This article was downloaded by: [University of Arizona]On: 03 June 2014, At: 04:38Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Drying Technology: An International JournalPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/ldrt20

NUMERICAL SIMULATION OF DRYING OF REFRACTORYCONCRETEZ. X. Gong a , B. Song a & Arun S. Mujumdar ba Mechanical Engineering Department , Tianjin Institute of Light Industry , Tianjin, P. R.,Chinab Chemical Engineering Department , McGill University , Montreal, Quebec, CanadaPublished online: 25 Apr 2007.

To cite this article: Z. X. Gong , B. Song & Arun S. Mujumdar (1991) NUMERICAL SIMULATION OF DRYING OF REFRACTORYCONCRETE, Drying Technology: An International Journal, 9:2, 479-500, DOI: 10.1080/07373939108916677

To link to this article: http://dx.doi.org/10.1080/07373939108916677

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

DRYING TECHNOLOGY, 9(2), 479-500 (1991)

NUMERICAL SIMULATION OF DRYING OF REFRACPORY CONLRETE

2 . X. Gong, B . Song Mechanical Engineering Department Tianjin Institute of Light Industry

Tianjin, P. R. China and

Arun S. Mujlrmdar Chemical Engineering Department

McGill University Montreal, Quebec, Canada

Key Words and Phrases: drying model; finite elements; volumetric heating; pore pressure.

The drying process of refractory concrete is simulated numerically using a newly-developed one-pint quadrature finite element algorithm. The effect of volumetric heat supply (e.g. simplified microwave heating) is also examined numerically. Time-dependent temperature, pore pressure and moisture profiles are presented at selected locations. Integrated water release curves are also given.

Refractory concrete is an essential material in the

fabrication of various industrial reactor vessels. Drying process

of monolithic refractory concrete is of significant practical

interest.

Copyright 1991 by Marcel Dekker, Inc.

Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

4:38

03

June

201

4

480 GONG, SONG, AND MUJUMDAR

Initial heating of refractory concrete may produce a signifi-

cant build-up of pore pressure which can cause explosive

"spalling". Although the causes are manifold, the influence of the

drying process is undoubtedly a main one. Therefore, rational

prediction of the response of refractory concrete to heating is of

great importance and much attention has been paid to it.

On the basis of Luikov's theory for coupled heat and mass

transfer, Bazant developed a mathematical model to describe the

drying process of refractory concrete [1 ,2 ,31 . Experiments to

measure the parameters used in his model were also carried out.

Recently, Dhatt and his co-workers solved the mathematical model

for one dimension using a finite element method [ 4 1 . The results

were reported to be in agreement with the experimental ones.

In this paper the Luikov model is solved numerically in two

dimensions using a newly-developed one-point quadrature finite

element procedure. The drying process of increasing temperature at

the boundary as well as a simplified volumetric heat supply is

simulated and analyzed. The volumetric supply term does not

strictly simulate dielectric heating.

FIELD EQUATIONS

One important characteristic of heat and moisture transfer in

porous bodies is their coupling. Thus, the fluu of moisture J

should consist of a flux due to the gradient of moisture content

W, as well as a flux due to the temperature gradient. Similarly,

the heat flux q should consist of a flux due to the temperature

gradient and a flux due to the gradient of moisture content.

Accordingly,

J = -a,,,grad W - a YT grad T q : -aT&-& W - aTTgrad T

in which W = mass of all free (not chemically bound) water per m3 of concrete. Coefficients a,,,,, a,,, , aTU, and aTT depend on W and T.

Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

4:38

03

June

201

4

NUMERICAL SIMULATION OF DRYING OF REFRACTORY CONCRETE 481

Because W is a function of temperature T and pore pressure P,

W = W(P,T), Eq. 1 can be rearranged as

a aW in which g = 9.806 m/s2, gravity acceleration; and = a,, (&,, a = a aw (-1 . Experiments show that a, is rather small [21.

1 ,I+ m P Therefore, the thermal gradient contribution, atgrad T, can be

neglected. Thus, Eq. 3 can be simplified as

in which a = permeability of moisture, in m/s. In fact, grad P

in Eq. 4 has already, included a part of the contribution of the

temperature gradient because P is related to temperature.

The coupled heat flux, aTygrad W, in Eq. 2 can also be

neglected. As a result, setting aTT= k = heat conductivity, we may write

q = -k grad T ( 5 ) According to the conservation laws of heat and moisture in

concrete, the simplified field equations are as follows:

awd *--divJ+- at - at

aw aw ap aw a in which = - - + - -, Wd= water liberated by dehydration aP at a~ at during heating (see Figure 1 141); p , C = mass density and isobaric heat capacity of concrete; C = absorption heat of water; C,,= heat capacity of water; Q = volumetric heat supply; and the tern C J-grad T = rate of heat convection by moving water during rapid heating.

Moisture and heat transmission at the surface are given as

follow..

For moisture flux:

Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

4:38

03

June

201

4

GONG, SONG, AND MUJUMDAR

E

e = Cemant Content:

FIGURE 1 Dehydration Curve

and for heat flux:

n.q = B (T - Ten) + Can.J , (9 )

where By and BT are transfer coefficients, and n is the outward

normal.

FINITE !&EKENT MODEL

The semi-discrete finite element equations of Eq. 6 and Eq. 7

are

in which Pnand Tn stand for nodal pore pressure and temperature.

The matrices Cm(m=1,2,3,4), Km(m=1,2) are obtained by assembling

the elementary matrices. If the problem dealt with is axi-

symmetric, the elements in the elementary matrices can be

calculated as follows:

Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

4:38

03

June

201

4

NUMERICAL S I M U L A T I O N O F DRYING O F REFRACTORY CONCRETE

where A = A = -C aw ah' aw awd aP, , A ~ = z, A ~ = PC - C. 5, A d = - at

a aW A"= -, CUPl, ; and 3 can be calculated from Figure 2 which is

obtained from W = W(P,T) [3] . For Cartesian coordinates, r in the

equations above is equal to 1.

Because of the strong non-linearity and the existence of two

degrees of freedom in this problem, the calculation of the element

diffusion matrices occupies a large portion of the total work.

Generally, the calculation of the element diffusion matrices is

performed by numerical quadrature. For isoparametric bilinear

quadrilateral elements adopted in this paper, four-point

quadrature is often required. This is very timesonsuming, so, a

substantial amount of time is needed to complete a run. In order

to speed up the computation, a newly-developed one-point

quadrature procedure is introduced.

The major drawback of one-pint quadrature is known as

hourglass. Recent studies have indicated the difficulty can be

overcome by adding a stabilization matrix [5].

Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

4:38

03

June

201

4

GONG, SONG, AND MUJUMDAR

0.4

0.3

e - 2 I

0.2

0.1

0 1 .O i

h = P/PslT)

FIGUB 2 b a t i m of State I = I ( P , T I

e(l) . where [K,] is the element matrix by one-pint quadrature; the

stabilization matrix is given by

T [ K , ] : ~ , ~ = ;,A (71 (7) (m = 1, 2) (12)

T T where t y ) = [(h) - ( ( h ) (x))(bll - ((h) (YI )(b,) I

7 1 (bl) = - [yz4, YS1t Yq21 Yl31 2 A

T 1 (bZl = 2 Ax4,' Xlj* XZ4* x3,1

X = x - X r~ I J "1,' '1- YJ

1 A = 5 (x3,yZ4+ X,~Y.,~)

T [h} = 11, -1, 1, -11

Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

4:38

03

June

201

4

NUMERICAL SIMULATION OF DRYING OF REFRACTORY CONCRETE

for m = l in the equations above km =

for m = 2 For non-linear heat conduction problems, almost two-thirds of

the total solution time is saved when one-point quadrature is used

in [61. Comparison in terms of the muracy and solution time is

made between one-point and four-point quadrature.

TIME INTEGRATION

The time integration of Eq.10 is obtained by a predictor

-corrector scheme. The final discrete forms of Eq. 6 and Eq. 7 are

[K*(u;+~,)] {A'}={~(u:+~~)) ( 1 3 )

in which

[K*(U:+~~)I = [c(u:+At)l + aht[K(u:+At)l {r(ut+Ai)} = {F(U:+~~)) - [K(u:+AL)I{u~+AL}

- [c(u;+Ac)l{'t+At) where i designates ith iteration. For i = 0

With these two starting vectors, {AC) can be computed by

solving Eq. 13. The updating procedures are:

where a is a parameter, a = 0.25 N 0.75.

Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

4:38

03

June

201

4

GONG, SONG, AND MlJUMDAR

If convergence is achieved,

and

NUMERICAL SIMULATION

Based on the procedure described above, a computer code to

solve two-dimensional problems was written. This code can be used

for the analysis of the drying process of refractory concrete and

for the failure analysis of concrete reactor vessels in accidents

of structure due to rapid heating.

Using the computer code, calculations have been made for a

typical refractory concrete cross-section as shown in Figure 3.

The sylmnetry of the problem allows analysis of half of a quarter of the section. The finite element mesh is shown in Figure 4. Two

cases of the problem are simulated. One is direct heating, and the

other is for simplified volumetric heating.

Case 1: Direct heating

The boundary conditions are:

on side AB

PZv for t r; tm

IT(~)=T m a x fort > tm

where B"= 1.0~10'~s/m, P~~/P~(ZOOC) = 45% 1 3 1 ,

V = 125O~/h Along side CD,

where BY= 1 .~xl~"s/m, Pen/P, (BOOC) = 45%,

BT= 1.0 J/~'.s.K, Ten= 20°c [3 ] .

Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

4:38

03

June

201

4

NUMERICAL SIMULATION OF DRYINGeOF REFRACTORY CONCRETE

Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

4:38

03

June

201

4

GONG, S O N G , AND MlJUMDAR

FIGURE 5 Relative Change in Perneabil i ty '

The sides AD and BC are zero-flux boundaries due to sylmnetry.

Initial conditions are assumed to be uniform temperature of

T = 2 0 ' ~ and po/pS ( 2 0 " ~ ) = 90%. 0

The permeability in practice is strongly temperature-

dependent. The permeability-temperature relation is given in

Figure 5 [$I in which aois specified at 2 0 " ~ as ao= 8.5 x

m/s. Absorption heat Ca is also temperature-dependent. It can be

calculated from the following equation [21:

3 . 5 ~ 1 0 ~ ( 3 7 4 . 1 5 - T)"~ for T s 3 7 4 . 1 5 ' ~ c = { a 0

The following values are assumed constant [ 2 ] :

p = 2200 kg/m3 C = 1100 J/kg.K

k = 1.67 J/m.s.K Cy= 4100 J/k.K

Case 2: Simplified microwave volumetric heating

Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

4:38

03

June

201

4

NUMERICAL SIMULATION OF DRYING OF REFRACTORY CONCRETE 489

For comparison, a simplified microwave heating model is also

computed. The microwave energy input is assumed to be proportional

to the local moisture content according to:

Q = S u W + S C p

where SU and Sc are coefficients dependent on the material,

microwave intensity etc.. Here the following values are specified:

Su= 1040 J/kg.s, Sc= 13 J/kg.s.

Note that this case does not represent true microwave drying. With

proper specification of Q the code can, however, simulate the

realistic case.

RESULTS AND DISCUSSION

For the two cases studied, relatively rapid convergent

results were obtained by using a time step of At = 7--10 seconds. Figure 6.1 shows the temperature profiles along side BC at

different times for case 1. In this(and the following) figure

x/BC=O is point B and x/BC=I is point C. Initial temperature of

the concrete is uniform at 20°c. As the drying process begins the

interior surface, e.g. side AB, is heated at a rate of V=125 Oc/h

until 5 0 0 ~ ~ is reached. With time the temperature of the concrete

slab rises monotonously. As the thennal conductivity of the

concrete is small, temperature of the area far from the heated

surface, e.g. the area near point B, increases more slowly. It can

be seen that between tz2.0 hrs and tz2.5 hrs the temperature at

point B has risen significantly. The reason is due that before

tz2.0 hrs temperature of every point from x/BC=O to x/BC=l goes

through the phase change temperature of the free water, about

100-120~~, therefore, temperature changes slowly; after tz2.0 hrs

most of the free water is dried out, the heat transported in is

only used to raise the temperature of the concrete itself, not for

evaporation of free water. Therefore, the temperature rises

rapidly.

Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

4:38

03

June

201

4

GONG, SONG, AND MUJUMDAR

- - - I Side BC

Distance From The Heated Surfacs(x/EC)

Fig. 6.1

Distonce From The Heated Surface(x/BC)

fig. 6.2

200 -

- 150.- & 5 100.;

i E

50.-

FIGURE 6 Temperature Profiles Along Side BC A t D i f f e r e n t Times

Side BC Microwave Heating

3.Oh

'2.5h ' 2 m \I.% 1.0h

Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

4:38

03

June

201

4

NUMERICAL SIMULATION OF DRYING OF REFRACTORY CONCRETE 491

Figure 6.2 gives the temperature profiles along BC for case

2. Initially the temperature of the concrete is 20°c. Energy

needed in the drying process is provided by a microwave s o m e

described above. During the first hour temperature reaches 100'~

quickly, while between tz1.5 hrs and tz2.5 hrs it changes slowly.

After t-2.5 hrs rise rate of temperature is accelerated. This is

due to the fact that after tz2.5 hrs phase change is completed and

almost all the free water is removed, heat supplied is only used

for the sensible heat of the concrete. In this figure the

temperature difference at different points along BC is very small.

It is determined by the heating characteristics of microwave

radiation.

Figure 8.1 displays the pore pressure profiles along BC for 2

case 1. The initial pore pressure is unifonnly 2100 N/m . With increasing temperature during the drying process , the rate of evaporation of the free water increases constantly, leading to the

elevation of the pore pressure. As the steam pressure in the

environment is assmed constant and quite low, pore pressure near

the two boundaries remains low during the whole process. The pore

pressure profile is in the shape of a parabola, and the peak of

the parabola gradually rises and migrates from x/BC:O to

x/BC=l(e.g. the interior surface to the exterior). At around t:

2.0 hra it is at x/BC-0.7 and reaches its highest value, 2.7 atm.

Afterwards the pressures along the whole side fall off rapidly. In

fact, after t12.0 hrs there is only a small amount of free water

left inside the concrete which can be evaporated to maintain the

pore pressure.

F i m 8.2 gives the pressure profiles along BC for case 2. 2 The initial pore pressure in the concrete is uniformly 2100 N/m .

In the drying process, with increase of temperature, steam

pressure of the free water rises, resulting in the elevation of

the pore pressure. During the first 1.5 hours the pore pressure

increases more rapidly near the center area, but keeps quite low

at x/BC=O and x/BC:l. At t31.5 hrs the pore pressure at x/BC=O. 5

is at its maximum value, 1.8 aim. After t-1.5 hrs the pressure

Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

4:38

03

June

201

4

: GONG, SONG, AND MUJUMDAR

Distance From The Heated Surfoce(x/AO)

Fig. 7.1

0 0.0 0.2 0.4 0.6 0 8 1 .O

Distance From The Heoted Surface(x/AD)

Fig. 7.2

150

E e! 1 100.- e x E 2 50.-

FIGURE 7 Temperature Profiles Along Side AD A t Different Times

Side AD Microwave Heating

-- 3.0h 2.5h

\2.0h \1.5h - 1 .Oh

Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

4:38

03

June

201

4

NUMERICAL SIMULATION O F DRYING OF REFRACTORY CONCRETE

Distance From The Heated Surfoce(x/BC)

Fig. 8.1

4.0

4 . 0 Side BC I

h N E 3.0.-

Microwave Heating I

Side BC

~ = 1 2 5 ' ~

2.0h

Distance From The Heated Surface(x/BC)

Fig. 8.2

0

0.0 0 . 2 0.4 0.6 0.8

FIGURE 8 Pressure Profiles Along Side BC At Different Times

Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

4:38

03

June

201

4

GONG, SONG, AND MUJUIIDAR

Distance From The Heated Surface(x/AD)

Fig. 9.1

4.0 Side AD -

N Microwave Heating E 3.0.- \ <

2.0.- a 1.5h

0.0 0.2 0.4 0.6 0.8

Distance From The Heated Surface(x/AD)

Fig. 9.2

FIGURE 9 Pressure Profiles Along Side AD At Different Times

Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

4:38

03

June

201

4

NUMERICAL SIMULATION OF DRYING OF REFRACTORY CONCRETE

Oistonce From The Heoted Surface(x/BC)

Fig. 10.1

Microwave Healing A

Side BC RE 80.-

0.0 0.2 0.4 0.6 0 . 8

Distance From The Heated Surface(x/BC)

Fig. 1 0 . 2

FIGURE 10 Moisture Profiles Along Side BC At Different Times

Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

4:38

03

June

201

4

496 GONG, SONG, AND MUJUMDAR

begins to decrease. At t-3.0 hrs little free water is left in the

concrete and the pore pressure decreases to very low values.

Figure 10.1 displays the moisture profiles along side BC for

case 1. The initial free water content is about 100kg/rn3. At the

beginning moisture in the area near the heated surface.diffuses

quickly towards the environment under the driving force of the

pore pressure and the free water content near x/BC=O decreases

rapidly to very low values. With the drying process proceeding

pressure gradients are set up gradually in the concrete; free

water migrates from the interior to the boundaries and then

diffuses to the environment. The integrated free water contained

in the concrete decreases continuously. The moisture profiles are

in the form of a parabola. The zeniths of the parabolas are at

x/BC-0.78. At tS.5 hrs the free water content decreases to nearly

zero, e.g. , the material is dry. Figure 10.2 shows the moisture profiles for side BC for case

3 2. The initial free water content is again about 100 kg/m In a

very short time after drying starts the free water content at

x/BC=O and x/BC=l decreases to very low levels. At tz1.O hr the

moisture content has a bow-form distribution. Later the moisture

profile evolves into a parabola. After t-3.0 hrs the moisture

content is nearly zero, e.g., the material is dry.

Figure 7, 9 and 11 are for side AD. The results are analogous

to those for side BC.

Figure 12 shows the integrated water release curves for the

two cases. The initial free water content is 0.375kg/m3. The rate

of water release begins to decrease around t-2.0 hrs for V-125 OC

but at t-1.5 hrs for microwave heating. It tends to zero at

t-3.0 hrs for microwave heating and at t-3.5hrs for V-125 OC. The

amount of water released in case 1 is obviously more than that of

the initial free water contained, but that in case 2 is not. The

reason is that in case 1 when temperature is above 200 OC, there

is free water that is released into the pores by dehydration, but

in case 2 this is not so because the maximum temperature reached

is only 130~~.

Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

4:38

03

June

201

4

NUMERICAL SIMULATION O F DRYING OF REFRACTORY CONCRETE

Distance From The Heated Surface(x/AD)

Fig 11.1

OD- Microwave Heating Side AD

m- E 80.- 1.0 h

0 . 0 0.2 0 . 4 0.6 0.8

Distance From The Heated Surface(x/AD)

Fig. 1 1.2

FIGURE 11 Moisture Profiles Along Side AD At Different Times

Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

4:38

03

June

201

4

GONG, SONG, AND HUJUHDAR

Time (hour)

FIGURE 12 Integrated Water Release Curve

Fl GURE 13 Maxi mum Pressure Curve

A comparison of the maximum pressure-time curves for the two

cases is given in Figure 13. It can be seen that in completing the

same drying process using the same time, the maximum pre pressure

is less when employing volumetric heating than hen using

direct boundary heating. The direct boundary heating curve shows

some numerical oscillations. This is caused by high temperature

gradients.

It was also found in other numerical experiments that the

influence of the boundary conditions of moisture transfer on the

Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

4:38

03

June

201

4

NUMERICAL SIMULATION OF DRYING OF REFRACTORY CONCRETE 499

variation of pore pressure is not obvious. This is due to the fact

that the resistance of moisture transfer is mainly in the interior

since the permeability of the refractory concrete is very small.

For comparison the two cases are computed not only with

one-point quadrature, but also with four-point quadrature. The

superiority of one-point quadrature in solution time is clear from

the following table:

The difference of the solution between one-point and four

-pint quadrature methods is detected only in the fourth digit.

CASE

1

2

This work is being extended to predict the drying-induced

stress field; in order to further speed up the computation a

highly efficient three-time-level scheme will be intmduced into

the finite element model. Effects of intermittent thennal energy

input as well as spatially varying permeability will be evaluated

numerically.

CONCLUSIONS

Four-pint

CPU time

8159 sec

4475 sec

The numerical experiments presented here lead to the

following conclusions. (1) It seems that the efficiency of

volumetric heating is higher than that of direct heating for thin

materials. (2) The permeability of the refractory concrete is the

main factor that controls the variation of pore pressure, not the

One-point

CPU time

4486 sec

2631 sec

Percent of

Time Saved

45.02 %

44.90 %

Type of

Computer

HITACHI

M 240-D

Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

4:38

03

June

201

4

500 GONG, SONG, AND MUJUMDAR

boundary condition of moisture transfer. (3) he-point quadrature

can save about 45% CRI time compared with four-point quadrature

with no loss of accuracy in finite element modeling..

REFERENCES

[I] Bazant, Z.P.and Najjar, L.J., 1972, Nonlinear water diffusion in nonsaturated concrete, Materials and Structures: Research and Testing, 5(25), 3-20.

[21 Bazant, Z.P., ASCE, M. and Thonguthai, W., 1978, Pore pressure and drying of concrete at high temperature, Proceedings of the American Society of Civil Ehgineers, 104(EM5), 1059-1079.

[31 Bazant,Z.P., and Thonguthai, W., 1979, Pore pressure in heated concrete walls: theoretical prediction, Magazine of'Concrete Research, 31(107), 67-76.

[41 Dhatt, G., Jacquemier, M. and Kadje, C., 1986, Modelling of drying refractory concrete, Drying'86 (Edited by A. S. Mu.idr). Vol. 1. 94-104.

[5] L ~ U , w.K;, and &lytschko, T., 1984, Efficient linear and nonlinear heat conduction with a quadrilateral element, Int. J. N u n . Meth. Eng., 20, 931-948.

[61 Gong, Z.X. and Zhang, Y.F., 1987, Application of one-point quadrature to non-linear heat conduction (in Chinese), J. Tianjin Institute of Light Industry, 2, 71-76.

Dow

nloa

ded

by [

Uni

vers

ity o

f A

rizo

na]

at 0

4:38

03

June

201

4