Nonlinear transient heat conduction problems for a class of inhomogeneous anisotropic materials by BEM

Size: px

Start display at page:

Download "Nonlinear transient heat conduction problems for a class of inhomogeneous anisotropic materials by BEM"

Helen Boyd
5 years ago
Views:

Engineering Analysis with Boundary Elements 32 (28) 154 16 www.elsevier.com/locate/enganabound Nonlinear transient heat conduction problems for a class of inhomogeneous anisotropic materials by BEM M.

1 Tamalanrea, Makassar 9245, Indonesia b School of Mathematics, University of Adelaide, SA 55, Australia Received 2 March 27; accepted 26 April 27 Available online 2 February 28 Abstract A class of

1 Engineering Analysis with Boundary Elements 32 (28) Nonlinear transient heat conduction problems for a class of inhomogeneous anisotropic materials by BEM M.I. Azis a, D.L. Clements b, a Jurusan Matematika, Fakultas MIPA, Universitas Hasanuddin, Jl. P. Kemerdekaan KM.1 Tamalanrea, Makassar 9245, Indonesia b School of Mathematics, University of Adelaide, SA 55, Australia Received 2 March 27; accepted 26 April 27 Available online 2 February 28 Abstract A class of nonlinear transient heat conduction problems for anisotropic inhomogeneous materials is considered. The problems under consideration are reduced to a boundary integral euation which may be solved numerically using standard techniues. Some numerical examples are considered in order to test the accuracy of the numerical procedure. r 28 Elsevier Ltd. All rights reserved. Keywords: Boundary integral methods; Heat transfer; Nonlinear differential euations; Numerical methods 1. Introduction Boundary integral euation methods have been used by a number of authors to consider steady-state and transient heat conduction problems for inhomogeneous materials. For example Divo and Kassab in [1] considered the linear elliptic euation governing the steady-state temperature field in inhomogeneous isotropic media while Clements et al. [2] obtained a boundary integral euation procedure which may be used to determine the steady-state temperature field for a wide class of inhomogeneous anisotropic materials. For time dependent problems Abdullah et al. [3] and Sutradhar et al. [4] employed the Laplace transform and a boundary integral formulation to solve problems for particular classes of isotropic inhomogeneous materials. Nonlinear heat conduction problems have been investigated by, for example, Clements and Budhi [5] who considered steady-state problems for a class of materials for which the thermal conductivity varied with the temperature and the three spatial coordinates. In the case of transient nonlinear problems boundary integral euation methods have been employed by a number of authors Corresponding author. addresses: mazis@indosat.net.id (M.I. Azis), david.clements@adelaide.edu.au (D.L. Clements). including Bialecki and Kuhn [6], oto and Suzuki [7] and Kikuta et al. [8]. This paper is concerned with nonlinear transient heat conduction problems for a class of anisotropic inhomogeneous materials. An outline of the paper is as follows. The Kirchhoff transformation is employed to convert the governing nonlinear euation to a linear second order partial differential euation with variable coefficients. A further transformation is then applied in order to transform the variable coefficients euation to a linear euation with constant coefficients. A boundary integral euation formulation is obtained for the solution of the constant coefficients euation. Then use of the inverse transformation provides a boundary integral euation for the solution of the linear variable coefficients euation. nce numerical values have been obtained from this boundary integral euation use of the inverse of the Kirchhoff transformation provides the temperature field for particular boundary value problems. Some numerical examples are considered in order to assess the effectiveness of the numerical procedure. Essentially the present work may be considered to be a generalisation of the paper of Clements and Budhi [5] to include transient effects and, more particularly, a generalisation of the paper by oto and Suzuki [7] to include a restricted class of thermal coefficients which vary with position /$ - see front matter r 28 Elsevier Ltd. All rights reserved. doi:1.116/j.enganabound

2 M.I. Azis, D.L. Clements / Engineering Analysis with Boundary Elements 32 (28) Basic euations Referred to the Cartesian frame x 1 x 2 x 3 this paper is concerned with obtaining the solution of the heat conduction problems governed by the euation Tðx; tþ Tðx; tþ k ij ðx; TÞ ¼ rðx; TÞcðx; TÞ x i x j for i; j ¼ 1; 2; 3, (1) T is the temperature, k ij are the heat conductivities, r is the density, c denotes the heat capacity, x ¼ðx 1 ; x 2 ; x 3 Þ, t is the time variable and the repeated suffix summation convention is employed. The analysis employed by oto and Suzuki in [7] is used and extended to derive a boundary element method (BEM) for the solution of (1). The matrix of the heat conductivities k ij is assumed to be a real symmetric positive definite matrix. Also the heat conductivities k ij are assumed to take the form k ij ðx; TÞ ¼m ij ðxþhðtþ, (2) m ij are differentiable functions and h is an integrable function. The analysis is specially relevant to an anisotropic medium but it eually applies to the isotropic case. For isotropic media the coefficients k ij in (1) take the form k 11 ¼ k 22 ¼ k 33 and k 12 ¼ k 13 ¼ k 23 ¼ and use of these euations in the following analysis immediately yields the corresponding results for isotropic materials. 3. The initial-boundary value problem A solution to (1) is sought in a domain for time tx. The boundary of consists of a finite number of piecewise smooth closed surfaces. In the initial temperature Tðx; Þ is given (see Fig. 1). n the temperature Tðx; tþ is specified and on the flux P T ðx; tþ is specified Flux P T (x,t) specified Temperature T(x,) specified Fig. 1. The initial-boundary value problem. Temperature T(x,t) specified ¼ [, P T ¼ k ij ðt=x j Þn i with n i denoting the ith component of the outward pointing vector n normal to. Substitution of (2) into (1) yields Tðx; tþ Tðx; tþ m x ij ðxþhðtþ ¼ rðx; TÞcðx; TÞ. (3) i x j Use of the Kirchhoff s transformation Y ¼ hðtþ dt (4) in (3) gives Yðx; tþ Yðx; tþ m x ij ðxþ ¼½rðx; YÞcðx; YÞ=hðYÞŠ. i x j (5) The corresponding initial and boundary conditions for (5) are Yðx; Þ ¼YðTðx; ÞÞ for x 2, Yðx; tþ ¼YðTðx; tþþ for x 2, P Y ðx; tþ ¼P T ðx; tþ for x 2, (6) P Y ¼ m ij ðy=x j Þn i. (7) 4. Integral euation formulation The coefficients m ij ðxþ are reuired to take the form m ij ðxþ ¼k ij gðxþ, (8) the k ij are constants and gðxþ is reuired to satisfy 2 g 1=2 k ij ¼. (9) Substitution of (8) and use of the transformation cðx; tþ ¼g 1=2 ðxþyðx; tþ (1) in (5) yields 2 cðx; tþ cðx; tþ k ij Cðx; cþ ¼, (11) rðx; cþcðx; cþ Cðx; cþ ¼. (12) hðcþgðxþ Let ðx; tjn; tþ be the fundamental solution of the euation cðx; tþ 2 cðx; tþ C r k ij ¼, (13) C r is a constant reference value of Cðx; cþ. Hence ðx; tjn; tþ 2 ðx; tjn; tþ C r k ij ¼ dðx nþdðt tþ, (14) ðx; tjn; tþ 2 ðx; tjn; tþ C r þ k ij ¼ dðx nþdðt tþ. (15)

3 156 ARTICLE IN PRESS M.I. Azis, D.L. Clements / Engineering Analysis with Boundary Elements 32 (28) Multiply (11) by ðx; tjn; tþ and integrate 2 cðn; tþ ðx; tjn; tþ Cðn; cþ k ij dðnþ dt ¼ (16) and multiply (15) by cðn; tþ and integrate ðx; tjn; tþ cðx; tþ ¼ cðn; tþ C r 2 ðx; tjn; tþ þ k ij dðnþ dt. (17) Adding (16) to (17) yields cðx; tþ ¼ Cðn; cþðx; tjn; tþ ðx; tjn; tþ þ C r cðn; tþ dðnþ dt 2 ðx; tjn; tþ þ cðn; tþk ij 2 cðn; tþ ðx; tjn; tþk ij dðnþ dt. (18) Now ðx; tjn; tþ ½ðx; tjn; tþcðn; tþš ¼ cðn; tþ þ ðx; tjn; tþ so that ðx; tjn; tþ cðn; tþ ¼ ðx; tjn; tþ þ ½ðx; tjn; tþcðn; tþš. (19) Use of (19) in (18) provides cðx; tþ ¼ ½Cðn; cþ C r Šðx; tjn; tþ dðnþ dt 2 ðx; tjn; tþ þ cðn; tþk ij 2 cðn; tþ ðx; tjn; tþk ij dðnþ dt þ C r ½ðx; tjn; tþcðn; tþš dðnþ dt. (2) Now t C r ½cŠ dðnþ dt ¼ C r dðnþ ½cŠ dt ¼ C r ½cŠ t¼ dðnþ (21) since ðx; tjn; tþ¼ when t ¼ t. Also reen s theorem gives 2 2 c ck ij k ij dðnþ ¼ ½cP P c Š dðnþ, (22) ðx; tjn; tþ P ðx; tjn; tþ ¼k ij n j, x i (23) P c ðn; tþ ¼k ij n j. x i (24) Use of (21) and (22) in (2) gives ðxþcðx; tþ ¼ ½ðx; tjn; tþp c ðn; tþ cðn; tþp ðx; tjn; tþš dðnþ dt ½Cðn; cþ C r Šðx; tjn; tþ dðnþ dt þ C r ½ðx; tjn; tþcðn; tþš t¼ dðnþ, (25) ¼ ifxe [, ¼ 1ifx2, ¼ 1 2 if x 2 and has a continuously turning tangent at x. Consider the function Hðn; cþ ¼ ½Cðn; cþ C r Š dc, (26) ¼ c c ¼ðC C rþ c. (27) Substitution of (27) into the second integral in (25) yields ðc C r Þ c dðnþ dt ¼ dðnþ dt. (28) Use of integration by parts for the integral on the righthand side of (28) gives dt ¼½H Št¼t t¼ H dt. (29) The unknowns are now assumed to be constant, on each time element. Hence, following the work of oto and Suzuki in [7], if the unknown c (and hence the temperature T) is assumed to take on its value at the upper bound ðt ¼ tþ of the semi-open time interval ð; tš then H dt ½HŠ t¼t½š t¼t t¼. (3) Substitution of (3) into (29) provides dt ½H Št¼t t¼ ½HŠ t¼t½š t¼t t¼ ¼½HŠ t¼t t¼ ½Š t¼. (31)

4 M.I. Azis, D.L. Clements / Engineering Analysis with Boundary Elements 32 (28) So the integral euation (25) may be written in the form ðxþcðx; tþ ¼ ½C r cðn; Þ Hðn; cðn; tþþ þ Hðn; cðn; ÞÞŠðx; tjn; Þ dðnþ þ P c ðn; tþ ðx; tjn; tþ dt cðn; tþ P ðx; tjn; tþ dt dðnþ. (32) However, if the temperature is assumed to take on its values at the lower bound t ¼ of the semi-open time interval ½; tþ then dt ¼. (33) So the integral euation (25) may be written in the form ðxþcðx; tþ ¼ P c ðn; Þ ðx; tjn; tþ dt cðn; Þ P ðx; tjn; tþ dt dðnþ þ C r cðn; Þðx; tjn; Þ dðnþ. (34) In (32) and (34) the fundamental solution is given in Chang et al. [9]. For the case of two-dimensional problems it may be written in the form s ðx; tjn; tþ ¼ 4pzðt tþ exp C rr 2, (35) 4zðt tþ R and z are given by R 2 ¼ð _ x 1 _x 1 Þ 2 þð _ x 2 _x 2 Þ 2 and z ¼½k 11 þ 2_sk 12 þð_s 2 þ s 2 Þk 22 Š=2 with _x 1 ¼ x 1 þ _sx 2 ; _x 1 ¼ x 1 þ _sx 2, _x ¼ sx 2 ; _x 2 ¼ sx 2, _s and s are, respectively, the real and the positive imaginary parts of the complex root s of the uadratic k 11 þ 2k 12 s þ k 22 s 2 ¼. Now from (7), (8) and (1) it follows that the corresponding value of P Y is P Y ðn; tþ ¼ P g ðnþcðn; tþþp c ðn; tþg 1=2 ðnþ, (36) P g ðnþ ¼k ij g 1=2 x j n i. (37) Substitution of (1) and (36) in (32) and (34), respectively, gives ðxþg 1=2 ðxþyðx; tþ ¼ ½C r g 1=2 ðnþyðn; Þ Hðn; g 1=2 ðnþyðn; tþþ and þ Hðn; g 1=2 ðnþyðn; ÞÞŠðx; tjn; Þ dðnþ þ g 1=2 ðnþp Y ðn; tþ ðx; tjn; tþ dt Yðn; tþ g 1=2 ðnþ P ðx; tjn; tþ dt P g ðnþ ðx; tjn; tþ dt dðnþ (38) ðxþg 1=2 ðxþyðx; tþ ¼ C r g 1=2 ðnþyðn; Þðx; tjn; Þ dðnþ þ g 1=2 ðnþp Y ðn; Þ ðx; tjn; tþ dt Yðn; Þg 1=2 ðnþ P ðx; tjn; tþ dt P g ðnþ ðx; tjn; tþ dt dðnþ. (39) The integral euations (38) and (39) may be used to obtain the solution Y of E. (5) subject to the initial and boundary conditions (6). Solutions for the temperature T may then be obtained using (4). Es. (38) and (39) are both based on the assumption that for each time step the dependent variable c (and hence the temperature T) takes on a constant value on the time interval ð; tþ. In the case of E. (38) this constant value is taken to be the value at the upper bound of the semi-open time interval ð; tš and in the case of E. (39) the constant value is taken to be the value at the lower bound of the semi-open time interval ½; tþ. 5. Discretisation and numerical examples In this section only the integral euation (39) will be considered. It will be used to find numerical solutions to some two-dimensional problems. Let the domain be discretised into K cells k for k ¼ 1; 2;...; K, the boundary into J elements j for j ¼ 1; 2;...; J and the time t into M time intervals ½t m 1 ; t m Š for m ¼ 1; 2;...; M. If it is further assumed that the unknowns are constant along any boundary segment j ¼½ j 1 ; j Š and take on their values at the mid-point j, and also

158 ARTICLE IN PRESS M.I. Azis, D.L. Clements / Engineering Analysis with Boundary Elements 32 (28) 154 16 constant in any domain cell k taking on their values at the center point r k of the cell.

5 158 ARTICLE IN PRESS M.I. Azis, D.L. Clements / Engineering Analysis with Boundary Elements 32 (28) constant in any domain cell k taking on their values at the center point r k of the cell. Then the discretised form of (39) for the time element ½t m 1 ; t m Š may be written as ðxþg 1=2 ðxþyðx; t m Þ ¼ XK Yðr k ; t m 1 Þ C r g 1=2 ðnþðx; t m jn; t m 1 Þ dðnþ k¼1 k ( þ XJ tm P Y ð j ; t m 1 Þ g 1=2 ðnþ ðx; t m jn; tþ dt dðnþ j¼1 j t m 1 " m Yð j ; t m 1 Þ g 1=2 ðnþ P ðx; t m jn; tþ dt dðnþ j t m 1 #) tm P g ðnþ ðx; t m jn; tþ dt dðnþ. (4) j t m 1 For each time interval m, (4) is treated as a new problem. It should be noted that the variable Y on the lefthand side of (4) is associated with the time t m as on the right-hand side it is associated with the time t m 1. Thus in order to solve (4) for the boundary unknowns (Y on and P Y on ) using an explicit scheme of linear euations the source point x has to be located outside the domain so that ¼ and the left-hand side of (4) is zero. Having solved for the boundary unknowns, solutions Y at the interior points r k for k ¼ 1; 2;...; K need to be found. These are then used as initial conditions for the time interval m þ 1. The time integrals of the fundamental solutions and P in (4) are evaluated analytically. Specifically m t m 1 ðx; t m jn; tþ dt ¼ s 4pz E 1 m C r R 2 4zðt m t m 1 Þ, (41) P ðx; t m jn; tþ dt t m 1 ¼ s 4pzR 2 exp C r R 2 R k 2 ij n j, (42) 4zðt m t m 1 Þ x i C r is assumed to be a positive number and E 1 is the exponential integral function defined by E 1 ðzþ 1 e s z s ds ¼ g ln z þ X1 n¼1 ð 1Þ n 1 z n nn!, (43) g ¼ : is Euler s constant. For all the problems considered the boundary of the domain is divided into a number of segments of eual length. Simpson s 3 8 rule is applied for the evaluation of the line integrals. For the area integrals, the domain is divided into a number of rectangular cells of eual area. The number of cells is increased to ensure the reuired level of accuracy for the domain integral. A uniform time step Dt ¼ t m t m 1 is used for all m. Also, when solving for the boundary unknowns, each source point is located outside the domain at a distance of one and a half times the length of the boundary segment under consideration. The distance is measured from the mid point of each segment in the direction of the outward pointing normal vector n Problem 1: Quadratic spatial variation of the conductivity matrix Consider the heat conduction problem governed by (1) for an inhomogeneous anisotropic material for which the coefficients of heat conduction are independent of temperature and are given by E. (8). The spatial variation of the coefficients is determined by the function gðxþ which is reuired to satisfy E. (9). Here gðxþ to taken to assume the uadratic form (Fig. 2) gðx 1 ; x 2 Þ¼ð1þ:1x 1 Þ2, (44) x i ¼ x i=d (for i ¼ 1; 2) with d a reference length. Also the coefficients occurring in E. (3) take the forms: ½m ij ðx 1 ; x 2 ÞŠ ¼ ½k ij Šgðx 1 ; x 2 Þ¼ 3 1 ð1 þ :1x Þ2, hðt Þ¼1, c ðx 1 ; x 2 ; T Þ¼ 4:5 T g1=2 ðx 1 ; x 2 Þ, r ðx 1 ; x 2 ; T Þ¼1, m ij ¼ m ij=m and k ij ¼ k ij=m with m a reference coefficient of heat conduction, T ¼ T=T with T a reference temperature, c ¼ c=c with c is a reference heat capacity and r ¼ r=r with r ¼ m h t =c d 2 a reference density. The geometry, initial and boundary conditions for the test problem under consideration are given in Fig. 3 T ðx 1 ; x 2 ; Þ ¼1 :25ðx 1 þ x 2 Þ2 ð1 þ :1x 1 Þ for px 1 p1 and px 2 p1, T ðx 1 ; 1; t Þ¼ 1 :25ð1 þ x 1 Þ2 ð1 þ t Þð1 þ :1x 1 Þ g(x 1,x 2 ) x Fig. 2. Spatial variation in the heat conductivities. x 1 1

6 M.I. Azis, D.L. Clements / Engineering Analysis with Boundary Elements 32 (28) for px 1 p1 and t X, T ðx 1 ; ; t 1 :25x 2 1 Þ¼ ð1 þ t Þð1 þ :1x 1 Þ for px 1 p1 and t X, P T ð1; x 2 ; t Þ¼ :3 2:2ð1 þ x 2 Þþ:75ð1 þ x 2 Þ2 1 þ t for px 2 p1 and t X, P T ð; x 2 ; t Þ¼ :3 þ 2x 2 :75x þ t for px 2 p1 and t X, t ¼ t=t, t is a reference time and P T ¼ P T=P with P ¼ k T =d a reference flux. It may be readily verified using the method of separation of variables that, for this particular problem, E. (3) admits the analytical solution T ðx 1 ; x 2 ; tþ ¼ 1 :25ðx 1 þ x 2 Þ2 ð1 þ t Þð1 þ :1x 1 Þ. (45) The corresponding heat flux is given by P T ðx 1; x 2 ; tþ ¼ :1ð3n 1 þ n 2 Þ 1 þ t ½1 :25ðx 1 þ x 2 Þ2 Š :5ð4n 1 þ 5n 2 Þ 1 þ t ðx 1 þ x 2 Þð1 þ :1x 1Þ. (46) Fig. 3. The boundary and the initial conditions for Problem 1. Since hðt Þ¼1, the variable Y is given by Y ¼ T Y ¼ Y=T. The function Cðx; cþ in (12) is given by C ðx; cþ ¼4:5=ðg 1=2 T Þ C ¼ C=C, C ¼ r c. As pg 1=2 T p1 it follows that C ðx; cþx4:5. It is convenient to take the reference value C r as C r ¼ 4:5 C r ¼ C r=c. The number of boundary segments and domain cells used is 16 (i.e. 4 4) and 16 (i.e. 4 4), respectively. Table 1 shows a comparison between the BEM and the analytical solutions at the interior point x ¼ð:5; :5Þ for some different values of the time step Dt. The results indicate the rate of convergence of the BEM solution to the analytical solution as the time step Dt decrerases Problem 2: Linear temperature variation of the conductivity matrix Consider the heat conduction problem for a homogeneous isotropic material with coefficients m ij ðx 1 ; x 2 Þ¼ 1 gðx 1 1 ; x 2 Þ, hðt Þ¼1 þ T, c ðx 1 ; x 2 ; T Þ¼1 þ :5T, r ðx 1 ; x 2 ; T Þ¼1, gðx 1 ; x 2 Þ¼1. The initial and boundary conditions are given in Fig. 4. Since hðt Þ¼1þT (4) gives Y ¼ T þ 1 2 T 2. Solving for T and taking the positive suare root yields p T ¼ 1þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Y þ 1. The function Cðx; cþ is given as C ðx; cþ ¼ 1 þ :5T 1 þ T. In view of the boundary conditions the temperature T is expected to satisfy pt p1, so that 3 4 pc ðx; cþp1 and hence a reference value of C r ¼ :75 is appropriate. This particular problem is euivalent to the problem in Example 2 of the paper by oto and Suzuki [7] (for the case when k ¼ 1, b ¼ :5). oto and Suzuki showed that results they obtained using their boundary element techniue for t ¼ 1, x 2 ¼ :1 and values of x 1 given in Table 1 Solution T ð:5; :5; t Þ for Problem 1 Time BEM solution Analytical solution t Dt ¼ :1 Dt ¼ :2 Dt ¼ :4 Dt ¼ :1 Dt ¼ :

7 16 ARTICLE IN PRESS M.I. Azis, D.L. Clements / Engineering Analysis with Boundary Elements 32 (28) Conclusion Fig. 4. The initial and boundary conditions for Problem 2. Table 2 Solution T for Problem 2 A boundary element method for the solution of a certain class of non-linear parabolic initial-boundary value problems for a certain class of anisotropic inhomogeneous media has been derived. The method is generally easy to implement to obtain numerical values for particular problems. It can be applied to the time dependent heat conduction problems for inhomogeneous anisotropic materials and the analysis includes isotropy as a special case. The numerical results obtained using the method indicate that it can provide accurate numerical solutions for particular transient 2-D heat conduction problems. Position, x 1 oto and Suzuki, t ¼ 1 Present procedure, t ¼ 1 References Table 2 agreed with the numerical solution obtained for the same problem using several different numerical techniues. It is therefore reasonable to assume that the figures calculated by them provide a reference solution which is close enough to the exact solution to the problem for the values of x 1 ; x 2 and t indicated. By way of comparison the current techniue was employed to solve the same problem with the time step decreased and with the boundary elements and corresponding cells increased until convergence was achieved to the 1 values shown in Table 2 using a time step of 3, 96 boundary elements and 32 ð4 8Þ cells. The calculated values which are shown in Table 2 agree with the reference solution correct to two decimal places. For convergence to the given values the current explicit method based on the formula (39) reuires more boundary elements and cells than the implicit method of oto and Suzuki which is based on a special case of the formula (38). [1] Divo E, Kassab AJ. eneralized boundary integral euation for heat conduction in non-homogeneous media: recent developments on the sifting property. Eng Anal Bound Elem 1998;22: [2] Clements DL, Budhi WS, Azis MI. Boundary element methods for the solution of a class of elliptic boundary value problems for anisotropic inhomogeneous media. ANIAM J 23;44E: [3] Abdullah H, Clements DL, Ang WT. A boundary element method for the solution of a class of heat conduction problems for inhomogeneous media. Eng Anal Bound Elem 1993;11: [4] Sutradhar A, Paulino H, ray LJ. Transient heat conduction in homogeneous and non-homogeneous materials by the Laplace transform alerkin boundary element method. Eng Anal Bound Elem 22;26: [5] Clements DL, Budhi WS. A boundary element method for the solution of a class of steady-state problems for anisotropic media. ASME J Heat Transfer 1999;121: [6] Bialecki R, Kuhn. Boundary element solution of heat conduction prolems in multizone bodies of nonlinear material. Int J Numer Methods Eng 1993;36: [7] oto T, Suzuki M. A boundary integral euation method for nonlinear heat conduction problems with temperature-dependent material properties. Int J Heat Mass Transfer 1996;39: [8] Kikuta M, Togoh H, Tanaka M. Boundary element analysis of nonlinear transient heat conduction problems. Comput Methods Appl Math 1987;62: [9] Chang YP, Kang S, Chen DJ. The use of fundamental reen s functions for the solution of problems of heat conduction in anisotropic media. Int J Heat Mass Transfer 1973;16:

Numerical solution of hyperbolic heat conduction in thin surface layers

International Journal of Heat and Mass Transfer 50 (007) 9 www.elsevier.com/locate/ijhmt Numerical solution of hyperbolic heat conduction in thin surface layers Tzer-Ming Chen * Department of Vehicle Engineering,