EngOpt 2008 - International Conference on Engineering Optimization Rio de Janeiro, Brazil, 01 - 05 June 2008.
Solution of Inverse Radiative Transfer Problems in Two-Layer Participating Media with Differential Evolution Fran S´ ergio Lobato1 , Valder Steffen Jr2 , Antˆ onio Jos´ e da Silva Neto3 1,2
School of Mechanical Engineering, Universidade Federal de Uberlˆ andia ´ Av. Jo˜ ao Naves de Avila 2121, Campus Santa Mˆ onica, P.O. Box 593 38400-902, Uberlˆ andia, MG. Brazil 1
[email protected], 2
[email protected] 3 Department of Mechanical Engineering and Energy, Instituto Polit´ecnico IPRJ Universidade do Estado do Rio de Janeiro, Rua Alberto Rangel, s/n◦ , Vila Nova 28630-050, Nova Friburgo-RJ, Brazil 3
[email protected]
1. Abstract In the present work the Differential Evolution Approach (DE) is used for the estimation of radiative properties in two-layer participating media. This physical phenomenon is modeled by an integro-differential equation known as Boltzmann equation. A review of the optimization technique used is presented. The direct radiative transfer problem is solved by using the Collocation Method. Then, some studies are presented aiming at illustrating the efficiency of these methodologies in the treatment of an inverse problem of radiative transfer. The results obtained from the solution of the inverse problem are compared with those obtained from a hybridization of Simulated Annealing and Levenberg-Marquardt methods. The preliminary results indicate that the proposed approach characterizes a promising methodology for this type of inverse problem. 2. Keywords: inverse problem, differential evolution, radiative transfer, two-layer participating media. 3. Introduction The use of optimization numerical techniques for parameter identification has increased significantly due to the difficulty in building theoretical models that are able to represent physical phenomena under real operating conditions, satisfactorily. Basically, this problem consists of minimizing the difference between experimental and calculated values. In the literature, three research lines are proposed for the solution of the parameter identification by using optimization techniques: the Deterministic, the Non-Deterministic and the Hybrid Approach [1, 2, 3, 4]. In this sense, both the direct and inverse radiative transfer problems constitutes significant contributions in the context of one-dimensional plane-parallel [5, 6, 7], and two-dimensional media [8, 9], and radiative transfer in composite layer media [10, 11], which are devoted to applications in scientific and technological areas that are related to environmental sciences [12], parameter estimation [13], tomography [14], and others work. In the present contribution DE is used for the parameter estimation of a two-layer participating media, with diffusely reflecting boundary surfaces and interface. The results obtained with this methodology are compared with other approaches, namely the Simulated Annealing Algorithm (SA), the LevenbergMarquardt Method (LM) and the hybridization Simulated Annealing - Levenberg-Marquardt (SA-LM), as obtained by Soeiro and Silva Neto [15]. The problem of two-layer participating media and its mathematical formulation is presented in Section 4. The inverse problem is formulated in Section 5. A review of the Differential Evolution method is presented in Section 6. The results and discussion are described in Section 7. Finally, the conclusions and suggestions for future works conclude the paper. 4. The Transfer Problem in Two-Layer Participating Media Consider a two-layer participating media, with diffusely reflecting boundary surfaces and interface, as shown in Figure 1.
1
ρ1
f1 (µ)
ρ4
ρ2 ρ 3
f2(µ)
2
1 0
L1+ L2
L1
Figure 1: Two-layer semitransparent medium.
The medium is subjected to external irradiation on both sides with intensity f1 (µ) at x=0 and f2 (µ) at x=L1 +L2 , where µ is the cosine of the polar angle, L1 and L2 represent the thickness of layers 1 and 2, respectively and ρj are the diffuse reflectivities, with j=1, ..., 4. The mathematical formulation of the problem considered is briefly described below [15]: Region 1: µ
∂I1 (x, µ) σs1 + β1 I1 (x, µ) = ∂x 2
Z
1 −1
I1 (x, µ0 ) dµ0 , Z
1
I1 (0, µ) = f1 (µ) + 2ρ1
0 < x < L1 ,
I1 (0, −µ0 ) µ0 dµ0 ,
0
Z
1
I1 (L1 , µ) = (1 − ρ3 ) I2 (L1 , µ) + 2ρ2
−1 ≤ µ ≤ 1
µ>0
I1 (L1 , µ0 ) µ0 dµ0 ,
(1) (2)
µ0
(4) (5)
0 1
I2 (L1 + L2 , µ) = f2 (µ) + 2ρ4
I2 (L1 + L2 , µ0 ) µ0 dµ0 ,
µ
