1;2; . . . ; ð22Þ

where bK is the search step size in going from iteration K to iterationK + 1, and pK(t⁄) is the direction of descent (i.e., search direction)given by:

pKðt�Þ ¼ J0Kðt�Þ þ cK pK�1ðt�Þ; ð23Þ

which is conjugation of the gradient direction J0K(t⁄) at iteration K

and the direction of descent pK�1 (t⁄) at iteration K � 1. The conju-gate coefficient cK is determined from:

cK ¼R t�

ft�¼0½J

0Kðt�Þ�2dt�R t�f

t�¼0½J0K�1ðt�Þ�2dt�

with c0 ¼ 0: ð24Þ

The convergence of the above iterative procedure in minimizing thefunctional J is proved in Ref. [11]. To perform the iterations accord-ing to Eq. (22), we need to compute the step size bK and the gradientof the functional J

0K(t⁄).The functional J[q⁄K+1(t⁄)] for iteration K + 1 is obtained by

rewriting Eq. (7) as:

J½q�Kþ1ðt�Þ� ¼Z t�

f

t�¼0T�ðq�K � bK pKÞ � Y� r�m; t

�� �� �2dt�; ð25Þ

where we replace q⁄K+1 by the expression given by Eq. (22). If tem-perature T⁄(q⁄K � bKpK) is linearized by a Taylor expansion, Eq. (25)takes the form:

J½q�Kþ1ðt�Þ� ¼Z t�

f

t�¼0T�ðq�KÞ � bKDT�ðpKÞ � Y� r�m; t

�� �� �2dt�; ð26Þ

where T⁄(q⁄K) is the solution of the direct problem at r� ¼ r�m byusing estimated q⁄K(t⁄) for exact q⁄(t⁄) at time t⁄. The sensitivityfunction DT⁄(pK) are taken as the solution of Eqs. (8)–(12) at the

measured position r� ¼ r�m by letting Dq⁄ = pK [11]. The search stepsize bK is determined by minimizing the functional given by Eq.(26) with respect to bK. The following expression can be obtained:

bK ¼R t�

ft�¼0 DT�ðpKÞ½T�ðq�KÞ � Y��dt�R t�

ft�¼0½DT�ðpKÞ�2dt�

: ð27Þ

2.6. Stopping criterion

If the problem contains no measurement errors, the traditionalcheck condition is specified as:

Jðq�Kþ1Þ < e; ð28Þ

where e is a small specified number, can be used as the stopping cri-terion. However, the observed temperature data contain measure-ment errors; as a result, the inverse solution will tend toapproach the perturbed input data, and the solution will exhibitoscillatory behavior as the number of iteration is increased [21].Computational experience has shown that it is advisable to usethe discrepancy principle [19] for terminating the iteration processin the conjugate gradient method. Assuming T� r�m; t

�� �� Y�

r�m; t�� �ffi r, r is the standard deviation of the measurement error,

the stopping criteria e by the discrepancy principle can be obtainedfrom Eq. (7) as:

e ¼ r2t�f : ð29Þ

Then the stopping criterion is given by Eq. (28) with e determinedfrom Eq. (29).

2.7. Computational procedure

The computational procedures for the solution of this inverseproblem may be summarized as follows:

Suppose q⁄K(t⁄) is available at iteration K.

Step 1 Solve the direct problem given by Eqs. (1)–(5) for T⁄(r⁄, t⁄).Step 2 Examine the stopping criterion given by Eq. (28) with e

given by Eq. (29). Continue if not satisfied.Step 3 Solve the adjoint problem given by Eqs. (15)–(18) for

k⁄(r⁄, t⁄).Step 4 Compute the gradient of the functional J0 from Eq. (21).Step 5 Compute the conjugate coefficient cK and direction of

decent pK from Eqs. (24) and (23), respectively.Step 6 Set Dq⁄(t⁄) = pK(t⁄) and solve the sensitivity problem given

by Eqs. (8)–(12) for DT⁄(r⁄, t⁄).Step 7 Compute the search step size bK from Eq. (27).Step 8 Compute the new estimation for q⁄K+1(t⁄) from Eq. (22) and

return to Step 1.

2.8. Displacement and thermal stresses

To simplify the solution process, we introduce the followingdimensionless variables:

E� ¼ E=E0; x� ¼ x=x0; q� ¼ q=q0; u� ¼ u=x0Trr1;

r�r ¼ rr=x0TrE0; r�h ¼ rh=x0TrE0; r�z ¼ rz=x0TrE0; ð30Þ

where E0 and x0 are reference values of elastic modulus and ther-mal expansion coefficient, respectively. Moreover, in this study,we also assume the elastic modulus E and thermal expansion coef-ficient x change smoothly and continuously through the thicknessof the FGM hollow circular cylinder.

Then, the dimensionless equation of equilibrium along theradial direction for a symmetrical cylinder can be written as:

x*

T*

0 0.2 0.4 0.6 0.8 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2 t*=0.25, Wang et al.t*=0.5, Wang et al.t*=0.75, Wang et al.t*=0.25t*=0.5t*=0.75

Fig. 2. Comparison of temperature distributions by Wang et al. [23] and the presentcode at three different time steps in a one-dimensional hyperbolic heat conductionproblem.

Y.-C. Yang et al. / International Journal of Heat and Mass Transfer 60 (2013) 125–133 129

@r�r ðr�; t�Þ@r�

þ r�r ðr�; t�Þ � r�hðr�; t�Þr�

¼ q0a20

E0r21

� q� @2u�

@t�2; t� > 0 ð31Þ

and the equations of equilibrium along the two other directions aresatisfied identically.

The dimensionless mechanical boundary conditions are

r�r ðr�; t�Þ ¼ 0; at r� ¼ r�1; ð32Þr�r ðr�; t�Þ ¼ 0; at r� ¼ r�2; ð33Þ

which represent the traction-free conditions along the inner andouter surfaces, respectively.

The dimensionless stress and displacement relations are:

r�r ðr�; t�Þ ¼E�

ð1þ mÞð1� 2mÞ ð1þ mÞ @u�ðr�; t�Þ@r�

þ mu�ðr�; t�Þ

r�

� �

� E�x�

ð1� 2mÞ T�ðr�; t�Þ; ð34Þ

r�hðr�; t�Þ ¼E�

ð1þ mÞð1� 2mÞ m@u�ðr�; t�Þ

@r�þ ð1þ mÞu

�ðr�; t�Þr�

� �

� E�x�

ð1� 2mÞ T�ðr�; t�Þ; ð35Þ

r�zðr�; t�Þ ¼E�

ð1þ mÞð1� 2mÞ m@u�ðr�; t�Þ

@r�þ m

u�ðr�; t�Þr�

� �

� E�x�

ð1� 2mÞ T�ðr�; t�Þ: ð36Þ

Generally, the Poisson’s ratio m of FGM varies in a small range. Forsimplicity, we assume m to be a constant in this study.

By substituting stress-displacement relations (34) and (35) intoequilibrium Eq. (29), the dimensionless field equation of displace-ment for the FGM hollow circular cylinder can be formulated as:

1E�ðr�Þ

1r�

@

@r�E�ðr�Þr� @u�

@r�

� �þ m

1� mdE�ðr�Þ=dr�

E�ðr�Þ � 1r�

� �u�

r�

� 1þ m1� m

1E�ðr�Þ

@

@r�½x�ðr�Þ � E�ðr�ÞT��

¼ ð1þ mÞð1� 2mÞE�ðr�Þð1� mÞ

q0a20

E0r21

� q� @2u�

@t�2: ð37Þ

Eq. (37) can be solved numerically with the boundary conditions ofEqs. (32) and (33) once the temperature distributions T⁄(r⁄, t⁄) havebeen obtained by the inverse method.

3. Results and discussion

The validity of the mathematical model for the direct problemand the accuracy of the numerical approach are very crucial foran inverse estimation to be credible in a real application. In termsof the first issue, many numerical studies have proven that Eq. (1)is capable of capturing the wave-like behaviors of hyperbolic heatconduction; whereas, the second issue will be verified through thefollowing code validation process. The numerical procedure in thispaper is based on the unstructured-mesh, fully collocated, finite-volume code, ‘USTREAM’ developed by the corresponding author.This is the descendent of the structured-mesh, multi-block codeof ‘STREAM’ [22]. The case selected for code validation is a one-dimensional hyperbolic heat conduction problem reported inWang et al. [23], where the first and the second derivatives in timewere solved by finite differencing. The problem is a piece of mate-rial with an initial low temperature being subjected to a suddenhigh temperature at one end. The details of dimensionless param-eters and boundary conditions have been documented in [23] andwill not be repeated here. They reported that 1,000 time steps per

unit dimensionless time and 500 cells in a unit space are required forbackward differencing to achieve a time-step-interval and grid-sizeindependent and oscillation free solution. Fig. 2 shows the compar-ison between the solutions by Wang et al. and by the current code.Here, we used the same time step interval as in [23] but much coar-ser grid, only 120 cells. Despite of the coarser grid, the agreement be-tween the two solutions are excellent at all three time steps shownin Fig. 2. The reason for using fewer cells but achieving the samenumerical stability could be due to the employment of finite volumescheme in space, which is more stable than finite difference tech-nique, in the current study. To this end, this practice has validatedthe correctness of the current numerical code.

In the present study, a molybdenum/mullite functional gradedhollow cylinder is considered. The inner and outer surfaces arepure mullite and composite of molybdenum/mullite, respectively.Both molybdenum and mullite vary continuously from the innerto the outer surfaces of the cylinder. The values for q0,k0,a0,E0,x0,and m of the mullite taken from [24] are 3,000 kg m�3, 5.9 W m�1

K�1, 2.8 � 10�6 m2 s�1, 225 GPa, 4.8 � 10�6 K�1, and 0.3, respec-tively. It is further assumed that the variations of materialproperties throughout the thickness of the hollow cylinder obeythe following exponential laws [25]:

E�ðr�Þ ¼ exp m1 r� � r�1� �� �

; ð38Þx�ðr�Þ ¼ exp m2 r� � r�1

� �� �; ð39Þ

k�ðr�Þ ¼ exp m3 r� � r�1� �� �

; ð40Þa�ðr�Þ ¼ exp m4 r� � r�1

� �� �; ð41Þ

where m1,m2,m3, and m4 are material constants, and they are as-sumed 2.0,0.3,2.0, and 2.0, respectively. In terms of the geometricaland thermal parameters, r�2 is assumed to be 2.0, and h⁄ is set to 2.0.Before the test on the inverse method can begin, appropriate gridsize and time step interval to achieve grid-and time-step-interval-independent solutions have to be determined through a grid-inde-pendency test and a time-step-interval-independency test. Here,the time-dependent heat flux applied on the inner wall of the cylin-der is assumed as:

q�ðt�Þ ¼ 1� expð�10t�Þ: ð42Þ

First, three different grids, termed grids 1, 2, and 3 with 40, 100, and150 cells in radial direction, have been tested for grid independency.

r*

T*

1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

40 cells100 cells150 cells

t*=0.06

t*=0.3

t*=0.54

(a)

r*

T*

1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

500 steps1000 steps1250 steps

t*=0.06

t*=0.3

t*=0.54

(b)

Fig. 3. Grid and time-step-interval independency tests; (a) radial temperaturedistributions by three different meshes, and (b) radial temperature distributions bythree different time-step intervals.

r*

T*

1 1.2 1.4 1.6 1.8 2-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

t*=0.06

t*=0.3

t*=0.54

simulated, σ=0.0 simulated, σ=0.02 inverse, σ=0.0 inverse, σ=0.02

Fig. 4. Simulated and inverse temperature distributions in case 1 at measurementlocation with initial guesses q⁄0(t⁄) = 0.0 and r = 0.0 and 0.02 at three different timesteps.

130 Y.-C. Yang et al. / International Journal of Heat and Mass Transfer 60 (2013) 125–133

The time step interval is fixed at Dt⁄ = 0.001, which is fine enough togive rise to time-step-interval independent solutions for the one-dimensional problem in Wang et al. [23]. Fig. 3(a) shows the tem-perature distributions along radial direction by the three grids. Itcan be seen that the results demonstrate moderate sensitivity tocell density, and those by grids 2 and 3 are almost identical, sug-gesting that grid 2 is fine enough to yield grid independent solu-tions. Next, grid 2 is used to repeat the calculation with threedifferent time step intervals, namely Dt⁄ = 0.002, 0.001, and0.0008, corresponding to 500, 1000, and 1250 time steps, respec-tively per unit dimensionless time, and the results are shown inFig. 3(b). As seen, the results indicate that the temperature distribu-tions by 1000 steps and 1250 steps are almost the same; provingthat 1000 time steps per unit dimensionless time is also good en-ough to achieve time-step-interval independent solutions for thecurrent hyperbolic heat transfer problem. It can be concluded fromthe tests that grid-and time-step-interval-independent solutions forthis hollow cylinder problem can be obtained by using 100 cells inradial direction and a time step interval of Dt⁄ = 0.001.

The objective of this article is to validate the present inversehyperbolic analysis when used in estimating the unknown

time-dependent heat flux at the inner wall of the FGM hollow cyl-inder accurately with no prior information on the functional formof the unknown quantities, a procedure called function estimation.In the analysis, we do not have a real experimental set up to mea-sure the temperature Y� r�m; t

�� �in Eq. (7). Instead, we assume a

heat flux function q⁄(t⁄) and substitute it into the direct problemof Eqs. (1)–(5) to calculate the temperatures at the location wherethe thermocouple is placed. The results are taken as the simulatedtemperature Y�exact r�m; t

�� �; similarly, the displacement and stresses

calculated this way are termed simulated displacement and stres-ses. Nevertheless, in reality, the temperature measurements al-ways contain some degree of errors, whose magnitude dependsupon the particular measuring method employed. In order to takemeasurement errors into account, a random error noise is added tothe above computed temperature Y�exact r�m; t

�� �to obtain the mea-

sured temperature Y� r�m; t�� �

. Hence, the measured temperatureY� r�m; t

�� �is expressed as:

Y� r�m; t�� �¼ Y�exact r�m; t

�� �þ-r; ð43Þ

where - is a random variable within �2.576 to 2.576 for a 99% con-fidence bounds, and r is the standard deviation of the measure-ment. The measured temperature Y� r�m; t

�� �generated in such way

is the so-called simulated measurement temperature.To demonstrate the accuracy of the CGM in estimating the in-

ner-wall heat flux q⁄(t⁄) of the FGM cylinder in the present hyper-bolic problem, two different functional forms of q⁄(t⁄) areconsidered. In all cases, the following geometrical parametersand boundary conditions are adopted:

r�2 ¼ 2:0; h� ¼ 2:0: ð44Þ

Case 1 The unknown transient heat flux q⁄(t⁄) applied at inner wallis assumed the form of Eq. (42). In terms of the temporal domain,the total dimensionless measurement time is chosen as t�f ¼ 0:6.The inverse temperatures and the estimated values of the unknownfunction q⁄(t⁄) at three different time steps, t⁄ = 0.06, 0.3, and 0.54,respectively are given in Figs. 4 and 5. These are obtained with theinitial guess values q⁄0(t⁄) = 0.0 and measurement error of devia-tion r = 0.0,0.01,0.02. For a temperature of unity and 99% confi-dence, the standard deviations r = 0.01 and 0.02 corresponds tomeasurement error of 2.58% and 5.12%, respectively. Also in Figs. 4

t*

q*

0 0.1 0.2 0.3 0.4 0.5 0.60

0.2

0.4

0.6

0.8

1

exactinverse, σ =0.0 inverse, σ =0.02

Fig. 5. Exact and inverse heat flux variations in case 1 with initial guessesq⁄0(t⁄) = 0.0 and r = 0.0 and 0.02.

r*1 1.2 1.4 1.6 1.8 2

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

t*=0.06

t*=0.3

t*=0.54

simulatedinverse, σ =0.0 inverse, σ =0.02

σ ∗

Fig. 7. Simulated and inverse radial stress distributions in case 1 with initialguesses q⁄0(t⁄) = 0.0 and r = 0.0 and 0.02 at three different time steps.

-1

-0.5

0

0.5

1

1.5

2

2.5

t*=0.06

t*=0.3

t*=0.54

simulated inverse, σ=0.0 inverse, σ=0.02

σ ∗

Y.-C. Yang et al. / International Journal of Heat and Mass Transfer 60 (2013) 125–133 131

and 5, only the results of r = 0.02 are shown to avoid putting toomany curves in a single plot. The temperature Y� r�m; t

�� �is mea-

sured at r�m ¼ 1:0 (at inner wall). Fig. 4 suggests that inverse andsimulated temperature distributions are almost identical, indicat-ing that the inverse solutions are quite converged. In the case ofzero measurement error, the results in Fig. 5 show that the esti-mated heat flux is in excellent agreement with the exact value.The average relative error for the estimated heat flux isERR = 0.05%, where the relative average error ERR is defined as:

ERR ¼PN

i¼1q�exactðt

�Þ�q�inv ðt

�Þq�exact ðt

�Þ

��� ���N

� 100%; ð45Þ

where N is the number of time steps. It is also noticeable from theresults that the estimated heat flux by considering measurement er-rors only slightly deviates from the exact value with ERR = 2.58%and 5.11% for r = 0.01 and 0.02, respectively, which are slightly

r*

u*

1 1.2 1.4 1.6 1.8 2-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

t*=0.06

t*=0.3

t*=0.54

simulatedinverse, σ=0.0 inverse, σ=0.01

Fig. 6. Simulated and inverse displacement distributions in case 1 with initialguesses q⁄0(t⁄) = 0.0 and r = 0.0 and 0.02 at three different time steps.

r*1 1.2 1.4 1.6 1.8 2-1.5

Fig. 8. Simulated and inverse tangential stress distributions in case 1 with initialguesses q⁄0(t⁄) = 0.0 and r = 0.0 and 0.02 at three different time steps.

larger than the magnitude of measurement errors. This suggeststhat present algorithm is not sensitive to the measurement error,and reliable inverse solutions can still be obtained when measure-ment errors are considered. With the inner-wall heat flux well esti-mated, the distributions of stresses can be obtained accurately. Thisis illustrated through Figs. 6–9, where the distributions of displace-ment and radial, tangential, and axial stresses at three differenttime steps, t⁄ = 0.06, 0.3, and 0.54, respectively, are shown. Despitethe appearance of the second-order derivative, which is responsiblefor finite signal propagation speed, on the right hand side of Eq.(37), the displacement given in Fig. 6 does not show any sign ofwave-like behavior. This implies that, in terms of displacement,the signal propagation speed is almost infinite. The reason is thatthe parameter q0a2

0=E0r21 multiplied to this derivative is very small.

For example, given r1 = 0.01 m, the value of this parameter is4.46 � 10�27; and such a small value renders the effect of this sec-ond-order derivative negligible when compared with other terms in

r*1 1.2 1.4 1.6 1.8 2

-2

-1.5

-1

-0.5

0

0.5

1

t*=0.06

t*=0.3

t*=0.54

simulated inverse, σ=0.0 inverse, σ=0.02

∗

Fig. 9. Simulated and inverse axial stress distributions in case 1 with initial guessesq⁄0(t⁄) = 0.0 and r = 0.0 and 0.02 at three different time steps.

t*

q*

0 0.1 0.2 0.3 0.4 0.5 0.60

0.2

0.4

0.6

0.8

1

exactinverse, σinverse, σ=0.02

=0.0

Fig. 11. Exact and inverse heat flux variations in case 2 with initial guessesq⁄0(t⁄) = 0.0 and r = 0.0 and 0.02.

132 Y.-C. Yang et al. / International Journal of Heat and Mass Transfer 60 (2013) 125–133

Eq. (37). In Fig. 7, it can be seen that the radial stress is always com-pressive except at both walls (zero radial stress boundary condi-tion), whereas, in Figs. 8 and 9, the tangential and axial stressesare compressive near the heated side and tensile toward the cooledside. The behaviors of these stresses are similar to those reported in[8] under similar geometry and conditions. In terms of inverse accu-racy, excellent agreement between the simulated and inverse dis-placement and stress distributions can be seen for all three timesteps for r = 0.0, and only slight deviations are observed forr = 0.02.Case 2 The unknown inner-wall heat flux q⁄(t⁄) takes the form as astep function:

T*

Fig. 10.location

8

q�ðt�Þ ¼

0:5; for 0 6 t� 6 0:2;1:0; for 0:2 < t� 6 0:4;0:5; for 0:4 < n 6 0:6:

><>: ð46Þ

The step function defines a sudden jump at t⁄ = 0.2 and a suddendrop at t⁄ = 0.4, both signify discontinuities in the slop of the heat

r*1 1.2 1.4 1.6 1.8 2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

t*=0.06t*=0.3

t*=0.54

simulated, σ=0.0 simulated, σ=0.02 inverse, σ=0.0 inverse, σ=0.02

Simulated and inverse temperature distributions in case 2 at measurementwith initial guesses q⁄0(t⁄) = 0.0 and r = 0.0 and 0.02.

flux function. These are challenging features for the inverse meth-od because estimated quantities tend to display some degree ofoscillation near such features. The inverse temperatures and esti-mated values of the inner-wall heat flux, with the initial guess val-ues q⁄0(t⁄) = 0.0 and measurement error of deviation r = 0.0 and0.02, are shown in Figs. 10 and 11, respectively. In Fig. 10, theexcellent agreement between inverse and simulated temperaturesindicates the inverse solutions are well converged. The heat fluxvariations in Fig. 11 again indicate that the estimated results arein very good agreement with those of the exact values. The averagerelative errors for the estimated heat flux are 0.07%, 2.6%, and5.11% respectively for r = 0.0,0.01 and r = 0.02. The estimated heatflux variation without measurement error is almost identical to theexact variation, and the magnitude of relative error is similar tothat of measurement errors when they are considered.

According to Zuiker [26], FGMs properties in exponential formonly have limited application. Therefore, the current method istested with different property functions. Here, commonly used lin-ear functions from [27] are adopted:

E�ðr�Þ ¼ 1:0þ E�2 � 1:0� � r� � r�1

r�2 � r�1; ð47Þ

x�ðr�Þ ¼ 1:0þ x�2 � 1:0� � r� � r�1

r�2 � r�1; ð48Þ

k�ðr�Þ ¼ 1:0þ k�2 � 1:0� � r� � r�1

r�2 � r�1; ð49Þ

a�ðr�Þ ¼ 1:0þ a�2 � 1:0� � r� � r�1

r�2 � r�1; ð50Þ

where E�2;x�2; k�2, and a�2 are dimensionless elastic modulus, thermal

expansion coefficient, thermal conductivity, and thermal diffusivityat r�2, and they are assumed 10,1.35,10, and 10, respectively. The re-sults in terms of inverse temperatures and estimated values of theinner-wall heat flux, with the initial guess values q⁄0(t⁄) = 0.0 andmeasurement error of deviation r = 0.0 and 0.02, are shown inFigs. 12 and 13, respectively. The performance of the current meth-od is quite similar to the previous cases with the average relativeerrors for the estimated heat flux being 0.06%, 2.5%, and 5.12%respectively for r = 0.0, 0.01 and r = 0.02. This exercise proves thatthe proposed method also works very well for different kinds ofFGMs properties functions. To this end, the current inverse meth-od’s superb accuracy and the generalization of its application on dif-ferent FGMs is demonstrated.

r*

T*

1 1.2 1.4 1.6 1.8 2-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

t*=0.06

t*=0.3

t*=0.54

simulated, σ=0.0 simulated, σ=0.02 inverse, σ=0.0 inverse, σ=0.02

Fig. 12. Simulated and inverse temperature distributions in case 2 with differentproperty functions at measurement location with initial guesses q⁄0(t⁄) = 0.0 andr = 0.0 and 0.02.

t*

q*

0 0.1 0.2 0.3 0.4 0.5 0.60

0.2

0.4

0.6

0.8

1

exactinverse, σ =0.0 inverse, σ =0.02

Fig. 13. Exact and inverse heat flux variations in case 2 with different propertyfunctions and with initial guesses q⁄0(t⁄) = 0.0 and r = 0.0 and 0.02.

Y.-C. Yang et al. / International Journal of Heat and Mass Transfer 60 (2013) 125–133 133

4. Conclusion

An inverse algorithm based on the conjugate gradient methodand the discrepancy principle was successfully applied to the solu-tion of an inverse hyperbolic heat conduction problem to deter-mine the unknown time-dependent heat flux at the inner wall ofa hollow cylinder constructed with FGMs, while knowing the tem-perature history at the measurement location. Numerical resultsconfirm that the method proposed herein can accurately estimatethe time-dependent heat flux for the problem even some inevitablemeasurement errors are involved. The proposed inverse algorithmdoes not require prior information for the functional form of theunknown quantities to perform the inverse calculation, and excel-lent estimated values can be obtained for the considered numericaltest cases. With the heat flux accurately estimated, the evolution ofthe distributions of temperature, displacement, and thermal

stresses within the material can be calculated in great precision,allowing engineers to obtain vital information for the design andmanufacture of such components.

Acknowledgment

This work was supported by the National Science Council, Tai-wan, Republic of China, under the Grant Nos. 98IC04 and NSC101-2221-E-168-019.

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