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Finite-Element Technique Applied to HeatConduction in Solids with TemperatureDependent Thermal ConductivityG. AGUIRRE·RAMIREZAssistant Professor

J. T. ODENProfessorMem.ASME

Engineering Mechanics,University of Alabama atHuntsville, Huntsville, Ala.

In this paper, the finite-element method is applied to solid heat conduction with anonlinear constitutive equation for the heat flux. General discrete models that aredeveloped enable approximate solutions to be obtained for arbitrary three-dimensionalregions and three boundary and initial conditions: (a) prescribed surface temperature, (b)prescribed heat flux at the surface, and (c) linear heat transfer at the surface. Numericalexamples involve a homogeneous solid with a dimensionless temperature-diffusivity curve,The resulting system of nonlinear differential equations is integrated numerically.

Contributed by the Heat Transfer Division of The American Society of Mechanical Engineers forpresentation at the ASME Winter Annual Meeting, November 1&-20, 1969, Los Angeles, C~f.Manuscript received at ASME Headquarters July 16, 1969.Copies will be available until Scptember 1, 1970.

,Finite-Element Technique Applied to HeatConduction in Solids with TemperatureDependent Thermal Conductivity

G. AGUIRRE·RAMIREZ J. T. ODEN

NON-LINEAR HEAT CONDUCTION EQUATION

where 1is a vector valued function of 9, ~, grad 9which is linear in grad 9. Upon substitution of (2)

We shall suppose that the body B is anisotropic andinhomogeneous with the following constitutive equationfor the heat flux:

(2)

(1)-div ~ + q09pc at

h

We consider a thermally-conductive body, occupyinga region R, on which at each instant of time there isdefined a temperature field 9~,t). The temperaturefield 9 is assumed to satisfy the classical energyequation

elusive in the transient heat-conduction problemsinvolving irregular geometries and mixed boundaryconditions, since classical methods of analysis aregenerally ineffective for this class of problems. Alogical alternative is to seek approximate solutions.

This paper deals with the application of thefinite element technique to the analysis of non-linearproblems in transient heat conduction. Equations ofheat conduction for the finite-elements of a three-dimensional solid described by a non-linear consti-tutive equation for the heat flux are derived on thebasis of assumed temperature distributions over eachelement.

Finite element formulations of linear heatconduction problems have been given by Wilaon andNickell [lJ and Becker and Parr [2J. Nickell andSackman [3,4J developed a finite element formulationof the coupled thermoelastic problem by using appro-priate variational principles developed uaing Gurtin'swork [5] as a guide. Oden and Kross [6] developedgeneral finite element models for the analysis ofcoupled thermoelaaticity problems by using energybalances.

The present paper is concerned with the develop-ment of finite element formulations which can be usedto obtain approximate solutions of non-linear heatconduction problems. Emphasis is given to solidswhich exhibit temperature-dependent thermal conducti-vity. The resulting finite-element equations areapplicable to the analysis of solids of arbitrary shapeand subjected to general boundary and initial condi-tions. Specific forms of the equations for morecommon boundary conditions are given. These includeprescribed temperatures, prescribed heat flux at thesurface and linear boundary-layer transfer at thesurface.

ABSTRACT

pc : = div C!:.(9,~)[grad 9]) + q.

INTRODUCTION

Consider a solid heat conductor with a non-linear constitutive equation for the heat flux. Ifthe material ia anisotropic and inhomogeneous, theheat conduction equation to be satisfied by the tem-perature field 9(~,t) is

It is widely known that the thermal conductivityof many common materials is dependent on the tempera-ture. However, few attempts at assessing theimportance of this dependence for practical problemshave been made, for it leads to considerable non-linearities in the governing heat conduction equation.The evaluation of these' effects is particularly

NCl1ENCLATURE

Here 1(9,~[grad 9] is a vector-valued function of 9,x, grad 9 which is linear in grad 9. In the presentpaper, the application of the finite-element method tothe solution of this class of problems is demonstrsted.General discrete models are developed which enableapproximate solutions to be obtained for arbitrarythree-dimensional regions and the following boundaryand initial conditions: (a) prescribed surface temper-ature, (b) prescribed heat flux at the surface and(c) linear heat transfer at the surface. Numericalexamples involve a homogeneous solid with a dimension-loss temperature-diffusivity curve of the formK a Xc(l + aT). The resulting system of nonlineardifferential equations is integrated numerically.

a = reference length, ftc .. heat capacity, fea/hr2_deg F.l!. .. heat flux, BTU/ft2-hrK - thermal conductivity, BTU/ft-hr-deg Fq .. heat source, BTU/ftS-hrT .. 9/9. .. dimensionless temperaturet .. time, hr~ .. space coordinatesK .. thermal diffusivity, ft2/hrXc .. reference thermal diffusivity, ft3/hrp D density, Ib/ftS

9 .. temperature, deg FQ. = reference temperature, deg FT .. Xct/a2 .. dimensionless timecr = slope of the, dimensionless temperature-

diffusivity curve~ .. local coordinates

2

into (1) we obtain a non-linear heat conductionequation 9(~, t) (7)

div (K(9) grad 9) + q

diV( ~(9,~) [grad 9)) + qwhere 9. denotes the value of the t~perature ,at thecorresponding node m at time t and 'l'·~are functionsdefined only on the element r. and are chosen such that

(8)

(9)

•The functions 'l"~

We note that becausewhere 6: is the Kronecker's delta.are called Lagrangian functions.of (8)

(3)

(5)

(4)

reduces to(2)

h = -K(9) grad 9

When B is isotropic and homogeneous

In this case (3) reduces to

(10)

(13)

!:(9,;9[grad 9]}dR

Denoting by V the gradient with respect to the variable~ we obtain from the assumed temperature distribution(7) within the element

Pig.1

EQUATIONS OF THE DISCRETE HODEL

Through an elementary identify we cast this equationinto the form

We now develop a self-consistent approximation,scheme for the value of the temperature at the nodesof the element. To this end multiply both sides of(1) by an instantaneous variation 69t of the tempera-ture field 9:

pc ~ 69t = -(div h)69t + q6et . (11)ut _

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

1(pcR

(6)

Approximate solutions of (5) with 9 = 0 have beenobtained by Hays [7) and Hays and Curd L8) forrectangular two-dimensional regions and special formof K(9) through the direct method of calculus ofvariations. Ames [9) discusses techniques for numeri-cal solutions of (5) by finite difference schemes.Here we shall demonstrate how numerical finite elementmethods may be used to obtain approximate solutionsof (3).

THE FINITE ELEHENT METiiOD

Applications of the finite-element concept toproblems in solid mechanics are well-documented andextensive references to previous work on the subjectare available [10,11]. For a genera~ account of themethod, see [12,13). Here we give only a briefoutline for the sake of completeness. Suppose thatthe region of interest R can be approximated by anew region R I. We further suppose that in R' we haveidentified a finite number P of points. These points,called nodes, are labeled a(= 1,2, ... ,P). ~ is thepoint labeled a. For the value of the temperature atnode a, we write

1Lower case Greek subscripts or superscripts have the2range 1 to P.

The summation convention is in effect, i.e., repeatedsubscripts or superscripts along the diagonal implysummation over the range of the index.

We suppose that the region R' can be divided intoa finite number G of arbitrary subregions (see Fig. 1).These subregions are called finite elements. Thefinite element technique consists of approximating acontinuous function on R by a piecewise continuousfunction on R' in such a way that at the nodes a, thevalues of the approximating function coincide withthe values of the approximated function. Furthermore,these nodal values uniquely define the approximationof the function locally over each disjoint element andthe local approximating functions are selected so asto fulfill continuity requirements along the surfacesseparating adjacent elements. These observationsreveal that we may study approximations of thetemperature field within an element independent of thebehavior of the temperature field in other elements.

We consider a typical element r.(e = 1,2.··"G).In r. we identify a finite number m of points. It isconvenient to consider the points of r. to be referredto local coordinate axes and to denote these localcoordinates by S. The coordinates of the point marethen referred to as~. For an approximate tempera-ture distribution within the e~ement we take thesimplest distribution given by

3

Upon substitution of (7). (10). and (14) into (13) wearrive at

(24)

The system of first order non-linear ordinarydifferential equations (22) for the values of thetemperature at the points 0. of R' replaces the non-linear partial differential equation (3) for thetemperature field Q(x. t) over the region R. We nowexamine auxiliary conditions of the problem followingthe plan of Wilson and Nickell [1] and Becker and Parr[2].

BOUNDARY AND INITIAL CONDITIONS

Prescribed Surface TemperatureWe consider the case when the auxiliary conditions

to be satisfied arc:

unless internal heat sources exist.Eq. (22) is a system of G equations to be solved

for the value of the temperature at G points of theregion of interest R.

We remark that upon reassembling the system ofelements the node points of the element r. becameinterior node points of the assumbled system. exceptof course, for those nodes of the elements whichhappen to lie on the boundary of the region R I. Theselatter points are called boundary points. In addition,we observe that according to (20)2. the generalizedheat flux Qo. at node 0. of the region R' is the sum ofall the local generalized heat fluxes ~n at thecorresponding node n of all the elements joined at(and having in common) node 0. of R'. This leads us toconclude that at all interior node points 0. of R'

(17)

(18)

(16)

(15)o

~Dp(QO) =1 {pc(~n<,~)Qn)yp(1)vm(9}dVro

KpD(QO) = KDp(Qn) =1 {~D(~ . ~(YO(1>Qn •.vr.

[..yP(,Q] } dv

· fO IOQp .. 'fP(9~' ~ ds - 'fp(9q(~dv.Se re

• 0

cpa. Kpa are. respectively. the components of the heatcapacity and ~hermal conductivity matrix for theelement ro' and 01' is the generalized normal heatflux at node p for the element.

Since the instantaneous variation of the tempera-ture is arbitrary, it follows from (15) that

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

where ~ denotes the unit normal vector to the boundingsurface S of the region R occupied by the body B attime to

Let us evaluate (13) over the finite element ro'For the instantaneous variation bet over the eiementro we take. according to (7).

o *bet : '1'" (,9 be" (14)

(20)Q 00. 0E 0 Qn

.=1 Ii

(26)

(25)for all ~ in Rand t S toQ c f(x)

Q .. g(~. t) fO,r_all ~ on S and all t > to

where g is a prescribed function of (x.t) and f aprescribed function of x.

In order to obtain-the corresponding conditionsfor the discrete model (22) we make the agreement tolabel the boundary nodes 1,2.'·,N where N is thenumber of boundary points. Moreover, let 6. E havethe range 1,2.··.N and R.M the range N + 1.·· ••P.Then~ denotes a boundary point and ~ and interiorpoint. We write

(19)o

-QP• • 0

cpa(Qn)Qa + Kpa(Qn)Q.

This system of p non-linear equations describes heatconduction in the finite element in terms of thevalues of the temperature at the p nodes of theelement. For each of the G number of elements wehave an equation of the form (19).

Hav~ng obtained the equations for the element r••the next order ot business is to obtain the system ofequations for the entire assembled system, To dothis. we use incidence transformations of the form[12].

1where

if the node of element r. isidentical to node 0. of theregion R'.

o otherwise.

(21)

From (22) we obtain the equation for the interiorpoints

(27)

where

Here Qo. is the g~neralized heat flux at node 0. of theregion R'. Using the transformations (20) we arriveat the system of discrete equations for the assem-bled system:

is the effective generalized heat at node M. Weobserve that G~ is completely determined from surfaceconditions.

For the initial conditions we write

(28)GM (t)

(22)

fM = f(~M) . (29)

The problem is thUB reduced to the solution of the

(23) system (27) subject to the initial conditions

QM(tO) .. fM t .. to . (30)

4

Prescribed Reat FluxConsider the case of a prescribed heat flux

across the bounding surface:

will then be

(42)

~ D ~(~,t) for all ~ on Sand t ~ to

• a.Observe \that the only QP contributing to Q are thosecorresponding to boundary points.

For the initial conditions we write

For a typical boundary element we compute thegeneralized heat at a boundary point through (18):

QP(t) = f h(i)l(i>t) . ~ ds - 1h(i)q(~dV (32)se re

generalized heat is then obtained

(43)

G

Hal3 .. ~ n~Up· n~0=1

R~ .. t O~ RP.

where

• aWe observe that the contributions to HP., RP of

all nodal surfaces of the boundary elements which areinterior to R' is zero.

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

SPECIAL FORM

(31)for ~ in Rand t ~ to .

~ ....L n~Q·(t) (33)e=l

e .. f(.!9

The transformedfrom (20)a

Qo.(t)

(34) h -K(e) grad e (44)

The problem is thus reduced to the solution of thesystem

(35)

In addition, the density p and specific heat c of thesolid are assumed to be constant. These terms, whencombined with the temperature dependent thernlalconductivity, leads to defining a temperature depend-ent thermal diffusivity

where

Linear Boundary Layer Heat Transfer at the SurfaceThe case of linear heat transfer at the surface

corresponds to the assumed rate of heat flux acrossthe boundary

where eo is the known temperature outside the boundary.To account for this effect we write the surfaceintegral appearing in (13) as

f.e . ~ C9tds .. - f H(e - eo)OQtds. (38)s s

(46)

(45)

(47)

T .. ~a

= fillp c

110(1 + 0 T)

1'.(e)

T

where 0 is the slope of the dimensionless temperature-diffusivity curve. The fundamental equations for theelement (19) become

It is convenient to introduce the change in variables

where e. is a reference temperature, 110 a referencethermal diffusivity, and a a reference length. T andTare dLmensionless.

The thermal diffusivity will be assumed to be alinear function of temperature

(36)

(37)-H(e - eo)

for a., ~ boundary points and a. = ~.

otherwise.

r

For the boundary elements, we write for thecorresponding surface integral

e. 0 eHPDTD + a KPlnT.Tn + KPDT. (48)

(49)

(50)

where

The corresponding equations for the assumbled systemare then

(39)

(41)

(40)a

-RP

modified form of the

where

° ° Ia aRPI .. H"P .. H 'fP(,9'fD(9dssa

° fa 1 aRP .. Heo ·fP(,.9ds- q(.z.)'fP<,~,>dv5e r.

This leads to the followingdiscrete equation (19)

° . (0 a )cPI(e.)el + KPI(e.) - HP. e.

\

'1, The corresponding equations for the assembled system

5

'.'~,

whereG

Hc$ = ~ O~ Hpm O~G

Kc$ = ~ 00. Kpm oSp :

0=1

G (51)

KaAP = ~ 0u eA 0 0o n 01' Kp·nI' m n \I)

0=1(I)

GI

G.... f-<

Qo. ~ O~ Qpg ,12

= '... p"

u: Et: <II

~E-<

NU}nffiICAL RESULTS 0

0 10

(bJ

z

o20

Fig.2

(c)

Dimensionless Time T

10(a)

2

For a numerical problem using the finite elementrepresentation (50) we consider an infinite slab offinite thickness L at a reference temperature Q•.The slab is given a sudden increase in temperature to2 Q. at one of its surfaces, Two cases are consid-ered: (1) the increase in temperature is maintainedfor all times (Fig. 2a), and (2) the increase intemperature is maintained for 10 units of thedimensionless time (Fig. 2b). Cases were run inwhich 50 finite elements, corresponding to 50simultaneous nonlinear differential equations in thenodal temperatures, were employed. The resultsindicated in the figures were obtained using 20-element models. The slab shown (Fig. 2c) was dividedalong L into 20 equal spacings. The reference lengthwas chosen as a = L/20. The resulting 20 nonlinearsimultaneous equations were solved numerically usingthe Runga-Kutta-Gill integration scheme in the UNIVAC1108. Figures 3 and 4 show the variation in thedimensionless temperature for various values of cr atnodes four and fifteen units from the boundary, forthe input shown. It can be observed that as crincreases, the slab reaches the steady state tempera-ture 2 9. more rapidly. Figure 5 shows the tempera-ture distribution of the slab throughout its thicknessfor various times for the input shown, while Fig. 6shows the diffusion of the pulse (Fig. 2b) forcr = 0.0 and cr= 0.5.

T

o' ...(I)

t:<II

!0,

oDimensionless Time T

Temperature Variatioll at Node 15

Fig.1f

o 50 100 150 zooDimensionless Time T

'teml'eratun, Variation at Node 4

Fig.'

6

2 Becker, E. B. and Parr, C. H., "Applicationof the Finite Element Method to Heat Conduction inSolids," Technical Report S-1l7, Rohm and HaasRedstone Research Labo~atories, Huntsville, 1967,

3 Nickell, R. E. and Sackman, J. J" "Approxi-mate Solutions in Linear, Coupled TheI1lloelasticity,"Journal of Applied Mechanics, Vol, 35, Series E ,No.2, June 1968, pp. 255-266,

4 Nickell, R, E, and Sackman, J, J., "TheExtended Ritz Method Applied to Transient, CoupledThermoelastic Boundary-Value Problems," Structuresand Materials Research Report No. 67-3, University ofCalifornia at Berkeley, Feb. 1967.

5 Gurtin, M, E" "Variational Principles forLinear Initial-Value Problems," Quarterly of AppliedMathematics, Vol. 22, No, 3, 1964, pp. 252-256.

6 Oden, J, T, and Kross, D, A" "Analysis ofGeneral Coupled Thermoelasticity Problems by theFinite Element tfethod," Proceedings, Second Conferenceon Matrix Methods of Structural Mechanics, Wright-Patterson AFB, Dayton, Ohio, Oct. 1968.

7 Hays, D. F" "Variational Formulation of theHeat Equation: Temperature Dependent ThermalConductivity," Symposium on Non-Equilibrium Thermo-dynamics, Variational Techniques and Stability,University of Chicago Press, Chicago, 1965.

8 Hays, D, F, and Curd, H, N" "Heat Conductionin Solids," International Journal of Heat and HassTransfer, Vol. 11, No.2, Feb. 1968, pp, 285-295.

9 Ames, W, F" Nonlinear Partial DifferentialEquations in Engineering, 1st ed., Academic Press,New York, 1965.

10 Przemieniecki, J, S" Theory of MatrixStructural Analysis, McGraw-lIill Book Co" New York,1968,

11 Singhal, A, C" "775 Selected References onthe Finite Element Method and Methods of StructuralAnalysis," Report S-12, Civil Engineering Dept"Laval University, Quebec, January 1969.

12 Oden, J. T., "A General Theory of FiniteElements. 1. Topological Cons iderations," Internat iona IJournal for Numerical Methods in Engineering, June 1968.

13 Oden, J, T., "A General Theory of FiniteElements. II. Applications," International Journalfor Numerical Hethods in Engineering, (to appear).

20

20/5

/5

---------- -

------ ----- ---

/0

----7"0,$

-""0.0

~~o 10'"

----

Distance lC into 1,{,111

If)

Distance into Wall

s

5

-7'0,0- ----7'·0,5

l'emperature Distrihutioll for Various Times

Fig.5

o

f-<

r:i.E<ll

f-<<Il..<ll....."0....~..e...Cl

L0.

2f-<

r:i.~H

'"'"<ll.....~I~

0....'"r:;<lle....CI

Diffusion of Pulse

Fig. 6

ACKNOWLEDGMENTS

The preparation of this paper was sponsored bythe Research Institute, University of Alabama inHuntsville under NASA Grant NGL 01-002-001.

REFERENCES

1 Wilson, E. L. and Nickell, R. E., "Applicationof the Finite Element Method to Heat ConductionAnalysis," Nuclear Engineering and Design, Vol. 4,No.3, Oct. 1966, pp. 276-286.

7