HT14 A COMPARISON OF DIFFERENT PARAMETER ESTIMATION TECHNIQUES FOR THE IDENTIFICATION OF THERMAL CONDUCTIVITY COMPONENTS OF ORTHOTROPIC SOLIDS

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1 Inverse Problems in Engineering: heory and Practice rd Int. Conference on Inverse Problems in Engineering June -8, 999, Port Ludlow, WA, USA H4 A COMPARISON OF DIFFEREN PARAMEER ESIMAION ECHNIQUES FOR HE IDENIFICAION OF HERMAL CONDUCIVIY COMPONENS OF ORHOROPIC SOLIDS M. M. Meias and H. R. B. Orlande Department of Mechanical Engineering, EE/COPPE Federal University of Rio de Janeiro, UFRJ Cid. Universitária, Cx. Postal:6850 Rio de Janeiro, RJ, Brasil M. N. Ozisi Department of Mechanical and Aerospace Engineering North Carolina State University, NCSU PO Box: 790 Raleigh, NC, USA ABSRAC In this paper we examine the inverse problem of identification of the three thermal conductivity components of an orthotropic cube. For the solution of such parameter estimation problem, we consider two different versions of the Levenberg-Marquardt method and four different versions of the conugate gradient method. he techniques are compared in terms of rate of reduction of the obective function with respect to the number of iterations, CPU time and accuracy of the estimated parameters. Simulated measurements with and without measurement errors are used in the analysis, for three different sets of exact values for the parameters. NOMENCLAURE I number of transient measurements per sensor J sensitivity coefficients J sensitivity matrix defined by equation (7),, thermal conductivities in the x, y and z directions, respectively M number of sensors P vector of unnown parameters S least-squares norm defined by equation (5) vector of estimated temperatures t h, t f heating time and final time Y vector of measured temperatures GREEKS σ standard deviation of the measurement errors INRODUCION In nature, several materials have direction-dependent thermal conductivities including, among others, woods and crystals. his is also the case for some man-made materials, for example, composites. Such ind of materials is denoted anisotropic, as an opposition to isotropic materials, in which the thermal conductivity does not vary with direction. A special case of anisotropic materials involve those where the offdiagonal elements of the conductivity tensor are null and the diagonal elements are the principal conductivities along three mutually orthogonal directions. hey are referred to as orthotropic materials (Ozisi, 99). As a result of the importance of orthotropic materials in nowadays engineering, a lot of attention has been devoted in the recent past to the estimation of their thermal properties, by using inverse analysis techniques of parameter estimation (Sawaf and Ozisi, 995, Sawaf et al, 995, ata et al, 99, ata, 99, Dowding et al, 995, 996, Meias et al, 999). In this paper we present a comparison of different methods of parameter estimation, as applied to the identification of the three thermal conductivity components of an orthotropic solid, by using simulated experimental data. Such a physical problem was chosen for comparison of the methods because it requires non-linear estimation procedures, since the sensitivity coefficients are functions of the unnown parameters. Experimental variables used in the analysis, such as the duration of the experiment, location of sensors and boundary conditions, were optimally chosen (Meias et al, 999). he methods examined in this wor include: the Levenberg-Marquardt Method (Levenberg, 944, Marquardt, 96, Bec and Arnold, 977, Moré, 977, Meias et al, 999, Ozisi and Orlande, Copyright 999 by ASME

2 999) and the Conugate Gradient Method versions of Fletcher- Reeves, Pola-Ribiere, Powell-Beale and Hestenes-Stieffel (Alifanov, 994, Daniel, 97, Ozisi and Orlande, 999, Powell, 977, Hestenes and Stiefel, 95, Fletcher and Reeves, 964, Colaço and Orlande, 999). Such methods are compared in terms of the rate of reduction of the obective function, accuracy of estimated parameters and effects of the initial-guess on the convergence. Different sets of values are used for the thermal conductivity components in order to generate the simulated measurements, thus representing several practical engineering materials (Meias et al, 999), as described next. DIREC PROBLEM he physical problem considered here involves the threedimensional linear heat conduction in an orthotropic solid, with thermal conductivity components, and in the x, y and z directions, respectively. he solid is considered to be a parallelepiped with sides a, b and c, initially at zero temperature. For times t > 0, uniform heat fluxes q ( t ), q ( t ) and q ( t) are supplied at the surfaces x = a, y = b and z = c, respectively, while the other three remaining boundaries at x = 0, y = 0 and z = 0 are supposed insulated. he mathematical formulation of such physical problem is given in dimensionless form by + + = x y z t (.a) in 0 < x < a, 0 < y < b, 0 < z < c ; x t > 0 = 0 at x = 0; = q( t) at x = a, for t > 0 x = 0 at y = 0; = q ( t) at y = b, for t > 0 y y (.b,c) (.d,e) = 0 at z = 0 ; = q( t) at z = c, for t > 0 (.f,g) z z = 0 for t = 0 ; in 0 x a, 0 y b, 0 z c (.h) In the direct problem associated with the physical problem described above, the three thermal conductivity components, and, as well as the solid geometry, initial and boundary conditions, are nown. he obective of the direct problem is then to determine the transient temperature field (x, y, z, t) in the body. By following the same approach of ata et al (99), we assume the boundary heat fluxes to be pulses of finite duration t h, that is, q, for 0 < t th q ( t) = for =,, () 0, for t > th where q is the dimensionless magnitude of the applied heat flux. he solution of problem () with boundary heat fluxes given by equation (), can be obtained analytically as a superposition of three one-dimensional solutions by using the split-up procedure (Mihailov and Ozisi, 994) for 0 < t t h. We obtain for 0< t t h q a x t ( x, y, z, t) = θ, + a a (.a) q b y t q c z t + θ + θ,, b b c c where ξ θ( ξ, τ) = + + τ + 6 (.b) ( i+ ) + ( ) cos( i π ξ) exp[ τ( i π) ] i= After the heating period, problem () becomes homogeneous with initial temperature distribution given by equation (.a) for t = t h. Hence, it can easily be solved by separation of variables (Ozisi, 99). Since the initial condition at t = t h obtained from equation (.a) is a superposition of three one-dimensional solutions, such is also the case for the solution for t > t h. he temperature field for t > t h is given by q q q ( x, y, z, t) = θ ( x, t, ) + θ ( y, t, ) + θ ( z, t, ) a b c (4.a) where θ ( ξ, t, ) = t + i= ( ) i h cos( i π ξ) exp[ ( t t h ) ]{ exp[ t h ]} (4.b) INVERSE PROBLEM For the inverse problem considered here, the thermal conductivity components, and are regarded as unnown, while the other quantities appearing in the formulation of the direct problem described above are assumed to be nown with high degree of accuracy. For the estimation of the vector of unnown parameters P = [,, ], we assume available the transient readings of M temperature sensors. Since it is desirable to have a nonintrusive experiment, we consider the sensors to be located at the insulated surfaces x = 0, y = 0 and z = 0. We note that the temperature measurements may contain random errors. Such errors are assumed here to be additive, uncorrelated, and normally distributed with a zero mean and a nown constant standard-deviation σ (Bec and Arnold, 977). herefore, the solution of the present parameter estimation problem can be obtained through the minimization of the ordinary least-squares norm S ( P) = [ Y ( P)] [ Y ( P)] (5) where Copyright 999 by ASME

3 H H H H H H [ Y ( P)] = [ Y ( P), Y ( P),, YI I ( P) ] (6.a) and each element [ Y H i H i ( P)] is a row vector of length equal to the number of sensors M, that is, H H [ Yi i ( P)] = [ Yi i ( P), Yi i ( P),, YiM im ( P)] for i =,..,I (6.b) We note that Y im and im (P) are the measured and estimated temperatures, respectively, for time t i, i=,,i, and for the sensor m, m=,,m. he estimated temperatures are obtained from the solution of the direct problem given by equations (,4), by using the current available estimate for the vector of unnown parameters P = [,, ]. he methods considered in this paper for the minimization of the least squares norm (5) mae use of the sensitivity matrix. For the present case, involving multiple measurements to estimate the vector of unnown parameters P = [,, ], the sensitivity matrix is defined as H H H H H H ( P) J ( P) = (7) P H H H I I I he elements of the sensitivity matrix are denoted Sensitivity Coefficients. Due to the analytical nature of the solution of the direct problem given by equations (,4), we can also obtain analytic expressions for the sensitivity coefficients. Since the solution of the direct problem is obtained as a superposition of one-dimensional solutions, we note that the sensitivity coefficient with respect to is a function of x, but not of y and z. he expressions for the sensitivity coefficients with respect to are obtained as q a x t J = θ, + a a π π + q t + i i x ( t)( i ) ( ) cos exp (8.a) a i= a a for 0 < t t h and ( ) cos( π ) exp[ ( π) ] = i q i x i t J a i= (8.b) { {exp[ th ]( t th ) t} + for t > t h + exp[ t ] } h and analogous expressions can be obtained for the sensitivity coefficients J and J with respect to and, respectively. We note that the present estimation problem is non-linear, since the sensitivity coefficients are functions of the unnown parameters. MEHODS OF SOLUION For the minimization of the least squares norm (5), we consider here the Levenberg-Marquardt Method (Levenberg, 944, Marquardt, 96, Bec and Arnold, 977, Meias et al, 999, Ozisi and Orlande, 999) and the Conugate Gradient Method versions of Fletcher-Reeves, Pola-Ribiere, Powell- Beale and Hestenes-Stieffel (Alifanov, 994, Daniel, 97, Ozisi and Orlande, 999, Powell, 977, Hestenes and Stiefel, 95, Fletcher and Reeves, 964, Colaço and Orlande, 999). hese methods can be suitably arranged in iterative procedures of the form P + = P + P where P is the increment in the vector of unnown parameters at iteration. he computation of P for each of the methods considered here are addressed next.. Levenberg-Marquardt Method he so-called Levenberg-Marquardt Method was first derived by Levenberg (944) by modifying the ordinary least squares norm. Later Marquardt (96) derived basically the same technique by using a different approach. Marquardt s intention was to obtain a method that would tend to the Gauss method in the neighborhood of the minimum of the ordinary least squares norm, and would tend to the steepest descent method in the neighborhood of the initial guess used for the iterative procedure. he increment P for the Levenberg-Marquardt Method can be written as where µ P = [( J ) (9) J + µ Ω ] ( J ) [ Y ( (0) is a positive scalar named damping parameter, and Ω is a diagonal matrix. he purpose of the matrix term µ Ω in equation (0) is to damp oscillations and instabilities due to the ill-conditioned character of the problem, by maing its components large as compared to those of J J, if necessary. he damping parameter is made large in the beginning of the iterations, since the problem is generally ill-conditioned in the region around the initial guess used for the iterative procedure, which can be quite far from the exact parameters. With such an approach, the matrix J J is not required to be non-singular in the beginning of iterations and the Levenberg-Marquardt Method tends to the Steepest Descent Method, that is, a very small step is taen in the negative gradient direction. he parameter µ is then gradually reduced as the iteration procedure advances to the solution of the parameter estimation problem, and then the Levenberg-Marquardt Method tends to the Gauss Method (Bec and Arnold, 977, Ozisi and Orlande, 999). he following criteria are used to stop the iterative procedure of the Levenberg-Marquardt Method: Copyright 999 by ASME

4 + (i) S( P ) < ε (ii) ( J ) [ Y ( < ε (.a) (.b) + (iii) P P < ε (.c) where ε, ε and ε are user prescribed tolerances and. is the vector Euclidean norm, i.e., x = ( x x), where the superscript denotes transpose. he criterion given by equation (.a) tests if the least squares norm is sufficiently small, which is expected to be in the neighborhood of the solution for the problem. Similarly, equation (.b) checs if the norm of the gradient of S(P) is sufficiently small, since it is expected to vanish at the point where S(P) is minimum. Although such a condition of vanishing gradient is also valid for maximum and saddle points of S(P), the Levenberg-Marquardt method is very unlie to converge to such points. he last criterion given by equation (.c) results from the fact that changes in the vector of parameters are very small when the method has converged. Different versions of the Levenberg-Marquardt method can be found in the literature, depending on the choice of the diagonal matrix Ω and on the form chosen for the variation of the damping parameter µ. We illustrate here the computational algorithm of the method with the matrix Ω taen as Ω = diag [( J ) J ] () Suppose that temperature measurements are given at times t i, i=,...,i. Also, suppose that an initial guess P 0 is available for the vector of unnown parameters P. Choose a value for µ 0, say, µ 0 = and set = 0. hen, Step. Solve the direct heat transfer problem given by equations () with the available estimate P. Step. Compute S(P ) from equation (5). Step. Compute the sensitivity matrix J defined by equation (7) and then the matrix Ω given by equation (). Step 4. Compute the increments for the unnown parameters by using equation (0). Step 5. Compute the new estimate P + as + P = P + P () Step 6. Step 7. Step 8. Solve now the direct problem () with the new estimate P + in order to find (P + ). hen compute S(P + ), as defined by equation (5). If S(P + ) S(P ), replace µ by 0µ and return to step 4. If S(P + )< S(P ), accept the new estimate P + and replace µ by 0.µ. Step 9. Chec the stopping criteria given by equations (.a-c). Stop the iterative procedure if any of them is satisfied; otherwise, replace by + and return to step. In another version of the Levenberg-Marquardt method due to Moré(977) the matrix Ω is taen as the identity matrix and the damping parameter µ is chosen based on the so-called trust region algorithm. he subroutines of the IMSL (987) are based on this version of the Levenberg-Marquardt Method.. Conugate Gradient Method he Conugate Gradient Method is a straightforward and powerful iterative technique for solving inverse problems of parameter estimation. In the iterative procedure of the Conugate Gradient Method, at each iteration a suitable step size is taen along a direction of descent in order to minimize the obective function. he direction of descent is obtained as a linear combination of the negative gradient direction at the current iteration with directions of descent from previous iterations. he Conugate Gradient Method with an appropriate stopping criterion belongs to the class of iterative regularization techniques, in which the number of iterations is chosen so that stable solutions are obtained for the inverse problem (Alifanov, 994, Ozisi and Orlande, 999). he increment in the vector of unnown parameters at each iteration of the conugate gradient method is given by P = β d (4) he search step size β is obtained by minimizing the obective function given by equation (5) with respect to β. By using a first-order aylor series approximation for the estimated temperatures, the following expression results for the search step size(ozisi and Orlande, 999): [ J d ] [ Y ( β = (5) [ J d ] [ J d ] he direction of descent d is given in the following general form d q = S( P ) + γ d + ψ d (6) where γ and ψ are conugation coefficients and S( P ) is the gradient vector given by S( P ) = ( J ) [ Y ( (7) he superscript q in equation (6) denotes the iteration number where a restarting strategy is applied to the iterative procedure of the conugate gradient method. Restarting strategies were suggested for the conugate gradient method of parameter estimation in order to improve its convergence rate (Powell, 977). Different versions of the Conugate Gradient Method can be found in the literature depending on the form used for the 4 Copyright 999 by ASME

5 computation of the direction of descent given by equation (6) (Alifanov, 994, Daniel, 97, Ozisi and Orlande, 999, Powell, 977, Hestenes and Stiefel, 95, Fletcher and Reeves, 964, Colaço and Orlande, 999). In the Fletcher-Reeves version, the conugation coefficients γ and ψ are obtained from the following expressions [ S( γ = with γ 0 = 0 for = 0 [ S( (8.a) ψ = 0 for = 0,,, (8.b) In the Pola-Ribiere version of the Conugate Gradient Method the conugation coefficients are given by [ S( γ = with γ 0 = 0 for =0 (9.a) [ S( ψ = 0 for = 0,,, (9.b) For the Hestenes-Stiefel version of the Conugate Gradient Method we have = [ S( γ with γ 0 = 0 for =0 (0.a) [ d ] [ S( P )] ψ = 0 for = 0,,, (0.b) Powell(977) suggested the following expressions for the conugation coefficients, which gives the so-called Powell- Beale s version of the conugate gradient method = [ S( γ with γ 0 = 0 for =0 (.a) [ d ] [ S( P )] q+ q [ S( ψ = with ψ 0 = 0 q q+ q [ d ] [ S( (.b) for = 0 In accordance with Powell(977), the application of the conugate gradient method with the conugation coefficients given by equations () requires restarting when gradients at successive iterations tend to be non-orthogonal (which is a measure of the local non-linearity of the problem) and when the direction of descent is not sufficiently downhill. Restarting is performed by maing ψ = 0 in equation (6). he non-orthogonality of gradients at successive iterations is tested by using: ABS( [ S( ) 0.[ S( (.a) where ABS (.) denotes the absolute value. A non-sufficiently downhill direction of descent (i.e., the angle between the direction of descent and the negative gradient direction is too large) is identified if either of the following inequalities are satisfied: [ d ] [ S( -.[ S( (.b) or [ d ] [ S( -0.8[ S( (.c) In Powell-Beale s version of the conugate gradient method, the direction of descent given by equation (6) is computed in accordance with the following algorithm for : SEP : est the inequality (.a). If it is true set q = -. SEP : Compute γ with equation (.a). SEP : If = q+ set ψ = 0. If q+ compute ψ with equation (.b). SEP 4: Compute the search direction d (X,Y,τ) with equation (6). SEP 5: If q+ test the inequalities (.b,c). If either one of them is satisfied set q = - and ψ =0. hen recompute the search direction with equation (6). he iterative procedure given by equations (9,4-) does not provide the conugate gradient method with the stabilization necessary for the minimization of the obective function (5) to be classified as well-posed. herefore, as the estimated temperatures approach the measured temperatures containing errors, during the minimization of the function (5), large oscillations may appear in the inverse problem solution. However, the conugate gradient method may become wellposed if the Discrepancy Principle (Alifanov, 994) is used to stop the iterative procedure. In the discrepancy principle, the iterative procedure is stopped when the following criterion is satisfied + S( P ) < ε () where the value of the tolerance ε is chosen so that sufficiently stable solutions are obtained. In this case, we stop the iterative procedure when the residuals between measured and estimated temperatures are of the same order of magnitude of the measurement errors, that is, Y - σ (4) im im where σ is the standard deviation of the measurement errors, which is supposed constant and nown. By substituting equation (4) into equation (5), we obtain ε as ε = I M σ (5) If the measurements are regarded as errorless, the tolerance ε can be chosen as a sufficiently small number, since the expected minimum value for the obective function (5) is zero. he iterative procedure of the conugate gradient method can be suitably arranged in the following computational algorithm. Suppose that temperature measurements and an initial guess P 0 is available for the vector of unnown parameters P. Set = 0 and then Step. Solve the direct heat transfer problem () by using the available estimate P and obtain the vector of estimated temperatures (P ). 5 Copyright 999 by ASME

6 Step. Chec the stopping criterion given by equation (). Continue if not satisfied. Step. Compute the sensitivity matrix J defined by equation (7). Step 4. Compute the gradient direction S( P ) from equation (7) and then the conugation coefficients from equations (8), (9), (0) or (). Note that Powell-Beale s version requires restarting if any of the inequalities (.a-c) are satisfied. Step 5. Compute the direction of descent d by using equation (6). Step 6. Compute the search step size β from equation (5). Step 7. Compute the increment P with equation (4) and then the new estimate P + with equation (9). Replace by + and return to step. RESULS AND DISCUSSIONS For the results presented below, we assumed the solid with unnown thermal conductivities to be available in the form of a cube, so that a = b = c =. Also, the heat fluxes applied on the boundaries x = a =, y = b = and z = c = were assumed to be of equal magnitude during the heating period 0 < t t h, so that q = in equation (), for =,,. Different experimental variables, such as the number and locations of sensors, heating time and final time, optimally chosen by Meias et al (999), were used here for the analysis of test-cases involving different values for the unnown thermal conductivity components. he test-cases examined here are summarized in table, together with their respective optimal values of heating and final times. he readings of three sensors (M=) located at the positions (0,0.9,0.9), (0.9,0,0.9) and (0.9,0.9,0) were used for the estimation of the unnown thermal conductivity components. One hundred simulated measurements per sensor, containing additive, uncorrelated and normally distributed errors with zero mean and constant standard-deviation, were assumed available for the estimation procedure. he computational algorithms presented above for the Levenberg-Marquardt method, as well as for the different versions of the conugate gradient method, were programmed in FORRAN 90 and applied to the estimation of the parameters shown in table, by using simulated measurements with standard deviations of σ = 0 (errorless measurements) and σ = 0.0 max, where max is the maximum measured temperature. For the comparison presented below, we also considered the IMSL (987) version of the Levenberg- Marquardt method in the form of the subroutine DBCLSJ. Upper and lower limits for the unnown parameters, required by such subroutine, were taen as 0 - and 0 4, respectively. he same limits were also considered in the implementation of the other estimation techniques. able summarizes the techniques used in the present paper. ables to 5 present the results obtained for the CPU time, average rate of reduction of the obective function with respect to the number of iterations (r) and RMS error (e RMS ), obtained with each of the techniques summarized in table, for the test-cases to, respectively. he computations were performed in a Pentium 00 MHz, under the Fortran PowerStation platform. he results presented in tables and 4 were obtained with an initial guess P 0 =[0.,0.,0.] for the unnown parameters, while an initial guess of P 0 =[0.5,0.5,0.5] was required for convergence of any of the methods for testcase (table 5). For those cases involving simulated measurements with random errors, the results shown in tables - 5 were averaged over 0 different runs, in order to reduce the effects of the random number generator utilized. he average rates of reduction of the obective function (r) were obtained from the following approximation for the variation of S(P) with the number of iterations (N): S where C is a constant depending on the data. r (P) = C N (6) he RMS errors were computed as e RMS = ( ex, est, ) (7) = where the subscripts ex and est refer to the exact and estimated parameters, respectively. able.. est-cases with respective heating and final times. t h t f Case Exact Parameters able. echniques used for the estimation of the unnown parameters. echnique Method Version A Levenberg-Marquardt his paper B Levenberg-Marquardt IMSL A Conugate Gradient Method Fletcher-Reeves B Conugate Gradient Method Pola-Ribiere C Conugate Gradient Method Hestenes-Stiefel D Conugate Gradient Method Powell-Beale he tolerances for the stopping criteria of technique A were taen as ε = ε = ε =0-5 in equations (.a-c), for cases involving errorless measurements, as well as measurements with random errors. For those cases involving errorless measurements, the tolerances for the stopping criterion of techniques A-D were taen as ε = 0-5 in equation (). For those cases involving measurements with random errors, such tolerances for techniques A-D were obtained from 6 Copyright 999 by ASME

7 equation(5) based on the discrepancy principle. We note that the tolerances for the stopping criteria of technique B were set internally by the subroutine DBCLSJ of the IMSL (987). able. Results for test-case echnique σ CPU time (s) r e RMS A 0.0 max B 0.0 max A 0.0 max B 0.0 max C 0.0 max D max able 4. Results for test-case echnique σ CPU time (s) r e RMS A 0.0 max B 0.0 max A 0.0 max B 0.0 max C 0.0 max D max able 5. Results for test-case echnique σ CPU time (s) r e RMS A 0.0 max B 0.0 max A 0.0 max B 0.0 max C 0.0 max D max.6 0. Let us consider first in the analysis of tables -5 the cases involving errorless measurements (σ = 0). ables -5 show that all techniques were able to estimate exactly the three different sets of unnown parameters examined, resulting in e RMS =. he highest rates of reduction of the obective function were obtained with the Levenberg-Marquardt method and, for such method, technique B had a better performance than technique A, except for test-case (table ) where both techniques were equivalent. he rates of reduction of the obective function were of the order of 0 (minimum of 8) or higher for the Levenberg- Marquardt method, while such rates were of the order of 0 (maximum of 4) for the conugate gradient method. he CPU times were generally smaller for the Levenberg-Marquardt method (techniques A and B) than for the conugate gradient method (techniques A-D). ables -5 show a strong reduction of the rate of reduction of the obective function when measurements with random errors were used in the analysis, instead of errorless measurements. Such a behavior was also observed in a nonlinear function estimation problem (Colaço and Orlande, 999). As for the cases with errorless measurements, the Levenberg- Marquardt method had a performance superior than that of the conugate gradient method in terms of CPU time and rate of reduction of the obective function, for the cases with measurements with random errors. All techniques resulted in accurate estimates for the unnown parameters when measurements with random errors were used in the analysis; but relative high RMS errors were noticed with techniques A and C for test-case. We note that the use of the discrepancy principle was not required to provide the Levenberg-Marquardt method with the regularization necessary to obtain stable solutions for those cases involving measurements with random errors. he computational experiments revealed that the Levenberg- Marquardt method, through its automatic control of the damping parameter µ, reduced drastically the increment in the vector of estimated parameters, at the iteration where the measurement errors started to cause instabilities on the inverse problem solution. he iterative procedure of the Levenberg- Marquardt method was then stopped by the criterion given by equation (.c). For some cases we also noticed that the iterative procedure of the Levenberg-Marquardt method was stopped by the criterion (.b) when measurements with random errors were used in the analysis, that is, the norm of gradient vector became very small. able 6 is prepared to illustrate the effect of the initial guess for the parameters, i.e., P 0, over the rate of reduction of the obective function. hree different initial guesses were considered for test-case, by using errorless measurements in the analysis. able 6 shows that, although the rate of reduction umped to 4 for technique A for the initial guess P 0 =[,,], the values of such rate were insensitive to the three initial guesses, with all techniques examined in this paper. We note that techniques A-D, based on the conugate gradient method, did not converge to the exact parameters for initial guesses larger than P 0 = [,,]. On the other hand, techniques A and B, based on the Levenberg-Marquardt method, were able to converge to the exact parameters even with initial guesses as large as P 0 = [0,0,0]. 7 Copyright 999 by ASME

8 able 6. Effect of the initial guess on the rate of reduction of the obective function. echnique r P 0 =[0.,0.,0.] P 0 =[,,] P 0 =[,,] A B A 4 B 4 4 C 4 D 0 CONCLUSIONS In this paper we compared 6 different parameter estimation techniques, based on the Levenberg-Marquardt method and on the conugate gradient method, for the identification of the three thermal conductivity components of orthotropic solids. he techniques were compared in terms of rate of reduction of the obective function, CPU time and accuracy of the estimated parameters. he foregoing analysis reveals that, besides having the highest rates of reduction of the obective function, the use of the Levenberg-Marquardt method also resulted in the smallest CPU times and in the smallest RMS errors of the estimated parameters. Such method was able to converge to the exact parameters even for initial guesses quite far from the exact values. Hence, the Levenberg-Marquardt method appears to be the best, among those tested, for the estimation of the three thermal conductivity components of orthotropic solids. ACKNOWLEDGMENS he solution of the direct problem, given by equations () and (4) as a superposition of three one-dimensional solutions, was derived by Prof. M. D. Mihailov, who is currently a Visiting Professor in the Department of Mechanical Engineering of the Federal University of Rio de Janeiro. REFERENCES Alifanov, O.M., 994, Inverse Heat ransfer Problems, Springer-Verlag. Bec, J.V. and Arnold, K.J., 977, Parameter Estimation in Engineering and Science, Wiley, New Yor. Colaço, M.J. and Orlande, H.R.B., 999, A Comparison of Different Versions of the Conugate Gradient Method of Function Estimation, Num. Heat ransfer, Part A (in press). Daniel, J.V., 97, he Approximate Minimization of Functionals, Prentice-Hall, Englewood Cliffs. Dowding, K.J., Bec, J.V. and Blacwell, B., 996, Estimation of Directional-Dependent hermal Properties in a carbon-carbon composite, Int. J. Heat Mass ransfer, Vol. 9, No. 5, pp Dowding, K., Bec, J.V., Ulbrich, A., Blacwell, B. and Hayes, J., 995, Estimation of hermal Properties and Surfaces Heat Flux in Carbon-Carbon Composite, J. of ermophysics and Heat ransfer, Vol.9, No., pp Fletcher R. and Reeves, C.M., 964, Function Minimization by Conugate Gradients, Computer J., Vol. 7, pp Hestenes, M.R. and Stiefel, E, 95., Method of Conugate Gradients for Solving Linear Systems, J. Research Nat. Bur. Standards, Vol. 49, pp IMSL Library, 987, Edition 0, Houston exas. Levenberg, K., 944, A Method for the Solution of Certain Non-linear Problems in Least Squares, Quart. Appl. Math.,, pp Marquardt, D.W., 96, An Algorithm for Least Squares Estimation of Nonlinear Parameters, J. Soc. Ind. Appl. Math,, pp Meias, M.M., Orlande, H.R.B. and Ozisi, M.N., 999, Design of Optimum Experiments for the Estimation of the hermal Conductivity Components of Orthotropic Solids, Hybrid Methods in Engineering, Vol., No, pp Mihailov, M. D. and Ozisi, M.N., Unified Analysis of Heat and Mass Diffusion, Dover, New Yor, 994. Moré, J.J., 977, Numerical Analysis, Lecture Notes in Mathematics, Vol. 60, G.A Watson, ed., Springer-Verlag, Berlin, pp Ozisi, M.N. and Orlande, H.R.B., 999, Inverse Heat ransfer: Fundamentals and Applications, aylor & Francis, Pennsylvania. Özisi, M.N., 99, Heat Conduction, nd ed., Wiley, New Yor. Powell, M.J.D., 977, Restart Procedures for the Conugate Gradient Method, Mathematical Programming, Vol., pp Sawaf, B.W., and Ozisi, M.N., 995, Determining the Constant hermal Conductivities of Orthotropic Materials, Int. Comm. Heat Mass ransfer, Vol., pp. 0-. Sawaf, B., Özisi, M.N. and Jarny, Y., 995, An Inverse Analysis to Estimate Linearly emperature Dependent hermal Conductivity Components and Heat Capacity of an Orthotropic Medium, Int. J. Heat Mass ransfer, Vol.8, No. 6, pp ata, R., Bec, J.V. and Scott, E.P., 99, Optimal Experimental Design for Estimating hermal Properties of Composite Materials, Int. J. Heat Mass ransfer, Vol.6, No., pp Copyright 999 by ASME