Farkas, H.

Noszticzius, Z.

English

Periodica Polytechnica (Budapest University of Technology and Economics)

A VARIATIONAL METHOD FOR SOLVING HEAT CONDUCTIONAL PROBLEMS* By H. FARKAS and Z. NOSZTICZIUS Institute of Physics, Technical University, Budapest Received September 15, 1977 Presented by Prof. Dr. A. KONYA 1. Outlines of the method Let A denote the set of functions y i) defined on the finite closed interval [a, b]; ii) piecewise continuously differentiable there; iii) satisfying the boundary conditions y(a) = Ya; y(b) = Yb' (1) Let the functional F{y} defined for all yEA take its unique relative minimum at Yo E A. The function Yo minimizing the functional F is sought for. The numerical calculation of Yo may be carried out in the following manner. Let us divide the interval [a, b] into n subintervals (n > 1). The points of subdivision are Xo = a, Xl' ••• , xn = b. A trial function Y(O) is taken arbitrar-ily: it will be the zeroth approximation. To the kth approximationY(k)(x) a function y* is constructed: where otherwise. __ Ll __ (x - Xi-I)' if x E [Xi-I' Xi], Xi - Xi-l (2) (3) Here Ll is an arbitrary (but fixed) value, and i E {I, 2, ... , n - I} That is the function Y(k)(X) is "varied" in the neighborhood of point Xi (Lagrange's method of the calculus of variations [1]). Next the values F{Y(k)} and F{y*} are compared. If F{y*} < F{Y(k)} then y* is a "better" approximation than y<k); therefore the procedure is con-tinued using y* instead of Y(k). For F{y*} > F{Y(k)} the values i or/and Ll are changed and another y* is sought for. * Paper commemorating the 60th birthday of Prof. Dr. A. Konya. 3* 240 H. FARKAS and Z. NOSZTICZIUS Conveniently, a linear function may be used for y<O). In this case, the approximations produced by the above procedure are polygonal functions, and elements of set A. Hence, this method is essentially a combination of the Lagrange method and the Euler method [1] of the calculus of variations. Making full use of the change of L1 and i, the "best" approximation of y 0 "with respect to the functional F is found among the polygonal functions in A which consist of n linear parts. 2. A heat conduction problem Let us consider a rod along the axis x. with the terminals x = 0 and x = L. Let the heat conducthity } .. depend on the space co-ordinate x and the temperature T : }. = }.(x, T). Assume }. to be piecevvise continuously differen-tiable "With respect to both its variables. The temperatures of the terminals are kept constant in time: (4) No,,!, our aim is the solution of the one-dimensional steady-state equation of heat conduction: - }.(x, T)- = 0 d [ dT) dx dx (5) , .. ith boundary conditions (4). In general, the solution cannot be given in closed fonll, so approximate calculations are justified. Our method ,\'ill be illustrated on the variational problem [2] L F{T} = J }.(x; T) (~:r dx - (~2 1 Tl)2 = minimum o 2j ---dx },(x, T) (6) taken from Gyarmati's Governing Principle of Dissipative Processes [3]. The functional F {T} in (6) is seen to satisfy the requirements in item 1. further-more, the desired stationary temperature distribution Toto be the unique solution of the e:x"tremum problem (6). Hence the method outlined in 1. can be applied for the numerical computation of To' 3. Numerical calculations The method in 1. is suitable for computer processing. It is easy to program and a significant advantage is to require a small storage capacity: in case of n points of subdivision only n + 2 or 2n + 4 storage units are needed SOLVING HEAT CONDUCTIONAL PROBLEMS 241 (depending upon the applied program), while n2 would be necessary in case of the more usual method (reducing the problem to solving a system of linear algebraic equations). Our method suits even a small programmable calculator in case of 8 to 10 points of subdivision. As an illustrative example let us consider a rod of length 10 units. Let the boundary conditions (4) now be T(O) = 0; T(10) = 1000. Let the heat conductivity depend on x and T as follows: x T A=l...L-...L--I 10 I 1000 (7) (8) The temperature values at integer values of x(T(l), T(2), ... , T(9)) are wanted. Starting from the linear temperature distribution as a trial function, compute the values (L1F)i = F{T + L1i} - F{T} belonging to variations L1 = 0.1 and L1 = -0.1, respectively, for i = 1,2, ... , 9. Only the negative L1F '\vill be used for each i. A new trial function is constructed as follows: increase (or decrease) the value of the trial function by 1 in the point where I L1FI is maximum and by a less amount, proportional to the local value of I L1FI, in the other points. Of course, the change follows the negative direction of L1F. If the function T** constructed in such a way meets F{T**} < F{T}, then the procedure is repeated using T** as a new trial function. Otherwise the original trial function is the first approximation. The second approximation is obtained after decreasing both the varia-tion L1 and the change value by one order of magnitude (to 0.01 and 0.1, respectively) and using the first approximation as trial function. Approxi. mations of higher order are obtained in the same manner. Table 1 shows the numerical data. The exact values have been obtained from the analytic solution of the problem. Approximations closer than the fourth one are seen to be possible only by increasing the number of the points of subdivision. Table 1 Trial 1st 2nd i 3rd I 4th 5th Exact function approx. approx. approx. I approx. approx. solution I i 16~:~31 T(O) 0.0 0.0 0.0 I 0.0 0.0 0.0 T(l) 100.0 131.92 165.25 168.73 168.73 168.94.-T(2) 200.0 247.10 301.07 306.19 I 306.66 306.67 306.94.-T(3) 300.0 353.51 418.42 ! 424.55 I 425.12 425.13 425.40 T(4) 400.0 i 455.56 522.93 529.28 529.88 529.88 530.13 T(5) 500.0 ' 552.62 617.57 623.74 624.32 624.33 624.55 T(6) 600.0 650.72 705.05 710.18 710.66 710.67 710.85 T(7) 700.0 738.68 785.49 789.99 790.42 790.43 790.56 T(8) 800.0 834.58 861.89 864.47 864.71 864.71 864.81 T(9) 900.0 914.93 932.48 934.19 934.35 934.36 934.40 T(10) 1000 1000 1000 1000 1000 1000 1000 242 H. FARKAS and Z. IVOSZTICZIUS Summary A combination of the Lagrange method and the Euler method of the calculus of vari-ations has been developed for computer processing. As an example, a nonlinear differential equation has been considered: the equation of steady-state heat conduction in an inhomogeneous rod of temperature-dependent conductivity. References 1. K6sA A.: Calculus of Variations (in Hungarian) Tankonyvkiado Budapest 1973. 2. FARKAS H., NOSZTICZIUS Z.: Annalen der Physik 27, 341 (1971). 3. GYARlIIATI 1.: Non-equilibrium Thermodynamics, Springer Verlag, Berlin 1970. Dr. Henrik FARKAS } Dr. Zoltan NOSZTICZIUS H-1521 Budapest 

A VARIATIONAL METHOD FOR SOLVING HEAT CONDUCTIONAL PROBLEMS

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A VARIATIONAL METHOD FOR SOLVING HEAT CONDUCTIONAL PROBLEMS

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Periodica Polytechnica (Budapest University of Technology and Economics)