Some remarks on transformation techniques for transient nonlinear problems

Technical Note Some remarks on transformation techniques for transient nonlinear problems Ryszard Bialecki and Andrzej J. Nowak Institute o f Thermal Technology, Silesian Technical University, 4 4 - - 1 0 1 Gliwice, Konarskiego 22, Poland

INTRODUCTION Many thermal problems of great practical importance may not be analysed as linear boundary value problems. The most common example occurs in heat conduction when the thermal properties of the material are temperature dependent. Thus, one has to consider a non-linear governing equation having the form 1

-

temperature thermal conductivity density specific heat time

The most efficient and elegant approach to solve equation (1) is based on the Kirchhofl's transformation. This is equivalent to converting of the equation (1) into a simpler form by introducing new variable U(u) defined as dU du

-

Using (6) one can write equation (4) in the form

(1)

V[k(u)Vu] = pc Ot

(2)

k(u)

or, in the integral form

(6)

Or = aOt

V2U -

c3u where: u k p c t

To deal with equation (4) Kadambi and Dorri 2 and later on Wrobel and Brebbia 3'4 applied another transformation defined as

OU 0z

Again, it should be stressed that transformation (6) does not linearise the problem. Variable z being proportional to diffusivity a varies with temperature U and is not an independent variable. The aim of this note is to point out some 'hidden traps' originated with transformations (2) and (6). The possible consequences are being presented studying simple 1-D example.

Example

Let us consider a transient heat conduction problem in which thermal diffusivity (after applying KirchholTs transformation) is a linear function of transformed temperature U. Thus, equation (4) takes form t~2U

U = Uo +

(3)

k(u) du

(7)

Ox 2

1

tgU

m U + n Ot

(8)

o

Notice that u o and U o are arbitrary reference values which do not affect obtained results. Upon transformation equation (1) can be written as

O2U

~U

~ x 2 -- ~T

lOU V2U a ~t

(4)

Although equation (4) looks as differential equation for linear transient conduction, it is still a non-linear one, as the thermal diffusivity a is temperature dependent k(u) a = a(u) - - -

(5)

p(u)c(u)

Paper accepted August 1989. Discussion closes February 1991.

© 1990Computational Mechanics Publications

where m, n parameters of linear function a = a(U). Applying transformation (6) one arrives at (9)

For the sake of simplicity let us prescribe the following initial and boundary conditions U(x, r = 0) = sin(nx)

U(x =

o, r ) = o

(lo)

U ( x = 1, "r) = 0

The very simple form of equation (9) suggests that its solution can be obtained without any difficulties. In fact, if equation (9) were linear, one would immediately write

Engineering Analysis with Boundary Elements, 1990, Vol. 7, No. 3

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Some remarks on transJormation techniques J or transient nonlinear problems: R. Bialecki and A. J. Nowak the function satisfying this equation and the conditions (10), i.e. U = sin(nx)exp( - ~2z)

(11)

Such solution implies the following relationship between variables t and z (cf. equation (6)) e x p ( - g2z) =

n e x p ( - ngZt) n + m sin(gx)[1 - e x p ( - n g 2 t ) ]

(12)

After simple algebra manipulation one arrives at the expression U =

n sin(rcx)exp( -- nrc2t) n + m sin(nx)[1 - e x p ( - nn2t)]

V2U * + 6(r - ri)tS(t --

1 ~U* ti)

CONCLUSIONS All transformation techniques should be utilized very carefully. Usually they only convert a differential equation into the simpler form but they do not linearize it. As a consequence the analytical solution of the transformed equation cannot be obtained. This may lead to some problems in the Boundary Elements Method which

--

a Ot

(14)

where 6 stands for Dirac's delta function. However, since equation (14) is not linear, the usual linear solution 1

(13)

which seems to be the solution of the original problem (8). However, calculating derivatives of U with respect to time t and space x, it is not difficult to check that function (13) does not satisfy equation (8). This simply means that function (11) is not a solution of the problem (9) or even more, equation (9) may not be solved analytically.

146

applies a fundamental solution, being the analytical solution of the differential equation adjoint to the original equation. For example, when formulation with time dependent fundamental solution is used for solving nonlinear problem, one requires to solve the equation adjoint to equation (4)

U*=

[4~a(t

1

ti)]s/2exp

I f

z ] 4a(i-- tl)

(15)

does not satisfy it (s is a dimensionality of the problem). This is why formulation with the time independent fundamental solution is recommended. REFERENCES

"1 Ozisik, M. N. Boundary Value Problems of Heat Conduction, International Textbook Company, Scranton, 1968 2 Kadambi,V. and Dorri, B. Solution of thermal problems with nonlinear material properties by the Boundary Integral Method, in BETECH 85 (C. A. Brebbia, Ed.), Springer-Verlag,Berlin, 1985 3 Wrob¢l, L. C. and Brebbia, C. A. The dual reciprocity boundary element formulation for non-linear diffusion problems, Computer Methods Appl. Mech. Eno., (1987) 65, 147--164 4 Wrobel, L. C. and Brebbia, C. A. Boundary elements for non-linear heat conduction, Communications in Applied Numerical Methods, 1988, 4, 617-622

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