Simulation of Steady-State Nonlinear Heat Transfer Problems Through the Minimization of Quadratic Functionals

Simulation of Steady-State Nonlinear Heat Transfer Problems Through the Minimization of Quadratic Functionals

R. M. S. da Gama

Lab. Nacional de Computação Científica

Av. Getulio Vargas, 333 Quitandinha

25651-070 Petrópolis, RJ. Brazil

[e-mail]

In this work it is presented a systematic procedure for constructing the solution of a large class of nonlinear conduction heat transfer problems through the minimization of quadratic functionals like the ones usually employed for linear descriptions. The proposed procedure gives rise to an efficient and easy way for carrying out numerical simulations of nonlinear heat transfer problems by means of finite elements. To illustrate the procedure a particular problem is simulated by means of a finite element approximation.

Keywords: Nonlinear heat transfer; variational formulation; sequence of functions

Introduction

Let us consider the bounded open set W with boundary ¶W and the known functions k, S, g₁ and g₂. The functional I[w] defined as

is quadratic, convex and coercive (Berger, 1977), provided k and g₁ are positive-valued, for any admissible w (wÎ H¹(W )) (Gama, 1992). In other words this functional admits one, and only one, minimum which is reached when w º u, being u the solution of the following problem (Euler-Lagrange equation and natural boundary condition)

Problem (2) represents the steady-state conduction heat transfer process in a body (represented by the open set W with boundary ¶W and unit outward normal n) with thermal conductivity k and internal heat supply S which exchanges energy with the environment following Newton's law of cooling with a (positive) coefficient g₁ and a (nonnegative) temperature of reference g₂/g₁ (Slattery, 1972). The unknown of (2), in this case, is the temperature u.

Unfortunately, a problem like (2) can not describe accurately many important heat transfer processes, in particular those ones in which the boundary conditions are not linear (Gama, 1990)¹.

The main objective of this work is to provide a way for simulating a large class of nonlinear heat transfer problems by means of the minimization of functionals like the one defined in (1). In other words, it will be presented a procedure for evaluating g ₁ and g ₂ in such a way that the minimization of the functional defined in (1) be equivalent to the solution of the following nonlinear problem

in which S is nonnegative-valued, is such that

and

Nomenclature

F = function depending on X and u

g ⁱ = function obtained from I i

h = heat transfer coefficient

I = quadratic functional

I ⁱ = quadratic functional whose minimum is reached for fi

I* = a particular case of I

J = an auxiliary quadratic functional

k = thermal conductivity

L, L_x and L_y = interval length

n = unit outward normal

r = dimensionless spatial variable

s = dimensionless spatial variable

S, S_A, S_B and S_C = heat source per unit volume

u = unknown (usually the temperature)

v = unknown of an auxiliary problem

w = admissible field

w_p = approximation for w at the node p

X º (x,y,z) = spatial position

a = positive constant

Dx and Dy = length and width of a given rectangular finite element

fⁱ = the field which minimizes Iⁱ

gⁱ = a field which depends on fⁱ

g¹ = positive coefficient

g² = known function

s = Stefan-Boltzmann constant

W = open set representing the body

¶W = boundary of W

Nonnegativeness of the Solution of (3)

Since S ³ 0 the nonempty subset ¶W* Í ¶W defined by

is such that

in which "inf" denotes the infimum. Therefore, taking into account (3), we have

and, thus, since (4), (5) and (6) hold, we are able to conclude that

An Auxiliary Problem

Let us consider the linear problem below (a particular case of (2) and (3))

whose unknown is v (here F º v).

The solution of (9) corresponds to the minimum of

and, since F º v, is nonnegative everywhere. It is to be noticed that I*[w] corresponds to I[w] when g₁ =0 and g₂ =0 and, consequently, the same procedure employed for minimizing I[w] may be employed for minimizing I*[w].

An Upper Bound for u on ¶W

The procedure to be proposed in this work demands an a priori estimate for the supremum of u on ¶W . This may be done with the aid of problem (9).

Combining (9) with (3) we have

Defining the subset ¶W**. as (see Appendix I)

we have that

and, thus

Since F is an increasing function of u, any constant satisfying

will satisfy the following inequality

So, taking into account that

we may conclude the following

which represents an upper bound estimate for u on ¶W , based on the solution of (9).

The Quadratic Functionals Ii [w] and the Solution of (3)

The solution of (3), denoted here by u, can be represented by

in which the fields fⁱ are obtained from the minimization of the quadratic functionals I ⁱ[w] given by (for i³ 1)

in which a is a constant such that

where u⁺ is any constant satisfying

while gⁱis given by

In other words, the solution of (3) is the field u that minimizes the functional I ^¥ [w] given by

Proving that fi ³ fi-1 in W

Since g⁰º 0 on ¶W , the natural boundary condition associated with I¹[w] is given by

and, therefore, taking into account (5) and considering the argument employed for proving (8), the nonnegativeness of f¹ is ensured.

Since f¹³ f⁰º 0 in W and a satisfies (19) we have (see Appendix II)

and, thus, the result of Appendix I enables us to write

Repeating this procedure we conclude the following

Proving that u ³ fi in W

Problem (3) admits one, and only one, solution u [5]. So, this solution must be the one which minimizes the functional J[w] below

in which g is given by

and, in (28), is regarded as a known function.

It is easy to see that

and, hence, since f⁰ º 0 and u³0, we conclude that (see Appendix II)

So (see Appendix I), taking into account that fⁱ minimizes I ⁱ[w], we conclude that

and, consequently, from (30), we have that g ³ g¹. Repeating this procedure, we are lead to conclude that

In other words, the solution of (3) is an upper bound for the fields fⁱ. In addition, combining (33) with (27), we may write

Proving that (18) Holds

From the definition of gⁱ and from (34) we have that

and, consequently, the sequence [g¹, g², g³,... ] is increasing and bounded. This ensures the convergence for all XÎ ¶W . So, we may define the field f^¥ as the one which minimizes the functional I^¥ [w] given by

In addition, since f^¥ minimizes I ^¥ [w], the following holds

and f^¥ coincides with u. In other words, (18) holds.

An Application - Numerical Simulation of a Heat Transfer Problem

Now, the procedure proposed in this work will be employed for simulating a nonlinear heat transfer process by means of a finite element approximation (for instance, a conduction/radiation heat transfer problem, a conduction/convection heat transfer problem with temperature-dependent coefficients, etc...).

Let us suppose that the set W is given by

that k is constant and that F (F = (x,y,u)) satisfies (4) and (5) being given as

where f₁, f₂, f₃and f₄ are continuous functions of x or y.

In this case, problem (3) yields

It will be considered the following finite element approximation for the admissible fields w ( w = (x,y))

in which w_p represents the approximation for w at the node p, p and j are related by ("int" denotes the integer part of)

and

In the above equations M represents the number of nodes in the x direction while N is the number of nodes in the y direction.

It is to be noticed that represents the position of center of the element j. The position of the node p is given by (x_p,y_p) where

Inserting the approximation (40) into functional I ⁱ[w], defined by (19), we obtain the following function

Assuming that S is given by (piecewise continuous in W )

and employing a piecewise linear approximation for g^i-1 between each two adjacent nodes on the boundary, we have

in which is given by

where f₁ and f₂ are used for evaluating while f₃ and f₄ are used for evaluating , as indicated above.

The minimum of gⁱ is reached when

in which ¶gⁱ/¶w_q is given by

for the interior nodes, where j=q-int (q/M).

for the nodes on the boundary, where j=q-int (q/M).in which b₁, b₂ , b₃ and b₄ are such that

So, (54) is a system of NM linear equations. The solution of this system (for each i³1) provides the approximation for fⁱ at each node q. In other words, the minimum of g ⁱ is reached for

Evaluating a

Problem (9) may be simulated with the same finite element approximation used for (39). Aiming to this it is sufficient to consider, in (55) and (56), that a = 1 and (for all q). In this case, the solution of system (54) will be the approximation for v (solution of (9)) and the constant u⁺ can be easily estimated as well as a .

Some Particular Results

Aiming to illustrate the convergence process of the sequence Zf¹, f², f³, ...[ to the solution u, some results are presented in Figures 2, 3, 4 and 5. Each of these figures present fⁱ versus (x,y) (for 16 different values of i), each one displayed at the right upper corner) employing M=N=11 (that means 100 elements and 121 nodes).

The following cases are simulated

Case 1 represents the conduction heat transfer process in a body with a variable internal heat supply which exchanges energy by thermal radiation following the Stefan-Boltzmann law (Sparrow and Cess, 1978). In (62) and (63) s is the Stefan-Boltzmann constant (in the S.I. s =5.668E-8 Wm^-2K^-4).

Case 2 differs from case 1 because there exists a convective heat transfer from the side y=L to the environment (assumed at a temperature u_¥ ). In (63) h represents the convection heat transfer coefficient (in the S.I. the units of h are Wm^-2K^-1) assumed constant.

In Figures 4 and 5 the convection heat transfer boundary condition at y=L gives rise to low temperature levels on this region. This fact is not observed in Figures 2 and 3 since we have only radiation boundary conditions.

The results presented in each figure required (approximately) 200 CPU seconds on a Pentium II 333MHz that means, for each fⁱ, 0.33 seconds.

Final Remarks

A Gauss-Seidel iterative scheme (Todd, 1962) was used for solving system (54) in all the considered simulations. It is remarkable that the convergence of this scheme increases as "i" increaseas, since is naturally employed as the initial shoot for reaching .

The procedure proposed in this work may be employed for a large class of nonlinear heat transfer problems avoiding the use of sophisticated and/or special tools. In any case, the tools used for simulation will be the same ones used for linear problems.

Nevertheless, the original problem (problem (3)) may be approximated by some finite difference scheme or by the classical Galerkin procedure. In both cases we shall depend on tools for nonlinear problems

Appendix I - An Useful Result

Let us consider the following functionals

in which l₁ and l₂ may depend only on the position X.

Since k and S do not depend on w we may write

in which U₁ and U₂ minimize, respectively, I₁[w] and I₂[w].

Defining now the subsets ¶W _- and ¶W ₊ as

we have

and, since k>0, the boundary condition of (AI-3) enables us to write

Therefore, the following statement holds

and, so,

Appendix II - The Increasing Function a w-F

Since the constant a is the defined in such a way that

the function defined by

has the following feature

Manuscript received: February 2000, Technical Editor: Angela Ourivio Nieckeie.

• Berger, M.S., 1977, "Nonlinearity and Functional Analysis", Academic Press, London, UK, 522 p.
• Gama, R.M.S., 1990, "A New Mathematical Model for Energy Transfer Problems with Radiative Boundary Conditions", Appl. Math. Modelling, Vol. 14, pp. 96-103.
• Gama, R.M.S., 1992, "Simulation of the Steady-State Energy Transfer in Rigid Bodies, with Convective/Radiative Boundary Conditions, Employing a Minimum Principle", J. Comp. Phys., Vol. 99, pp. 310-320.
• Maz'ja, V.G., 1985, "Sobolev Spaces", Springer-Verlag, Berlin, Germany, 471 p.
• Slattery, J.C., 1972, "Momentum, Energy and Mass Transfer in Continua", McGraw-Hill, New York, 679 p.
• Sparrow, E.M. and Cess, R.D., 1978, "Radiation Heat Transfer", McGraw-Hill, Washington, 365 p.
• Todd, J., 1962, "Survey of Numerical Analysis", McGraw-Hill, New York, 492 p.

Publication Dates

• Publication in this collection
  18 Sept 2002
• Date of issue
  Mar 2002

History

• Received
  Feb 2000