Three-Phase Inverse Design Stefan Problem

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1 Three-Phase Inverse Design Stefan Problem Damian S lota Institute of Mathematics Silesian University of Technology Kaszubska 23, 44-1 Gliwice, Poland d.slota@polsl.pl Abstract. The method of the convective heat transfer coefficient identificationinathree-phaseinversedesignstefanproblemispresentedin this paper. The convective heat transfer coefficient will be sought in the form of a continuous function, non-linearly dependent on the parameters sought. A genetic algorithm was used to determine these parameters. The direct Stefan problem was solved via a generalized alternating phase truncation method. Stability of the whole algorithm was ensured by applying the Tikhonov regularization technique. Keywords: Inverse Stefan problem, solidification, genetic algorithm, Tikhonov regularization. 1 Introduction A majority of available studies refer to the one- or two-phase inverse Stefan problem [1,2,3,4,8,11,16,18], whereas studies regarding the three-phase inverse Stefan problem are scarce [5,6,7,15]. In paper [15], three tasks are identified, successively in a liquid, solid and mush zone, on the basis of which the sought convective heat transfer coefficient is then determined. The method described in papers [6,7] consists of minimization of a functional, whose value is the norm of difference between the given positions of phase-change front and the positions reconstructed based on the selected function describing the convective heat transfer coefficient. In paper [5], a solution is found in a linear combination form of functions satisfying the heat conduction equation. The coefficients of the combination are determined by the least square method to minimize the maximal defect in the initial-boundary data, thus making it possible to find the temperature distribution, the heat flux, or the convective heat transfer coefficient for the boundary. The method of the convective heat transfer coefficient identification in a threephase inverse design Stefan problem is presented in this paper. The convective heat transfer coefficient will be sought in the form of a continuous function, nonlinearly dependent on the parameters sought. A genetic algorithm was used to determine these parameters. The direct Stefan problem was solved via a generalized alternating phase truncation method [9, 14]. Stability of the whole algorithm was ensured by applying the Tikhonov regularization technique [1,17]. The featured example calculations show a very good approximation of the exact solution. Y. Shi et al. (Eds.): ICCS 27, Part I, LNCS 4487, pp , 27. c Springer-Verlag Berlin Heidelberg 27

2 2 Three-Phase Problem Three-Phase Inverse Design Stefan Problem 185 Now, we are going to describe an algorithm for the solution of a three-phase inverse design Stefan problem. Let the boundary of the domain D =[,b] [,t ] R 2 be divided into seven parts (Figure 1), where the boundary and initial conditions are given, and let the D domain be divided into three subdomains D 1, D 2 and D 3 (D = D 1 D 2 D 3 ). Let Γ 1,2 will designate the common boundary of domains D 1 and D 2, whilst Γ 2,3 will mean the common boundary of domains D 2 and D 3. Let us assume that boundary Γ k,k+1 is described by function x = ξ k,k+1 (t). For the given partial position of interfaces Γ 1,2 and Γ 2,3, we will determine function α(t) defined on boundaries Γ 2k (k =1, 2, 3) and temperature distributions T k, which inside domains D k (k =1, 2, 3) fulfil the heat conduction equation: c k ϱ k T k t (x, t) = 1 x ( x on boundary Γ, they fulfil the initial condition: λ k x T k (x, t) x ), (1) T 1 (x, ) = T, (2) on boundaries Γ 1k (k =1, 2, 3), they fulfil the second-kind homogeneous condition: T k (x, t) =, (3) x on boundaries Γ 2k (k =1, 2, 3), they fulfil the third-kind condition: T k λ k x (x, t) =α(t) ( ) T k (x, t) T, (4) whereas on moving interfaces Γ 1,2 and Γ 2,3, they fulfil the condition of temperature continuity and the Stefan condition (k =1, 2): ( T k ξk,k+1 (t),t ) ( = T k+1 ξk,k+1 (t),t ) = Tk,k+1, (5) dξ k,k+1 (t) T k (x, t) L k,k+1 ϱ k+1 = λ k T k+1 (x, t) dt x + λ k+1 Γk,k+1 x, (6) Γk,k+1 where c k, ϱ k and λ k are: the specific heat, the mass density and the thermal conductivity in respective phases, α is the convective heat transfer coefficient, T is the initial temperature, T is the ambient temperature, Tk,k+1 is the phase change temperature, L k,k+1 is the latent heat of fusion, and t and x refer to time and spatial location, respectively. Function α(t), describing the convective heat transfer coefficient, will be sought in the form of a function dependent (linearly or non-linearly) on n parameters: α(t) =α(t; α 1,α 2,...,α n ). (7) Let V mean a set of all functions in form (7). In real processes, function α(t) does not have an arbitrary value. Therefore, the problem of minimization with

3 186 D. S lota D 1 Γ Γ 21 b t p1 x Γ 11 Γ 22 t p2 Γ 1,2 t k1 D 2 Γ 12 Γ 2,3 Γ 23 t k2 t t Γ 13 D 3 Fig. 1. Domain of the three-phase problem constraints has some practical importance. Let V c mean a set of those functions from set V,forwhichα i [α l i,αu i ], αl i <αu i, αl i,αu i R, fori =1, 2,...,n. For the given function α(t) V c, the problem (1) (6) becomes a direct Stefan problem, whose solution makes it possible to find the positions of interfaces ξ 1,2 (t) andξ 2,3 (t) corresponding to function α(t). By using the interface positions found, ξ k,k+1 (t), and the given positions ξk,k+1 (t) (k =1, 2), we can build a functional which will define the error of an approximate solution: J ( α(t) ) = 2 k=1 ξk,k+1 (t) ξ k,k+1(t) 2 + γ α(t) 2, (8) where γ is the regularization parameter. The discrepancy principle proposed by Morozov will be used to determine the regularization parameter [17, 1]. The above norms denote the norms in a space of square-integrable functions in the interval (,t ). When the exact position of the interface is given only in selected points (hereafter called the control points ), the first norm present in functional (8) will be calculated from the following dependence: ξk,k+1 (t) ξ k,k+1 (t) 2 = M [ ( A i ξk,k+1;i ξk,k+1;i i=1 ) 2 ], (9) where A i are coefficients dependent on the chosen numerical integration method, M is the number of control points, and ξ k,k+1;i = ξ k,k+1 (t i)andξ k,k+1;i = ξ k,k+1 (t i ) are the given and calculated points respectively, describing the interfaces positions.

4 Three-Phase Inverse Design Stefan Problem Genetic Algorithm For the representation of the vector of decision variables, a chromosome was used in the form of a vector of real numbers (real number representation) [12, 13]. The tournament selection and elitist model were applied in the algorithm. This selection is carried out so that two chromosomes are drawn and the one with better fitness, goes to a new generation. There are as many draws as individuals that the new generation is supposed to include. In the elitist model the best individual of the previous generation is saved and, if all individuals in the current generation are worse, the worst of them is replaced with the saved best individual from the previous population. As the crossover operator, arithmetical crossover was applied, where as a result of crossing of two chromosomes, their linear combinations are obtained: α 1 = r α 1 +(1 r) α 2, α 2 = r α 2 +(1 r) α 1, (1) where parameter r is a random number with a uniform distribution from the domain [, 1]. In the calculations, a nonuniform mutation operator was used as well. During mutation, the α i gene is transformed according to the equation: { α i = α i + Δ(τ,α u i α i), α i Δ(τ,α i α l i ), (11) and a decision is taken at random which from the above formulas should be applied, where: Δ(τ,x) =x ( 1 r (1 τ )d) N, (12) and r is a random number with a uniform distribution from the domain [, 1], τ is the current generation number, N is the maximum number of generations and d is a constant parameter (in the calculations, d = 2 was assumed). In calculations parameters used for the genetic algorithm are as follows: population size n pop = 7, number of generations N = 5, crossover probability p c =.7 and mutation probability p m =.1. 4 Numerical Example Now we will present an example illustrating the application of the method discussed. An axisymmetric problem is considered in the example, where: b =.6 [m], λ 1 =54[W/(mK)], λ 2 =42[W/(mK)], λ 3 =3[W/(mK)], c 1 = 84 [J/(kg K)], c 2 = 755 [J/(kg K)], c 3 = 67 [J/(kg K)], ϱ 1 = 7 [kg/m 3 ], ϱ 2 = 725 [kg/m 3 ], ϱ 3 = 75 [kg/m 3 ], L 1,2 = 2176 [J/kg], L 2,3 = 544 [J/kg], T1,2 = 1773 [K], T 2,3 = 1718 [K], T = 33 [K] andt = 183 [K]. Function α(t) is sought as an exponential function (Figure 2): ( t α(t) =α 1 exp ln α ) 2. (13) t 2 α 1

5 188 D. S lota α α 1 P 1 α 2 P 2 α 3 P 3 t 2 t 3 t Fig. 2. Function α(t) The parameters describing the exact form of function α(t) are: α 1 = 12, α 2 = 6, t 2 =9. Set V c is defined in the following way: { } V c = α(t) V : α 1 [1, 14], α 2 [4, 7], t 2 [5, 15]. (14) In the alternating phase truncation method, the finite-difference method was used, the calculations having been made on a grid of discretization intervals equal Δt =.1 andδx = b/5. A change (reasonable) of the grid density did not significantly affect the results obtained. The giving of two points: P 1 (,α 1 )andp 2 (t 2,α 2 ) (see Figure 2) explicitly determines the exponential function in form (13), however, the same function can be determined through defining points P 1 (,α 1 )andp 3 (t 3,α 3 ). Therefore, the problem of reconstructing function α(t) in form (13) based on three parameters: (α 1,α 2,t 2 ), has infinitely many solutions. For this reason, the accuracy of solution in this case will be determined for the entire function α(t), and not separately for each of the sought parameters. Thus, the relative percentage error will be calculated from the following relation: e α = ( αe (t) α a (t) ) ) 2 1/2 dt ( t ( αe (t) ) ) 2 1/2 dt 1%, (15) ( t where α e (t) is the exact value of function α(t), and α a (t) is an approximate value of function α(t). The calculations were made for an accurate moving interface position and for a position disturbed with a pseudorandom error of normal distribution. Results for 1%, 2% and 5% disturbance are presented in the paper. Also, the influence of the number of control points, i.e. the number of points where the interface position is known (including the addends in sum (9)), has been examined. The results of interface control carried out every.1,.2,.5 and 1 seconds are presented below. They correspond to a situation where M = 293, 147, 419, 21.

6 Three-Phase Inverse Design Stefan Problem 189 a) b).3.25 Per. % av. max Per. 1% av. max. Error % Error % c).1s.2s.5s 1s Control points d).1s.2s.5s 1s Control points Error % Per. 2% av. max. Error % Per. 5% av. max s.2s.5s 1s Control points.1s.2s.5s 1s Control points Fig. 3. Average and maximum errors of function α(t) reconstruction Α t t Fig. 4. Exact (solid line) and approximate (dot line) values of function α(t) for interface control every one second and for perturbation equal to 5% In each case, calculations were carried out for ten different initial settings of a pseudorandom numbers generator. Figure 3 presents the errors with which function α(t) was reconstructed for different disturbance values and a different number of control points. The mean error value for ten activations of the algorithm and the obtained maximum error value are shown in the figure. It should be noted that where the input data were given without disturbance, the convective heat transfer coefficient was reconstructed with minimal errors resulting from the chosen algorithm termination point, whereas for the disturbed input data, the result s error is much lower than the error at the beginning. For the highest disturbance equal to 5% and for

7 19 D. S lota.5 x.4.3 2, , t Fig. 5. Exact (solid lines) and reconstructed (dot lines) positions of interfaces Γ 1,2 and Γ 2,3 for control every one second and for perturbation equal to 5% interface controls every 1 second, the maximum error equals to.68%. With a larger amount of control points, errors become significantly lower. An exception is the result obtained for a 2% disturbance and for interface control carried out every second, where the errors are slightly higher than those obtained for a smaller number of control points. Figure 4 shows the exact and approximate values of function α(t) for interface control every one second and for perturbations equal to 5%. Figure 5 presents the exact and reconstructed positions of interfaces Γ 1,2 and Γ 2,3 for control carried out every one second and for perturbation equal to 5%. In the remaining cases, all curves were reconstructed equally well. 5 Conclusion This paper discussed the identification of the convective heat transfer coefficient in a three-phase inverse design Stefan problem. The problem consists in the reconstruction of the function which describes the convective heat transfer coefficient, where the position of the moving interfaces of the phase change are well-known. The convective heat transfer coefficient is sought in the form of a continuous function, non-linearly dependent on the parameters sought. A genetic algorithm was used to determine these parameters. The direct Stefan problem was solved via a generalized alternating phase truncation method. Stability of the whole algorithm was ensured by applying the Tikhonov regularization technique. The calculations made show stability of the proposed algorithm in terms of the input data errors and the number of control points. The application of genetic algorithms yields better results than the classical nonderivative optimization methods (e.g. the Nelder-Mead method). A comparison of the results for the two-phase problem is included in paper [16]. In that case, the scatter of the results obtained is also much smaller. In the future we are going to use the presented algorithm for the solution of the inverse Stefan problem in which measured temperatures are given at some points of the domain.

8 Three-Phase Inverse Design Stefan Problem 191 References 1. Ang, D.D., Dinh, A.P.N., Thanh, D.N.: Regularization of an inverse two-phase Stefan problem. Nonlinear Anal. 34 (1998) Briozzo, A.C., Natale, M.F., Tarzia, D.A.: Determination of unknown thermal coefficients for Storm s-type materials through a phase-change process. Int. J. Nonlinear Mech. 34 (1999) Goldman, N.L.: Inverse Stefan problem. Kluwer, Dordrecht (1997) 4. Grzymkowski, R., S lota, D.: One-phase inverse Stefan problems solved by Adomian decomposition method. Comput. Math. Appl. 51 (26) Grzymkowski, R., S lota, D.: Multi-phase inverse Stefan problems solved by approximation method. In: Wyrzykowski, R. et al. (eds.): Parallel Processing and Applied Mathematics. LNCS 2328, Springer-Verlag, Berlin (22) Grzymkowski, R., S lota, D.: Numerical calculations of the heat-transfer coefficient during solidification of alloys. In: Sarler, B. et al. (eds.): Moving Boundaries VI. Wit Press, Southampton (21) Grzymkowski, R., S lota, D.: Numerical method for multi-phase inverse Stefan design problems. Arch. Metall. Mater. 51 (26) Jochum, P.: The numerical solution of the inverse Stefan problem. Numer. Math. 34 (198) Kapusta, A., Mochnacki, B.: The analysis of heat transfer processes in the cylindrical radial continuous casting volume. Bull. Pol. Acad. Sci. Tech. Sci. 36 (1988) Kurpisz, K., Nowak, A.J.: Inverse thermal problems. CMP, Southampton (1995) 11. Liu, J., Guerrier, B.: A comparative study of domain embedding methods for regularized solutions of inverse Stefan problems. Int. J. Numer. Methods Engrg. 4 (1997) Michalewicz, Z.: Genetic algorithms + data structures = evolution programs. Springer-Verlag, Berlin (1996) 13. Osyczka, A.: Evolutionary algorithms for single and multicriteria design optimization. Physica-Verlag, Heidelberg (22) 14. Rogers, J.C.W., Berger, A.E., Ciment, M.: The alternating phase truncation method for numerical solution of a Stefan problem. SIAM J. Numer. Anal. 16 (1979) Slodička, M., De Schepper, H.: Determination of the heat-transfer coefficient during soldification of alloys. Comput. Methods Appl. Mech. Engrg. 194 (25) S lota, D.: Solving the inverse Stefan design problem using genetic algorithms. Inverse Probl. Sci. Eng. (in review) 17. Tikhonov, A.N., Arsenin, V.Y.: Solution of ill-posed problems. Wiley & Sons, New York (1977) 18. Zabaras, N., Kang, S.: On the solution of an ill-posed design solidification problem using minimization techniques in finite- and infinite-dimensional function space. Int. J. Numer. Methods Engrg. 36 (1993)

Computers and Mathematics with Applications. A new application of He s variational iteration method for the solution of the one-phase Stefan problem

Computers and Mathematics with Applications 58 (29) 2489 2494 Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa A new