Applications of gauss's principle of least constraint to the nonlinear heat-transfer problem

Abstract An approximate direct method for solving linear and nonlinear heat conduction problems, based on the Gauss's principle of least constraint is presented. In every particular case, the problem is reduced to the algebraic minimization of a quadratic form with respect to some complex of physical parameters. By the help of several concrete examples the efficiency and accuracy of this new method is demonstrated.


APPLICATIONS OF GAUSS'S PRINCIPLE OF LEAST CONSTRAINT TO THE NONLINEAR
HEAT-TRANSFER PROBLEM B. VUJANOVIC and B. BACLIC: Faculty of Technical Sciences, University of Novi Sad, 21000 Novi Sad, Yugoslavia Abstract-An approximate direct method for solving linear and nonlinear heat conduction problems, based on the Gauss's principle of least constraint is presented. In every particular case, the problem is reduced to the algebraic minimization of a quadratic form with respect to some complex of physical parameters. In this paper we shall study the possibility of applications of Gauss's principle of least constraint to the nonlinear heat transfer. This principle is a true minimum principle in contrast with the two aforementioned principles of D'Alambert and Hamilton, which are generally not the minimum principles. • Contrary to *The variational principle of Hamilton may be occasionally a true minimum principle. the general impression that the Gauss's principle is of purely theoretical interest in classical (point) mechanics, the purpose of this, and the previous article [2] is, to point out that this principle can lead to considerable practical advantages in nonlinear heat transfer.

GAUSS'S PRINCIPLE
For the sake of clarity we will briefly describe the Gauss principle of ordinary mechanics in a form suitable for applications in heat transfer.
Consider a free dynamical system of n particles, subject to impressed forces F;(i = I, 2, . . . , n). If m; are masses, r; position vectors and a; = d 2 r;/dt 2 accelerations, the differential equations of motion are (I) Let us introduce the quantity* Consider the nonlinear equation of heat conduction t' T div(k · grad T)-pc ---:. ; -= 0.
where p is the density, k(T) and c(T) are the thermophysical coefficients which are supposed to be the functions of temperature T.
Consider the heat-transfer analog of equation (2) : where Vis the volume which is engaged in the process of heat transfer and X = div(k ·grad T) y = pc oT (10) ar are spatial and temporal parts respectively. As in the previous case the following two variational rules for Z =!I (F;-m;a;) 2 (2) minimizing (9) are possible i=l bX=!-0, bY=O (11) and suppose that the configuration (r;), velocity (v;) and and forces (F;) of the system are prescribed at timet, i.e. and the remaining "complex"-m;a; is then the only one to be varied, i.e.

(4)
The Gauss's principle of least constraint states that under the conditions (3) and (4) the quantity Z can assume its absolute minimum, which is zero. The proof is simple. If ( -m;a;) represents the actual inertial force, and -m;a;+b(m;a;) represents any other possible inertial force, we have: As a second possibilityt of achieving the absolute minimum of "constraint" Z defined by (2), we can suppose that the configuration, velocity and acceleration of the system are prescribed at time t and the impressed force F; is then only one to be varied, i.e.
.5r, = 0, bv; = 0, b(m;a;) = 0, bF, =!-0. (7) If the system in question is subject to holonomic or nonholonomic constraints the constraint Z is in minimum again but introduction of Lagrange's undetermined multipliers is necessary. We will show now how one can treat the equations of nonlinear heat transfer in the same way as the dynamical systems discussed above. *In the original version of Gauss's principle the expression in the parenthesis of (2) is multiplied by the factor 1/m;. tFor more details see [2].
This variational principle may be formulated by means of the generalized coordinates instead of the components of the field itself. In numerous problems it is possible to guess that a solution belongs to a family of functions with one or more unknown parameters, which is more or less characteristic feature for all approximate methods. In each particular case we must be able to identify the characteristic complex of parameters which represent X or Y and minimization of Z has to be performed with respect to one of these complexes. The method proposed is very simple because the whole technique for obtaining approximate solution is of the algebraic nature-minimization of a quadratic form with respect to a complex. If some of generalized coordinates are coupled by one or more algebraic ·relations the minimization of Z should be performed using Lagrange's undetermined multipliers. The efficiency of the method can be illustrated by obtaining approximate solution of a simple linear heat-transfer problem.
Consider a one-dimensional thermally insulated semi-infinite body with constant thermo-physical coefficients p = p 0 , c = c0 and k = k0 . The body is initially at the temperature T = 0. At t = 0 the face of the body, located at x = 0, is suddenly brought to a constant temperature T = T0 • Mathematically the problem is to solve following boundary-value problem where rx = k0 /p 0 c0 , together with Following the concept ofthe penetration depth assume the solution in the form where q(t) is penetration depth.
Let us solve the problem minimizing the constraint: with respect to the temporal part aTjat. From (15) we have q is the "temporal complex" with respect to which we will minimize the expression (16). Substituting (17) into (16) integrating and retaining only the terms with W we have where a constant multiplicative factor has been omitted.

az(W) =O
aw and using ( 18) we find the following differential equation the solution of which, with respect to the initial condition q(O) = 0 is Finally let us use the second possibility, minimizing ( 16) with respect to the spatial complex a2T X = rt. ,; yz = 2KT0 i.e.
These two results (20) and (24)  It should be noted that the Gauss principle of least constraint has been applied on the linear heatconduction problems by Samoilovich in [3]. However, the author deals with a transformed form of the basic heat conduction equation similar with the Biot's quasivariational method.
It is the purpose of this note to show that the Gauss's principle can be applied directly on the governing heat-conduction equations, and in addition, the nonlinear problems are generally treated in the same way as the linear ones.
As was shown above, the applications of direct methods, using Gauss's principle are simple and straightforward. In the next section several more complex examples will be presented .

EXAMPLES (A) Unsteady two-dimensional nonlinear heat conduction through the prism-like infinite bodies with a given cross section
As the first example we shall study the temperature distribution through the prism-like infinite bodies. The thermal conductivity is supposed to be a linear function of temperature hence the differential equation is of the form where rt. and a are given constants.
Initially, the body is at the uniform temperature, i.e.
and the surfaces of the body are maintained at the zero temperature where Is is the symbol for the external surfaces of the prism. Consequently the problem is to find the approximate solution of (25) in the presence of initial and boundary conditions (26) and (27).
The suitable form of the trial solution will be : Let us consider the constraint in the same sense of Gauss: ay ay where(x0 ,y0 ) and (x 11 yJ) are to be selected in accordance with the contour in question.
The minimization of Z will be performed with respect to the temporal "complex" ¢. Introducing (28) into (29), integrating, and omitting all terms not containing ¢,we find: The condition az --...,-=0 o<P yields the differential equation The initial condition </J(O) will be determined by minimizing the initial square residual of the form: with respect to the arbitrary constant of the general solution of differential equation (32).* As the particular examples we will consider two characteristic shapes of cross section.  [ 4]. will yield 2 56 hence the solution is of the form Let   It is interesting to note that for the linear case a= 0, the corresponding results obtained from (38) and (41) are identical with those obtained by Tsoi in [5], who used the approximate method based on Laplace transforms. Tsoi reports that for the case of a rectangle his results are in good agreement with the exact solution. Unfortunately for more complicated geometry as triangular for example, the comparison is not possible because the exact analytical solution is not available.
It is reasonable to suppose that the solutions (38) and (41) are of some validity for the moderate range of parameter a.
The corresponding results for both cases (a) and (b) are presented graphically on Figs. 1 and 2, where the influence of nonlinearity is presented also.

(B) A melting problem
Consider a semi-infinite solid initially at the melting temperature eP whose surface x = 0 is raised suddenly to the temperature 80 and held there for t ~ 0. We will assume the temperature distribution only in the liquid phase. Such a simplification was proposed by Goodman [6] and greatly enhances the use of penetration depth concept in trial solution as far as the latter becomes identical with the location of the melting line ~(t).
Introducing dimensionless temperature T = (8 -8p)/ where only penetration depth ~(Fo) and the constant A remain for evaluation in accordance with the condition at the interface (45) and the governing heat-conduction equation (13).

(50)
It can easily be seen that in this manner all aforementioned conditions are fulfilled by (47) and (49). But as trial function (47) fails to satisfy the governing differential equation (13), we may proceed, by forming the constraint (16). The temperature profile (47)  The improvement of the accuracy for the approximate solutions obtained here, when compared with those obtained in [6] using the heat-balance integral, and in equation (5.28) of [8] by the help of a rather incorrect treatment of the problem, with a trial solution of the same form, is self-evident. We also conclude that the optimization with respect to temporal change of temperature field yields, in this case, much better results than the same with respect to spatial change of temperature.

(C) Semi-i'!finite body with an arbitrary heat flux input
As the last example, consider the case of the transient heat-conduction problem in the semi-infinite solid with constant thermal properties whose initial temperature is zero. In order to involve both linear and nonlinear boundary conditions the assumption will be made that the heat flux at the surface x = 0 is an arbitrary function of surface temperature  Following references [6] and [9] let us assume the cubic temperature profile in the form: where q = q(t) is the penetration depth and both q and T. are unknown functions of time.
From the boundary condition (59) it follows that penetration depth and surface temperature are dependent which indicates that we are going to deal with a constrained optimization problem in the appliance of the above exposed direct method to find T. and q. In the sense of this fact, we will use the Lagrange's multiplier technique, when dealing with the minimization with respect to physical components of temporal change of temperature.
Let us start by substituting the trial function, equation (60) where the dot represents differentiation with respect to time. The next step is to recognize the physical components of temporal and spatial change of temperature in this expression so that the minimization procedures may be performed straightforwardly. Since aT t.
On the other hand spatial change of temperature a 2 T r.
CJ.-2 = 6 2 (q-x)er: ax q has only one complex: q In order to demonstrate two possible solutions we will solve the problem minimizing corresponding Gauss's constraint with respect to temporal and spatial change of the assumed temperature distribution respectively. (i) Minimization with respect to temporal change. Omitting last term in (63) as irrelevant since it does not contain components W1 and W2 defined by (65), and using (61) whence where). is an unknown Lagrange's multiplier. Minimization with respect to oT /ot, i.e.
The method of handling (82) applies also to the case in which the surface flux is a sum of powers of the surface and environment temperatures, and thus to heat convection from bounding surface into a fluid at temperature T 1 , for which the boundary condition is (linear case) f=h (4-T 1 ), and also to the blackbody radiation into a medium at temperature  (T 1 -T,).
The discussion of the results given above will take place after solving the same problems using the second possibility, viz. minimizing the constraint (63) with respect to the component of spatial change of temperature. ( (92) As previously in the second case we choose f = fo · t N to get the solution of (92) Let us now proceed with the discussion and the comparison of the results. One thing is obvious-the closed form solutions obtained by minimization of Gauss's constraint with respect to the component of spatial change of temperature distribution are more attractive since they are of a rather less complicated form. To evaluate the accuracy of the results we are going to compare all of them with the exact solutions for surface temperature.
In the case of constant heat flux the exact solution is known to be T.  Table 1 for the values of power N from 0 to 12. In the same the corresponding ratios for the approximate solutions obtained by variational [9] and integral [6] method are also listed. Comparison of the approximate solutions with the numerical ones [10], for the case when the heat flux is a power low function of the surface temperature, is evident from Fig. 5. The dots represent approximation equations (96), (97) and (98) for m = 4, 2 a nd 5/4 respectively. If one superimposes the results of Goodman [6], or Vujanovic and Strauss [9] (available only for m = 4), or those obtained by minimization with respect to o Tjot in this paper, it may be concluded that all of them are less accurate. When the solution of equation (99) is compared with the exact one, and the percentage error E = 100('7approx. -'lexactl/'lexact is calculated it turns out that E = 3%, for (T./ TJ)approx = 0.8, which reduces to 1% for (T./ T.)approx. = 0.5, while for (T./Tf)approx. ~ 0.3, E ~ 1%. Doing the same with equation (86): E = 6.4% at (T./ T,)approx = 0.8; E = 1% at (T./Tf)approx. = 0.4 and less than 0.5% for (T./TJ)approx. < ~. 3. Thus, in all cases the solutions here obtained differ only slightly from the exact ones. Further, the important conclusion is that the result obtained by minimization of Gauss's constraint with respect to the component of spatial change of temperature distribution are always more accurate, although of rather less complicated form. DISCUSSION In conclusion several remarks would be of interest: 1. The method we have presented is quite useful in the search for approximate solutions of linear and nonlinear heat-conduction problems. Usually, in the case of nonlinear analysis, it pertains to a high accuracy when compared with the other approximate methods.
2. A common feature of all approximate methods in heat transfer, including the method presented, is that the solution of a problem should be selected to some extent a priori. The choice of the form of a solution which contains some parameters that should be determined by the help of the Gauss's method depends upon all the information available from empirical, experimental, intuitive, etc. data.
3. Our aim has been to demonstrate two possibilities for obtaining approximate solutions from the same Gauss's constraint. The question as to which of the two possible approximate solutions should be taken is to be decided by considering the mean square residual [11] (p. 388) in the form 'o Y Naturally, the value of Z defined by (101) is equal to zero for the exact solution. Hence, the better solution is that one for which Z has the smallest value. It should be pointed out that the same criteria can be employed for evaluating the accuracy of any approximate solution obtained by some other approximate method.
4. The extension of the method presented in this paper on some more elaborated mathematical schemes as for example the method of finite elements is possible and will be reported elsewhere.
5. This method can be extended also in a straightforward way to handle the numerous problems arising in transport phenomena.