INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 7, 345-355 (1973)

FINITE ELEMENT TECHNIQUE APPLIED TO HEATCONDUCTION IN SOLIDS WITH TEMPERATURE

DEPENDENT THERMAL CONDUCTIVITY

G. AGUIRRE-RAMIREZ AND J. T. ODEN

Department of Applied MClthematics, University of Louisville, Louisville Kelllucky alld Division of Engineering.Mechanics, University of Texas, Austin. Texas, U.S.A.

SUMMARY

Consider a solid heat conductor with a non-linear constitutive equation for the heat flux. If the materialis anisotropic and inhomogeneous, the heat conduction equation to be satisfied by the temperaturefield 8(x, t) is

pc ~8 = div (L(8, x) [grad 8])+qvi

Here L( 8, x) [grad 8] is a vector-valued function of 8, x, grad 8 which is linear in grad 8. In the presentpaper, the application of the finite element method to the solution of this class of problems is demon-strated. General discrete models are developed which enable approximate solutions to be obtained forarbitrary three-dimensional regions and the following boundary and initial conditions: (a) prescribedsurface temperature, (b) prescribed heat flux at the surface and (c) linear heat transfer at the surface.Numerical examples involve a homogeneous solid with a dimensionless temperature-diffusivitycurve of the form K = KO( I+ aT). The resulting system of non-linear differential equations is integratednumerically.

INTRODUCTION

It is widely known tbat the thermal conductivity of many common materials is dependent on thetemperature. However, few attempts at assessing the importance of this dependence for practicalproblems have been made, for it leads to considerable non-linearities in the governing heatconduction equation. The evaluation of these effects is particularly elusive in the transient heat-conduction problems involving irregular geometries and mixed boundary conditions, sinceclassical methods of analysis are generally ineffective for this class of problems. A logicalalternative is to seek approximate solutions.

This paper deals with the application of the finite element technique to tbe analysis of non-linear problems in transient beat conduction. Equations of heat conduction for the finite elementsof a three-dimensional solid described by a non-linear constitutive equation for tbe heat fluxare derived on the basis of assumed temperature distributions over each element.

Finite element formulations of linear heat conduction problems have been given by Wilsonand Nickell! and Becker and Parr.2 Nickell and Sackman3.4 developed a finite elementformulation of the coupled thermoelastic problem by using appropriate variational principlesdeveloped using Gurtin's work5 as a guide. Oden and Kross6 developed general finite elementmodels for the analysis of coupled thermoelasticity problems by using energy balances.

The present paper is concerned with the development of finite element formulations whichcan be used to obtain approximate solutions of non-linear heat conduction problems. Emphasis

Received 19 April 1973@ 1973 by Jobn Wiley & Sons, Ltd.

345

346 G, AGUIRRE-RAMIREZ AND J. T. ODEN

is given to solids which exhibit temperature-dependent thermal conductivity. The resultingfinite element equations are applicable to the analysis of solids of arbitrary shape and subjectedto general boundary and initial conditions. Specific forms of the equations for more commonboundary conditions are given. These include prescribed temperatures, prescribed heat flux atthe surface and linear boundary-layer transfer at the surface,

NON-LINEAR HEAT CONDUCTION EQUA nONWe consider a thermally conductive body, occupying a region R, on which at each instant oftime there is defined a temperature field O(x, t), The temperature field 0 is assumed to satisfythe classical energy equation

(I)

We shall suppose that the body B is anisotropic and inhomogeneous with the followingconstitutive equation for the heat flux

b = -L(O,x) [grad 0] (2)

(3)

where L is a vector-valued function of 0,x, grad 0 which is linear in grad (). Upon substitutionof equation (2) into equation (1) we obtain a non-linear heat conduction equation

pc ~~ = div {L( 0,x)[grad OJ} +q

When B is isotropic and homogeneous equation (2) reduces to

b = -K(O) grad 0In this case equation (3) reduces to

pc :~ = div [K( 0) grad 0] +q

(4)

(5)

Approximate solutions of equation (5) with q = 0 have been obtained by Hays? and Hays andCurd8 for rectangular two-dimensional regions and special form of K(O) through the directmethod of calculus of variations. Ames9 discusses techniques for numerical solutions of equation(5) by finite difference schemes. Here we shall demonstrate how numerical finite element methodsmay be used to obtain approximate solutions of equation (3).

THE FINITE ELEMENT METHODApplications of the finite element concept to problems in solid mechanics are well-documentedand extensive references to previous work on the subject are available.10•1l For a general accountof the method, see References 12and 13. Here wegive only a brief outline for the sake of complete-ness, Suppose that the rcgion of interest R can be approximated by a new region R'. We furthersuppose that in R' we have idcntified a finitc number P of points, These points, called nodes,are labelledt ex(= 1,2, ... ,P). xCI:is thc point ·Iabelled ex.For the value of the tcmpcrature atnode ex,we write

(6)

We suppose that the region R' can be divided into a finite number G of arbitrary subregions(see Figure 1). These subregions are called finite elements. The finite element technique consists

t Lower case Greek subscripts have the range 1 to P.

HEAT CONDUCTION IN SOLIDS 347

of approximating a continuous function on R by a piecewise continuous function on R' in sucha way that at the nodes 0:, the values of the approximating function coincide with the values ofthe approximated function. Furthermore, these nodal values uniquely define the approximationof the function locally over each disjoint element and the local approximating functions areselected so as to fulfil continuity requirements along the surfaces separating adjacent elements.These observations reveal that we may study approximations of the temperature field withinan element independent of the behaviour of the temperature field in other elements,

Figure I.

We consider a typical element re (e = 1,2, ... , G). In re we identify a finite number m of points.It is convenient to consider the points of re to be referred to local co-ordinate axes and to denotethese local co-ordinates by;, The co-ordinates of the point m are then referred to as ;m' For anapproximate temperature distribution within the element we take the simplest distributiongiven byt

0(;, t) = 'Fm(;) Orn(t) (7)

where Om denotes the value of the temperature at the corresponding node 111 at time t and'Fm(;) are functions defined only on the element re and are chosen such that

'Fm(;n) = S~ (8)

where S~I is the Kronecker's delta, The functions \}fm(;) are called Lagrangian functions. Wenote that because of equation (8)

O(;m> t) = 0m(t) (9)

Denoting by V the gradient witb respect to the variable; we obtain from the assumed temperaturedistribution equation (7) within the element

(10)

EQUA TIONS OF THE DISCRETE MODEL

We now develop a self-consistent approximation scheme for the value of the temperature at thenodes of the element. To this end multiply both sides of equation (1) by an instantaneous variationSO, of the temperature field °

pc :~ SO, = -(divh) SO,+qSO, (11)

t The summation convention is in effecl, i.e. repealed subscripts or superscripts along the diagonal implysummation over the range of the index.

348 G. AGUIRRE-RAMIREZ AND J. T. ODEN

Through an elementary identify we cast this equation into the form

pc ~~ 88,+div(SO,h)-h' grad SO,-qSO, = 0 (12)

Substituting into this equation the constitutive relation (2), integrating over the volume and usingthe divergence theorem we arrive at

where n denotes the unit normal vector to the bounding surface S of the region R occupied by thebody B at time I.

Let us evaluate equation (13) over the finite element 'e' For the instantaneous variation SO,over the element 'e we take, according to equation (7)

S8, = '1""'(1;) SO~I

Upon substitution of equations (7), (10) and (14) into equation (13) we arrive at

[cl"II(On) 8171 +KP"'(OJ 0m+QI') SO; = 0

where a superposed dot indicates a time rate-of-change and

epm(On) = eIllP(On) = L{pe['PIl(I;) On] 'FP(;) \l·"'(;)}dv

Kpm( O,J = K"'/I( On) = L{v\}-,m(;). LrF'"(;) 0", ;] [Vo/P(;))} dt,

QP = f \FP(;)h'nds-j \}-'P(;)q(;)dvJ Ie r.

(14)

(15)

(16)

(17)

(18)

eP"', Kpm are, respectively, the components of the heat capacity and thermal conductivity matrixfor the element 'e' and QP is the generalized normal heat flux at node p for the element.

Since the instantaneous variation of the temperature is arbitrary, it follows from equation (15)that

el''''(O ) fi +Kpm(o ) ° = - QPn m 11 m (19)

This system of p non-linear equations describes heat conduction in the finite element in terms ofthe values of the temperature at the p nodes of the element. For each of the G number of elementswe have an equation of the form (19).

Having obtained the equations for the element 'e' the next order of business is to obtain thesystem of equations for the entire assembled system. To do this, we use incidence transformationsof the form.12

where

On = n~0a.G

Qa.= ~Q~Q".=1

{I if the node of element 'e is identical to node exof the region R'

Qa. =n 0 otherwise

(20a)

(20b)

(21 )

HEAT CONDUCTION IN SOLIDS 349

Here QlX is the generalized heat flux at node (X of the region R'. Using the transformations (20)we arrive at the system of discrete equations for the assembled system

where

<"P( 8,) ~ c"'( 8,) ~ ,~~lP. c,m(fl~ 8,l fl1.. )

KlXf/(O),) = Kf/lX(O).} = ~ n~Kpm(n~ o).)n~ne=1

(22)

(23)

We remark that upon reassembling the system of elements the node points of the element rebecame interior node points of the assembled system except, of course, for those nodes of theelements which happen to lie on the boundary of the region R'. These latter points are calledboundary points. In addition, we observe that according to equation (20b), the generalized heatflux QlX at node (X of the region R' is the sum of all the local generalized heat fluxes Qn at thecorresponding node 11 of all the elements joined at (and having in common) node (X of R'. Thisleads us to conclude that at all interior node points (X of R'

QlX=O (24)unless internal heat sources exist.

Equation (22) is a system of G equations to be solved for the value of the temperature at Gpoints of the region of interest R.

BOUNDARY AND INITIAL CONDITIONS

The system of first-order non-linear ordinary differential equations (22) for the values of thetemperature at the points exof R' replaces the non-linear partial differential equation (3) for thetemperature field O(x, t) over the region R. We now examine auxiliary conditions of the problemfollowing the plan of Wilson and Nickell1 and Becker and Parr.2

Prescribed surface temperatureWe consider the case when the auxiliary conditions to be satisfied are

0= g(x, t) for all x on S and all t> to }(25)

0= f(x) for all x in R and t~ to

where g is a prescribed function of (x, 1) and f a prescribed function of x.In order to obtain the corresponding conditions for the discrete model [equation (22)] we make

the agreement to label the boundary nodes 1,2, ... , N where N is the number of boundary points.Moreover, let Ll, ~ have the range 1,2, ... , Nand R, M the range N + I, ... ,P. Then xd denotes aboundary point and Xm an interior point. We write

gA(t) = g(xA, t), gA(t) = g(xA, t)

From equation (22) we obtain the equation for the interior points

cllfR(Os) 8n+KMR(Os) Oil = -GM(t)where

(26)

(27)

(28)

350 G. AGUIRRE-RAMIREZ AND J. T. aDEN

is the effective generalized heat at node M. We observe that GM is completely determined fromsurface conditions.

For the initial conditions we write

(29)

The problem is thus reduced to the solution of the system [equation (27)] subject to the initialconditions

(30)

Prescribed heat flux

Consider the case of a prescribed heat flux across the bounding surface

h = g(x,t) for all x on S and t~to }(31)

B =f(x) for x in R and t~to

For a typical boundary element we compute the generalized heat at a boundary point throughequation (18)

The transformed generalized heat is then obtained from equation (20b)

GQiX(t) = ~ Q~ Q"(t)

e=1

Observe that the only Q1I contributing to QiX are those corresponding to boundary points.For the initial conditions we write

fa =f(xJThe problem is thus reduced to the solution of the system

cap(BA) Bp+KaP(BA) Bp = -6.pQPwhere

{I for a,fJ boundary points and a = fJ }

6.a =P 0 otherwise

(32)

(33)

(34)

(35)

(36)

Linear boundary layer heat transfer at the surfaceThe case of linear heat transfer at the surface corresponds to the assumed rate of heat flux

across the boundaryr = -H(B- Bo) (37)

(38)

where 80 is the known temperature outside the boundary. To account for this effect we writethe surface integral appearing in equation (13) as

f,h.n SO,ds = - f,H(O- 00) SO,ds

For the boundary elements, we write for the corresponding surface integral

r. h·n \!"tl(l;)ds80~ = - r Hrt·n1(S) 0".- 00]'YnI(S)ds80~.~ J~ (39)

HEAT CONDUCTION IN SOLIDS

This leads to the following modified form of the discrete equation (19)

cpm(On) Om+ (Kpm(On) - Hpm) Om = - R'lwhere

Hpm = Hmp = H f ,¥p® 'J'm(~ ds )

R' ~ HO, f.'Y'(~ds- Lq(~) 'Y'md.The corresponding equations for the assembled system will then be

caP(O),) 0p+ [KCtP( O.J- HCtP] °P = - Ll,RPwhere

Ho;p = fn;Hllmn~ )e~1

oRP= ~n:RP

e-1

351

(40)

(41)

(42)

(43)

We observe that the contributions to Hpm, RP of all nodal surfaces of the boundary elementswhich are interior to R' is zero.

SPECIAL FORM

To obtain specific forms of the finite element equations, we now consider an isotropic homogeneoussolid with constitutive equation for the heat flux of the form

h = -K(8) grad 8 (44)

In addition, the density p and specific heat c of the solid are assumed to be constant. These terms,when combined with the temperature-dependent thermal conductivity, leads to defining atemperature-dependent thermal diffusivity

K( 0) = K( O)f pc

It is convenient to introduce the change in variables

T = Of Os, T = KO tfa2

(45)

(46)

where Os is a reference temperature, KO a reference thermal diffusivity and a a reference length.T and T are dimensionless.

The thermal diffusivity will be assumed to be a linear function of temperature

(47)

where a is tbe slope of the dimensionless temperature-diffusivity curve, The fundamental equationsfor the element (19) become

2QI'Rpm t +aKpmn T. T. +Kpm T. --..!!.....-m m n m - pCK

OOs (48)

352

where

G. AGUlRRE-RAMlREZ AND J. T. ODEN

HPIIl = J' o/P(;) o/tIl(;) dvr.

KPlIln = a2 j' {Vo/P(;).V'I"/Il(;)} q;'n(;) dvr.

Kl)/Il = a2 J V,¥P(;). W'/Il(;) dvr.

(49)

The corresponding equations for the assembled system are then

H(Y.Pf,+aK(Y.),PT), Tp+K(Y.PTp = _ a2Q(Y.

pCo Oswhere

GHOtP = ~ QOt H pnw QP

~ P IIIe=l

GKOtP = ~ Q~KPIIl Q~

e=1

(]

QOt= ~.o~Ql'e=l

NUMERICAL RESULTS

(50)

(51)

For a numerical problem using the finite element representation [equation (50)] we consider aninfinite slab of finite thickness L at a reference temperature 88, The slab is given a sudden increasein temperature to 288 at one of its surfaces. Two cases are considered:

1. The increase in temperature is maintained for aU times [Figure 2(a)].2. The increase in temperature is maintained for 10 units of the dimensionless time [Figure

2(b»).

.. ~

..OJ..J~1-

~o: IJ=:=:. 0 " •DIMENSIOIIL,SS TIME T

10) (0)

(e)

Figure 2.

HEAT CONDUCTION IN SOLIDS 353

Cases were run in which 50 finite elements, corresponding to 50 simultaneous non-lineardifferential equations in the nodal temperatures, were employed. The results indicated in thefigures were obtained using 20-element models. The slab shown [Figure 2(c)] was divided alongL into 20 equal spacings. The reference length was chosen as a = L/20. The resulting 20 non-linear simultaneous equations were solved numerically using the Runge-Kutta-Gill integrationscheme in the UNIVAC 1108. Figures 3 and 4 show the variation in the dimensionless temperaturefor various values of u at nodes 4 and 15 units from the boundary, for the input shown, It can

2.0

I-,., 'CT

o 25 50 75 100 125DIMENSIONLESS TIME r

150 175

Figure 3. Temperalure variation al node 4

20]'5

~ I ~,"" 1"'--- 0' • 05~ T:: ''---cr-09~ .

""1 'I Iii. i • io ~ 50 n ~ ~ ~ ~ ~

DIMENS10lO_ESS TIME r

Figure 4. Temperature variation at node 15

I-

1.0

0.5

o

- 0" 0.0 (LINEAR)

----- a· 0.5

10

DISTANCE X INTO WALL

1:1

r

20

Figure 5. Temperature distribulion for various times

354 G, AGUIRRE-RAMIREZ AND 1. T. ODEN

T-50,0

- 0' -0.0 (LINEAR)

---- 0'-0.5 ~~o T 10

Il I 10

DISTANCE X INTO WALL

Figure 6. Diffusion of pulse

15 20

be observed that as a increases, the slab reaches the steady-state temperature 20s more rapidly.Figure 5 shows the temperature distribution of the slab throughout its thickness for varioustimes for the input shown, while Figure 6 shows the diffusion of the pulse [Figure 2(b)] fora = 0,0 and a = 0,5.

ACKNOWLEDGEMENTS

The preparation of this paper was sponsored by the Research Institute, University of Alabamain Huntsville under NASA Grant NGL 01-002-001.

APPENDIXNotation

a = Reference length, ftc = Heat capacity, ft2/hr2-oFh;= Heat flux, Btu/ft2-luK = Thermal conductivity, Btu/ft-hr-oFq = Heat source, Btu/ftS-hrT = 6/6s = dimensionless temperaturet = Time, hrx = Space co-ordinatesK = Thermal diffusivity, ft2/hr

Kll = Reference thermal diffusivity, ft2/hrp = Density, lb/ft3°= Temperature, OF

Os = Reference temperature, OFT = KO t/a2 = dimensionless timea = slope of the dimensionless temperature-diffusivity curvel; = Local co-ordinates

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Nuc/. Engng Design, 4, 276-286 (1966).2. E. B. Becker and C. H, Parr, 'Application of the finite elemenl method 10 heat conduction in solids',

Tech, Repl S-117, Rohm and Haas, Redslone Research Laboratories, Huntsville (1967).3, R. E. Nickell and J. J. Sackman, 'Approximate solutions in linear, coupled thermoelasticity', J. Appl.

Mech, 35, Ser. E, 255-266 (1968).

HEAT CONDUCTION IN SOLIDS 355

4. R, E. Nickell and J. J. Sackman, 'The extended Ril2 method applied to Iransient, coupled Ihermoelaslicboundary-value problems', Slruct. and Malls Res. Repl No. 67-3, Univ, of Calif. al Berkeley (1967).

5. M. E. Gurtin, 'Variational principles for linear initial·value problems', Q. Appl. Math. 22, 252-256 (1964).6. J. T. Oden and D. A. Kross, 'Analysis of general coupled thermoelaslicilY problems by Ihe finite elemenl

method', Proc. 2nd COllf Malrix Melh. Struct. Meek, Wright-Patterson Air Force Base, Ohio, (1968).7. D. F. Hays, 'Variationat formulation of the heat equation: temperature dependent thermal conductivity',

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1965.10. J. S. Przemieniecki, Theory of Matrix Structural Analysis, McGraw-Hili, New York, 1968.I!. A. C. Singhal, '775 selected references on the finite element method and methods of struclural analysis',

Rept. S-12, Civil Engng Dept., Laval Univ., Quebec (1969).12. J. T. Oden, 'A general theory of finite elements: I.Topological considerations', Int. J. 1II1m.Melh. Engng,

I, 205-221 (1969).13. J. T. Oden. Finite Elements of Nonlinear Continua, McGraw-Hili, New York, 1972.