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1 Numerical Heat Transfer, Part A, 50: , 2006 Copyright # Taylor & Francis Group, LLC ISSN: print= online DOI: / INVERSE RADIATION DESIGN PROBLEM IN A TWO-DIMENSIONAL RADIATIVELY ACTIVE CYLINDRICAL MEDIUM USING AUTOMATIC DIFFERENTIATION AND BROYDEN COMBINED UPDATE Ki Wan Kim and Seung Wook Baek Division of Aerospace Engineering, Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Yuseong-Gu, Taejon, Republic of Korea Inverse radiation design analysis in a two-dimensional, concentric, cylindrical, absorbing, emitting and scattering medium has been conducted, given desired boundary conditions on the design surface. The finite-volume method was adopted to solve the radiative transfer equation and the energy conservation equation in the direct problem, while the Levenberg- Marquardt method was used to solve a set of equations, which are expressed by errors between estimated and desired radiative heat fluxes on the design surface. In order to diminish the computational time required for calculating sensitivity matrix, automatic differentiation as well as the Broyden combined update were utilized. INTRODUCTION Inverse radiation analysis has been utilized to estimate radiative properties when the other measured radiative data were available or to deduce the optimal condition satisfying a design goal [1]. In the former case, medium properties such as extinction coefficient, absorption coefficient, single scattering albedo, phase function, optical depth, and temperature, as well as surface properties such as boundary emissivity and temperature, have been estimated from measured intensities exiting from the boundary or temperature inside the medium [2 6]. This concept has also been applied to optical tomography for examining cross-sectional images of the human body in medical science [7]. In the latter case, boundary or source design problems have been solved to find optimal distribution of boundary condition or source strength satisfying the desired condition on design surfaces in irregular as well as regular geometries containing either a nonparticipating or a gray participating medium [8 12]. However, cylindrical geometry problems have not attracted as much Received 1 October 2005; accepted 23 November The financial assistance by the Combustion Engineering Research Center at the Korea Advanced Institute of Science and Technology is gratefully acknowledged. Address correspondence to Seung Wook Baek, Division of Aerospace Engineering, Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Guseong-Dong, Yuseong-Gu, Daejeon , Republic of Korea. swbaek@kaist.ac.kr 525

2 526 K. W. KIM AND S. W. BAEK NOMENCLATURE D m ci direction weights ~e x ;~e y ;~e z unit vector in x, y, z directions E b emissive power ð¼pi b Þ; W=m 2 I radiation intensity, W=m 2 sr I b blackbody radiation intensity, W=m 2 sr L cylinder height, m ~n i unit vector normal to control-volume surface i n e number of elements on design surface n h number of heaters n p number of unknown parameters P a vector of unknown parameters q R radiative heat flux, W=m 2 ~r position vector of intensity r in inner cylinder radius, m r out outer cylinder radius, m _ s direction vector of intensity S equations set T temperature of participating medium, K T i boundary temperature at the surface i X sensitivity matrix b o extinction coefficient ð¼j a þ r s Þ; m 1 DA i ; DV ith surface area and volume of the control volume DX control angle boundary emissivity at the surface i e i h H j a r s u o u X U x o polar angle measured from the z direction, rad nondimensional medium temperature absorption coefficient, m 1 scattering coefficient, m 1 space variable in the azimuthal direction measured from x axis, rad angular variable in the azimuthal direction measured from x 0 axis, rad scattering phase function single scattering albedo Subscripts d design surface e estimated value e; w; n; s; t; b index of control-volume surfaces E; W; N; S; T; B index of neighboring nodal points h heater surface P calculating nodal point R radiation ref reference value w wall Superscripts k iteration number m; m 0 radiation direction m þ ; m boundaries of the control volume * nondimensional quantity attention [12], while the scattering term in the radiative transfer equation was not taken into account either [10, 11]. Various inverse methods have been adopted to obtain a stable solution in spite of the ill-posed characteristic of inverse problems. Even though matrix solver-based methods do not need an iteration procedure, special treatment is required to deal with the ill-conditioned nature of the equations [8, 9]. On the other hand, optimization technique-based methods are emerging as a stable inverse method due to multidimensionality as well as complexity in inverse problems [2 7, 10, 11]. A successful estimation of unknown properties depends on the magnitude of the sensitivity coefficient. The bigger its magnitude, the easier it is to estimate the inverse solutions. Furthermore, a reduction in computational time for calculating those values is one of the main issues in gradient-based optimization technique. The sensitivity method deals with the calculation of sensitivity coefficients. So far, equation-based sensitivity methods, such as sensitivity problems [1, 4, 5], boundary-value problems [1, 10], or adjoint problems [1], have been prevalent. Recently, code-based methods have received much attention due to their advantages [7]. These methods do not need to derive an adjoint problem or a sensitivity problem, so a discretization procedure of relevant equations is not necessary. Computational code

3 INVERSE RADIATION DESIGN PROBLEM 527 for the sensitivity matrix can be extracted directly from the code of the direct problem. Automatic differentiation belongs to this category. The goal of this study is to reduce the computational time for calculating the sensitivity matrix. Our previous study shows that the effect of the precision of the sensitivity matrix on estimation accuracy is not crucial [13]. Because of this, the Broyden combined update, which is one of the secant methods, is used with automatic differentiation to further shorten the computational time required to calculate the sensitivity matrix. Therefore, the major contribution of this work is to propose automatic differentiation with Broyden combined update as an alternative sensitivity method and to extend the inverse radiation design problem to a two-dimensional cylindrical enclosure as a successful application. ANALYSIS AND MODELING Model Description Figure 1 shows a concentric cylindrical enclosure which is filled with an absorbing, emitting, and scattering gray medium, and its simplified geometry with boundary conditions. The walls are diffusely emitting and reflecting gray walls. The direct problem is to calculate the radiative heat flux distribution on the design surface, given boundary conditions, such as temperature, wall emissivity, and medium properties such as absorption and scattering coefficient. The temperature distribution inside the medium is determined by solving the energy conservation equation in radiative equilibrium, since radiation is the only dominant heat transfer in this work. Radiative heat flux over the wall including the heater surface is obtained by integrating the dot product of intensity and the unit normal vector over all solid angles, while the intensity distribution is acquired by solving the radiative transfer equation. In this work, the finite-volume method is adopted for solving the radiative transfer equation. In the inverse problem, the goal is to find the radiative heat flux distribution over the heater surface which satisfies the desired radiative heat flux as well as temperature on the design surface. The Levenberg-Marquardt method is used to solve a set of equations expressed by errors between the estimated and desired radiative heat flux on the design surface. In order to reduce the computing time for the sensitivity matrix, a combined method of automatic differentiation and Broyden combined update is adopted in this study. Detailed description of mathematical formulations and numerical methods used in this work is given below. Radiative Transfer Equation The nondimensionalized radiative transfer equation (RTE) governing radiation intensity for a gray medium at any position ~r along a path _ s through an absorbing, emitting, and scattering medium is given by di ð~r; s _ Þ ds þ I ð~r; _ s Þ¼ð1 x o ÞH 4 ð~rþþ x Z o Uð _0 s ; _ s ÞI ð~r; _ s ÞdX 0 4p 4p ð1þ

4 528 K. W. KIM AND S. W. BAEK Figure 1. Schematic of the physical system and its simplified geometry: (a) physical system; (b) simplified geometry (L ¼ 4, r out ¼ 1, r in ¼ 0.5). where H ¼ T T ref I ¼ I I b ðt ref Þ x o ¼ r s b o s ¼ b o s b o ¼ j a þ r s Here, Uð _0 s ; _ s Þ is the scattering phase function for radiation from incoming direction _0 _ s to scattered direction s. It is approximated by a finite series of Legendre polynomials such that Uð s _0 ; s _ Þ¼Uðcos WÞ ¼ XJ j¼0 C j P j ðcos WÞ ð2þ where the C j s are the expansion coefficients, and J is the order of the phase function.

5 INVERSE RADIATION DESIGN PROBLEM 529 The boundary condition for a diffusely emitting and reflecting wall can be written as I ð~r w ; _ s Þ¼e w ð~r w ÞH 4 ð~r w Þþ 1 e Z wð~r w Þ I ð~r w ; _0 s Þj _0 s ~n w j dx 0 ð3þ p _0 s ~nw <0 where e w is the wall emissivity and ~n w, which has a positive value when the ray travels from the wall to medium, is the unit normal vector to the wall. Equation (3) can also be expressed as a form of prescribed radiative heat flux: Z I ð~r w ; _0 s Þj _0 s ~n w j dx 0 ð4þ I ð~r w ; s _ Þ¼q R ð~r w Þþ 1 p _0 s ~nw <0 Here, nondimensional heat flux is q R ¼ q R=rTref 4. The radiative heat flux on the design surface is calculated as follows: q R ð~r w;d Þ¼ 1 Z I ð~r w ; _ s Þð~n w;d _ s Þ dx p X¼4p ð5þ Finite-Volume Method for Radiation In solving the RTE, the finite-volume method (FVM) is adopted for its convenience in selecting the solid angle while guaranteeing an exact global conservation of radiative energy. Chui s way to treat the axisymmetric geometry is employed [14, 15]. To obtain the discretization equation, Eq. (1) is integrated over a control volume, DV, and a control angle, DX m, in the axisymmetric orthogonal grid as shown in Figure 2. By assuming that the magnitude of the intensity in a control volume and control angle is constant, the following finite-volume formulation can be obtained: X I ;m i DA i D m ;m ci ¼ð IP þ S ;m r;p ÞDVDXm ð6aþ where i¼e;w;n;s;t;b Z u mþ D m ci ¼ X u m X Z h mþ h m ð s _ ~n i Þ sin h dh du X ð6bþ _ s ¼ sin h cos ux ~e x þ sin h sin u X ~e y þ cos h~e z ð6cþ ~n e ¼ sin u o;p ~e x cos u o;p ~e y ð6dþ ~n w ¼ sin u o;p þ~e x þ cos u o;pþ ~e y ð6eþ ~n n ¼ ~n s ¼ cos u o;p ~e x þ sin u o;p ~e y ð6f Þ

6 530 K. W. KIM AND S. W. BAEK Figure 2. Angular control angle and spatial control volume: (a) angular control angle; (b) spatial control volume. ~n t ¼ ~n b ¼ ~e z ð6gþ S ;m r ¼ð1 x o ÞH 4 þ x Z o I ;m0 U m 4p 0!m dx 0 X 0 ¼4p ð6hþ DX m ¼ Z u mþ X u m X Z h mþ h m sin h dh du X ð6iþ To relate the intensities on the control-volume surfaces to a nodal one, the step scheme, which is not only simple and convenient but also ensures positive intensity, is

7 INVERSE RADIATION DESIGN PROBLEM 531 adopted. Then, the final discretized equation for the FVM becomes a m P I X ;m P ¼ a m I I ;m I þ b m P ð7aþ I¼E;W;S;N;T;B a m I ¼ DA i D m ci;in ð7bþ a m P ¼ X DA i D m ci;out þ DVDXm i¼e;w;s;n;t;b ð7cþ b m P ¼ðS;m r;p ÞDVDXm ð7dþ where D m ci;out ¼ ZDX m ð~n i s _ ÞdX ~n i s _ > 0 ð7eþ D m ci;in ¼ ZDX m ð~n i s _ ÞdX ~n i s _ < 0 ð7f Þ The spatial and angular domains are discretized into control volumes and 8 20 control angles, and temperature inside the domain is calculated by solving the energy conservation equation in radiative equilibrium as follows: rq R ¼ 4b oð1 x o Þ H 4 1 Z I dx ¼ 0 ð8þ 4p 4p Levenberg-Marquardt Method (LMM) In applying the quasi-newton method to inverse analysis, a set of equations has to be formulated using the difference between the estimated and desired heat flux on the design surface as follows: S i ðpþ ¼q R;e;i ðpþ q R;d;i for i ¼ 1;...; n e ð9þ Here, P is an unknown parameter, which is radiative heat flux on the heater surface, q R;h, in this work. In the above equation, S can be expressed in Taylor series, and should be zero to find appropriate parameters such that SðPÞ ffis 0 ðp 0 ÞþrSðPÞðP P 0 Þ¼0 ð10þ where rsðpþ ¼ qsðpþ qp which is the matrix of sensitivity coefficients, X. ¼ qq R;e ðpþ qp

8 532 K. W. KIM AND S. W. BAEK After some manipulation of Eq. (10), we can derive the following relation: P kþ1 ¼ P k ðx k Þ 1 S k ð11þ Here, it must be noted that the above relation may be applied only for the case that the number of equations is the same as that of the unknown variables. However, based on the fact that the number of equations can be greater than that of the unknown variables in the inverse analysis, Eq. (11) is modified as follows: P kþ1 ¼ P k ðx k Þ T X k 1 ðx k Þ T S k ð12þ The Levenberg-Marquardt method has a damping term to cope with ill-posed characteristics such that P kþ1 ¼ P k ðx k Þ T X k þ m k X k 1 ðx k Þ T S k ð13þ where m is a positive scalar called the damping parameter, and X k ¼ diag½ðx k Þ T X k Š is a diagonal matrix [1]. Automatic Differentiation (AD) No matter how complicated the objective function, it is executed on a computer as a sequence of code events containing only elementary operations such as addition, subtraction, multiplication, or division. By applying the chain rule on every code event, sensitivity coefficients can be calculated. Based on this fact, a program code for the calculation of sensitivity coefficients can be extracted from that of the direct problem. For this reason, this method does not need to derive an adjoint problem or a sensitivity problem, so a discretization procedure for these problems is not necessary either. Rather, we need the following two steps for automatic differentiation. First, the objective function for differentiation should be broken into its most elementary forms. Each elementary form permits only one computational operation, such as addition, subtraction, multiplication, or division. Second, the code event is sequentially differentiated in the form of total derivatives. There are two approaches to calculate a sensitivity matrix in AD, i.e., source transformation and operator overloading. The former is to change the source code following the previously described manner such that the source code is decomposed into elementary code events. The latter does not decompose the source code, but instead uses a derived type of modern complier. It needs a modern complier to overload elemental operators such as intrinsic functions in FORTRAN. Since the variables for differentiation are a derived type comprised of the function value, the first, and the second-order derivatives, redefinition of operators is needed. In this approach, all we have to do is define all variables involved in the calculation of differentiation in the derived type. This requires, however, additional computational resources, since all variables involved in the calculation of differentiation should be defined in derived type. We adopted ADIFOR as the source transformation type in this work [16].

9 There are two modes, i.e., forward and backward, according to the direction for applying the chain rule to code events. In this study, forward mode is adopted, and the general formulation for differentiation by the chain rule in forward mode is written by qs i qe j ¼ XN k¼n p þ1 where N is the number of code events. INVERSE RADIATION DESIGN PROBLEM 533 qs i qe k i ¼ 1;...; n e and j ¼ 1;...; n p ð14þ qe k qe j Broyden Combined Update (BC) The various secant methods have been studied by many researchers for their convenience in calculating the gradient value [17, 18]. They utilize the function values of present and previous iterations to obtain the gradient information. Broyden s method for nonlinear systems belongs to the present type. The Levenberg- Marquardt method finds the solution with the following iteration procedure: P kþ1 ¼ P k þ D k ð15þ D k ¼ ðx k Þ T X k þ m k X k 1 ðx k Þ T S k ð16þ The Broyden method approximates X k by B k. It does not involve computing derivatives at all. Moreover, B k is obtained from B k 1 using simple procedures, so called, update. Broyden s good method (BGM) is defined by B kþ1 ¼ B k þ ðy k B k D k ÞD T k D T k D k ð17þ Here, y k ¼ SðP kþ1 Þ SðP k Þ is the difference in function values of the previous and present iterations. Additionally, the definition of Broyden s bad method (BBM) is B kþ1 ¼ B k þ ðy k B k D k Þy T k B k y T k B kd k ð18þ The name of the method does not indicate that BGM is superior to BBM. In [17], a combined method was devised to choose BGM or BBM according to the test rule as defined by jd T k D k 1j jd T k ðx kþ 1 y k j < jyt k y k 1j y T k y k ð19þ If the above condition is satisfied, BGM is applied, otherwise, BBM is selected.

10 534 K. W. KIM AND S. W. BAEK The Broyden combined update turned out to be superior to BGM or BBM alone. In order to use this method, the first jacobian value, B 0, should be provided using finite-difference (FD) approximation or AD. Inverse Analysis Procedure In order to estimate the radiative heat flux distribution over the heater surface satisfying the desired radiative heat flux and temperature on the design surface, all properties are assumed to be known except the radiative heat flux distribution over the heater surface. The computational algorithm for the inverse problem can be summarized as follows. Suppose that the desired radiative heat flux q R;d ¼ðq R;d;1 ; q R;d;2 ;...; q R;d;n e Þ and temperature are given at each element on the design surface and an initial guess P 0 is available. The solution is sought by the following iterative steps. 1. Knowing P k, compute I ð~r; _ s Þ in Eq. (1) by solving the direct problem with Eq. (4) for the heater surface and Eq. (3) for other boundaries. 2. Compute internal temperature distribution H from Eq. (8) and radiative heat flux on the design surface q R;d given by Eq. (5). 3. Check the stopping criterion. If max½ðp k P k 1 Þ=P k Š10 6, all steps are terminated, otherwise, continue the inverse procedure. 4. Compute sensitivity matrix X with methods of AD from Eq. (14) or BC from Eqs. (17), (18), and (19). 5. Knowing X k, compute P kþ1 using Eq. (13). Replace k by k þ 1 and return to step 1. RESULTS AND DISCUSSION Validation of FVM Code for Radiation FVM code was applied for validation to infinite concentric cylinders with isothermal black boundaries in radiative equilibrium, since results obtained by the Monte Carlo method [19] and the discrete ordinates method (DOM) [20] are available in the literature. At radiative equilibrium, the temperature field is computed from Eq. (8), given the ratio of inner to outer radius. In order to simulate infinite concentric cylinders, cyclic condition was imposed on the east and west boundaries, that is, outward intensity on the east boundary was set as inward intensity on the west boundary if ray direction is the same, and vice versa, respectively. Emissive power distribution for various ratios of inner to outer radius was computed in comparison with other results as follows: E b ¼ H4 ðrþ H 4 ðr out Þ H 4 ðr in Þ H 4 ðr out Þ ð20þ Figure 3 shows the results acquired by various methods: Monte Carlo, DOM, and the present FVM. The best agreement between FVM and Monte Carlo is

11 INVERSE RADIATION DESIGN PROBLEM 535 Figure 3. Emissive power distribution inside medium for various ratios of inner to outer radius (s ¼ 2, black walls). obtained at radius ratios of 0.01 and 0.1, while FVM is better than DOM. Through this comparison, the computational code made here for radiation could be validated for future analysis. Inverse Problem In order to simulate desired radiative heat flux, the direct problem was solved using Eqs. (1), (3), (5), and (8), given boundary conditions and medium properties as shown in Figure 1. After obtaining simulated desired radiative heat flux, inversely we tried to estimate boundary temperature over the heater surface to see if the inverse method works properly, and then radiative heat flux over the heater surface was also estimated with boundary condition of Eq. (4) over the heater surface. As shown in Figure 4, the boundary temperature over the heater surface is estimated very accurately, while the radiative heat flux, which satisfies the desired heat flux on the design surface, could also be found. The result has the same trend as in [10]. However, in this study the downward direction is defined as positive for radiative heat flux over the design and heater surface. Comparison of computational time for the sensitivity matrix was carried out to investigate the efficiency of AD and BC. Finite-difference approximation was adopted as reference, while the forward approximation was used as follows

12 536 K. W. KIM AND S. W. BAEK Figure 4. Estimated temperature and radiative heat flux over heater surface and desired radiative heat flux on design surface. (e ¼ 10 6 is used): qs i qp j ¼ qq e;i qp j q e;i ðp 1;...; P j þ ep j ;...; P np Þ q e;i ðp 1;...; P j ;...; P np Þ ep j for i ¼ 1;...; n e and j ¼ 1;...; n p ð21þ Maximum optimization level was used in the FORTRAN compiler, using a Pentium IV 3.0-GHz processor. All sensitivity methods satisfied the stopping criterion at the same iteration number, while BC needed the shortest computational time as shown in Table 1. The result shows that BC is efficient as well as accurate enough to estimate unknown parameters, so we adopted BC as the sensitivity method for future inverse analysis, while AD is used for B 0. If iteration number increases, the difference in computational time among sensitivity methods will also increase. Table 1. Comparison of iteration number and computational time for three cases of LMM þ FD, LMM þ AD, and LMM þ BC LMM þ FD LMM þ AD LMM þ BC (AD used for B 0 ) Initial value Iteration no. CPU time (s) Iteration no. CPU time (s) Iteration no. CPU time (s) , , ,061.39

13 INVERSE RADIATION DESIGN PROBLEM 537 Figure 5. Comparison of estimated radiative heat flux over heater surface satisfying desired condition for various damping parameters at k ¼ 2 and radiative heat flux on design surface in case of L=r out ¼ 2. Effect of Ratio of Cylinder Length to Outer Radius The effect of the ratio of cylinder length to outer radius on the radiative heat flux distribution was considered at various dimensionless positions defined as z ¼ z=l over the heater surface. When the ratio was smaller than the reference value, L=r out ¼ 4, some damping was necessary to obtain a smooth and stable solution. For this reason, in order to select an appropriate damping parameter, various values were applied, especially to the case of L=r out ¼ 2, as shown in Figure 5. The results for values of 0.01 and 0.1 satisfied the desired radiative heat flux over the design surface. However, since design engineers usually prefer a more uniform distribution, the value of 0.1 is selected as damping parameter. Figure 6 shows radiative heat flux distributions over the heater and design surface for the various ratios of cylinder length to outer radius. As the ratio increases, the radiative heat flux distribution over the heater surface becomes more uniform. This result indicates that the desired radiative heat flux over the design surface can be achieved with a heater with more uniform heat flux if the length of the heater surface is extended. Effect of Emissivity at Side Walls When emissivity at side walls 3 and 4 is changed, the variation of radiative heat flux over the heater surface which minimizes the difference between estimated and desired heat flux over the design surface, is shown in Figure 7. The result shows that the magnitude of radiative heat flux over the heater surface near the side walls becomes bigger than that for the reference value, e 3;4 ¼ 0:8, when the wall emissivity

14 538 K. W. KIM AND S. W. BAEK Figure 6. Estimated radiative heat flux distributions over heater surface satisfying desired condition for various ratios of cylinder length to outer radius and resulting radiative heat flux on design surface. Figure 7. Estimated radiative heat flux distributions over heater surface satisfying desired condition for various wall emissivities and resulting radiative heat flux on design surface.

15 INVERSE RADIATION DESIGN PROBLEM 539 increases. Even though the emissivity was significantly changed, corresponding inverse solutions satisfying the desired heat flux were found. Effect of Temperature at Side Walls When temperature at side walls 3 and 4 increases, the magnitude of radiative heat flux over the heater surface near the side walls becomes smaller than that for the reference value, H 3;4 ¼ 0:5, as shown in Figure 8. The effect is smaller than that of wall emissivity, because the sensitivity of wall emissivity is greater than that of wall temperature. In the case of H 3;4 ¼ 0:7, damping is needed to obtain the stable inverse solution at k ¼ 2. It is also found that inverse solutions which satisfy the desired condition are obtained even when side wall temperatures are varied. Effect of Phase Function If intensities are scattered toward the forward direction, more energy is emitted to the design surface. For this reason, less radiative heat flux from the heater will be necessary to satisfy the desired condition for forward scattering, and vice versa. The effect of phase function on radiative heat flux over the heater surface is the smallest compared with previous cases, as shown in Figure 9. This means that the sensitivity of phase function with respect to radiative heat flux over the design surface is the smallest. Figure 8. Estimated radiative heat flux distributions over heater surface satisfying desired condition for various side wall temperatures and resulting radiative heat flux on design surface.

16 540 K. W. KIM AND S. W. BAEK Figure 9. Estimated radiative heat flux distributions over heater surface satisfying desired condition for various phase functions and resulting radiative heat flux on design surface. Effect of Single Scattering Albedo Additionally, the effect of variation of single scattering albedo was negligible, because in the case of radiative equilibrium the RTE with isotropic scattering does not contain single scattering albedo. This implies that the absorption and scattering coefficients do not affect the radiative heat flux distribution over the heater surface. Example of a Real Design Problem The methodology used in this study was applied to a real design problem in which an inverse solution is not known a priori. On the design surface, two boundary conditions of uniform dimensionless temperature, H 1 ¼ 0:5, and radiative heat flux, q R;d ¼ 0:4 or 0.7, are imposed. In order to inspect convergence more carefully, a damping parameter of m ¼ 0:1 was applied. Figure 10 shows the inverse solution satisfying each imposed condition. At early iteration stage of k ¼ 2, the distribution is smooth owing to damping. However, as iteration number increases, nonsmooth variation occurs near the edge position. Resulting radiative heat flux distributions over the design surface calculated with estimated heat flux over the heater surface are shown in Figure 11. Desired uniform heat fluxes are achieved at the final stage. However, even with the early-staged smooth solutions for k ¼ 2, desired heat flux distributions could be achieved except at both end regions. This minor shortcoming can be easily eliminated though the extension of heater length, as discussed previously. Additionally, the increment in boundary temperature would reduce the dramatic increase of radiative heat flux at end regions over the heat surface.

17 INVERSE RADIATION DESIGN PROBLEM 541 Figure 10. Estimated radiative heat flux distributions over heater surface satisfying various desired conditions on design surface. Figure 11. Resulting radiative heat fluxes on design surface.

18 542 K. W. KIM AND S. W. BAEK CONCLUSIONS Inverse radiation design analysis in a two-dimensional, concentric, cylindrical, absorbing, emitting, and scattering medium has been conducted, given desired boundary conditions on the design surface. By adopting the automatic differentiation and the Broyden combined update as sensitivity method, the computational time for calculating the sensitivity matrix could be reduced up to one-third compared with that required for the finite-difference approximation. Various parametric analyses have shown that in most cases the radiative heat flux over the heater surface satisfying the desired condition over the design surface could be well estimated using the current inverse analysis. Stable and smooth solution could be obtained by controlling the damping parameter used in the Levenberg-Marqurdt method, even for ill-posed cases. Finally, the inverse method was applied to a real design problem to get the radiative heat flux over a heater surface satisfying uniform heat flux and temperature distribution on the design surface. Even with the early-staged smooth solutions, the desired heat flux distributions could be achieved except at both end regions. This minor trouble could be successfully eliminated though extension of the heater surface length. REFERENCES 1. M. N. Özisik and H. R. B. Orlande, Inverse Heat Transfer, chap. 1, Taylor & Francis, New York, H. Y. Li and C. Y. Yang, A Genetic Algorithm for Inverse Radiation Problems, Int. J. Heat Mass Transfer, vol. 40, pp , N. R. Ou and C. H. Wu, Simultaneous Estimation of Extinction Coefficient Distribution, Scattering Albedo and Phase Function of a Two-Dimensional Medium, Int. J. Heat Mass Transfer, vol. 45, pp , H. Y. Li, An Inverse Source Problem in Radiative Transfer for Spherical Media, Numer. Heat Transfer B, vol. 32, pp , H. M. Park and T. Y. Yoon, Solution of the Inverse Radiation Problem Using a Conjugate Gradient Method, Int. J. Heat Mass Transfer, vol. 43, pp , K. W. Kim, S. W. Baek, M. Y. Kim, and H. S. Ryu, Estimation of Emissivities in a Two- Dimensional Irregular Geometry by Inverse Radiation Analysis Using Hybrid Genetic Algorithm, J. Quant. Spectrosc. Radiat. Transfer, vol. 87, pp. 1 14, D. K. Alexander and H. H. Andreas, Optical Tomography Using the Time-Independent Equation of Radiative Transfer Part 2: Inverse Model, J. Quant. Spectrosc. Radiat. Transfer, vol. 72, pp , M. R. Jones, Inverse Analysis of Radiative Heat Transfer Systems, J. Heat Transfer, vol. 121, pp , J. R. Howell, O. A. Ezekoye, and J. C. Morales, Inverse Design Model for Radiative Heat Transfer, J. Heat Transfer, vol. 122, pp , S. S. M. Hosseini, S. H. Mansouri, and J. R. Howell, Inverse Boundary Design Radiation Problem in Absorbing-Emitting Media with Irregular Geometry, Numer. Heat Transfer A, vol. 43, pp , S. S. M. Hosseini and S. H. Mansouri, Inverse Design for Radiative Heat Source in Two-Dimensional Participating Media, Numer. Heat Transfer B, vol. 46, pp , 2004.

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