Improved social spider optimization algorithms for solving inverse radiation and coupled radiation–conduction heat transfer problems

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Abstract

A novel bio-inspired swarm algorithm, social spider optimization (SSO), is introduced to solve the inverse transient radiation and coupled radiation–conduction problems for the first time. Based on the original model, five improved SSO (ISSO) algorithms are developed to enhance search ability and convergence velocity. The sensitivity analysis of measured signals with respect to the physical parameters of the medium are described. After which, the SSO and ISSO algorithms are applied to solve the inverse estimation problems in a one-dimensional participating medium. Two cases concerns radiative transfer problems are investigated, in which the radiative source term, extinction coefficient, scattering albedo, and scattering symmetry factor are reconstructed. Furthermore, the coupled radiation–conduction heat transfer model is considered and the main parameters such as the conduction–radiation parameter, boundary emissivity, and scattering albedo are retrieved. All retrieval results show that SSO-based algorithms are robust and effective in solving inverse estimation problems even with measurement errors. Findings also show that the proposed ISSO algorithms are superior to the original SSO model in terms of computational accuracy and convergence velocity.

Introduction

Radiation and coupled radiation–conduction heat transfer exist in various industrial fields. Inverse heat transfer problems have been widely studied in numerous research fields, such as the combustion diagnosis in high-temperature flame, reconstruction of the temperature distribution and the optical parameters in combustion chambers, remote sensing in atmospheric science, optical tomography in medical imaging, and inverse design of radiative enclosures [1], [2], [3], [4], [5], [6]. Meanwhile, a great quantity of inverse techniques have been proposed and developed to solve the problems of inverse heat transfer. Most of these techniques are accomplished by optimizing a certain objective function. The extinction coefficient, scattering albedo, boundary emissivity, scattering phase function, conduction–radiation parameter, particle size, and the pre-desired heat flux distribution on the specified boundary are successfully retrieved [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36].

The theoretical techniques for solving inverse heat transfer problems can be generally classified into two groups: the gradient-based method (or deterministic algorithm) and the random search-based method (or stochastic/evolutionary-based optimization algorithm). Numerous estimation problems have been solved by the gradient-based method due to its high computational efficiency. For example, the conjugate gradient method (CGM) was employed to solve the inverse radiation heat transfer problems for retrieving the radiative properties and temperature distribution of media by Li et al. [8] and Salinas [11]. Howell et al. [17] and Bayat et al. [19] applied CGM to the inverse design of radiative enclosures, and Daun et al. [18] compared the advantages and disadvantages between CGM and other regularization methods for solving inverse design problems. The retrieval results showed that the CGM needed less CPU time with less storage requirements. Neto et al. [14] and Ren et al. [15] studied the inverse estimation of radiative properties and temperature distribution of media through the Levenberg–Marquardt (L–M) method, respectively. Good agreements between measured and estimated measurement signals were obtained. The CGM and L–M methodhad also been extensively investigated for solving the parameter identification [20], [21] and inverse design problems [22], [23] involving coupled radiation–conduction heat transfer. Lots of studies demonstrated that the gradient-based method was robust and efficient in solving inverse heat transfer problems.

However, the process for retrieving the gradient in the above gradient-based optimization techniques is quite difficult. Also, the retrieval results strongly depend on the initially guessed values, and even an unfeasible solution can be obtained if the initial value is unsuitable. In the recent decades, many swarm intelligence algorithms, including differential evolution (DE) [24], [25], genetic algorithm (GA) [26], [27], particle swarm optimization (PSO) [28], [29], [30], ant colony optimization (ACO) [31], [32], [33], krill herd (KH) [34], [35], have been proposed and applied in various industrial fields. These algorithms can effectively overcome the drawbacks and limitations of the above conventional gradient-based techniques. Moreover, the intelligent algorithms can deal with lots of feasible solutions at each iteration and all the processes are performed in parallel. These algorithms are significantly superior to the gradient-based method in terms of achieving global optimal values and computational stability, especially for higher dimensional problems [36], [37]. Li et al. [26] applied GA to estimate the single scattering albedo, optical thickness and phase function in an azimuthally symmetric, absorbing, anisotropically scattering parallel slab simultaneously. All the parameters were accurately reconstructed. The authors pointed out that the inverse radiation problem considered in this study was difficult to be solved by traditional optimization methods, whereas GA was quite robust in solving this optimization task even with measurement errors, demonstrating the superiority of the intelligent algorithm. Bokar [38] studied the simultaneous estimation of the optical thickness and the spatially varying albedo in a one-dimensional (1D) parallel slab filled with inhomogeneous gray medium. The direct problem was solved by the discrete ordinate method (DOM), and the artificial neural networks (ANN) algorithm was utilized to estimate the determined optical thickness and the varying scattering albedo that was expressed in a polynomial form. The retrieval results showed that accurate estimation results could only be obtained in small optical thickness through the ANN algorithm. Qi et al. [30] firstly introduced the PSO algorithm in solving the inverse radiation problems and applied the stochastic PSO (SPSO) algorithm to estimate the source term, extinction coefficient, scattering coefficient, and non-uniform absorption coefficients in a 1D radiating gray plane with gray boundaries. All the above radiation parameters were accurately estimated even with noisy data, which provided a new robust tool for solving inverse radiation problems.

Meanwhile, the intelligent algorithms had also been extended and applied to various inverse coupled radiation–conduction heat transfer problems. Das et al. [27] investigated the simultaneous estimation of the emissivity, temperature distribution, and heat flux on the left boundary in a 1D transient radiation–conduction heat transfer problem through the GA. The effects of the measurement errors and control parameters in the GA such as the population size and the iteration number on the estimation results were also discussed. Chopade et al. [25] simultaneously estimated the extinction coefficient, scattering albedo, emissivity, and conduction–radiation parameter in the participating medium with diffused gray boundaries using the DE algorithm. The DE algorithm achieved superior performance compared with GA in terms of computational accuracy. Qi et al. [39] proposed a hybrid KSM–PSO algorithm, in which the K-means clustering and the simplex method were combined with the standard PSO. The conduction–radiation parameter, scattering albedo and boundary emissivity in 1D semi-transparent medium were simultaneously estimated using PSO algorithms, and the proposed KSM–PSO was proved to be more efficient and accurate than PSO and simplex bare-bones PSO algorithms. However, most of the above intelligent algorithms have the common drawbacks of time–consuming calculation and slow convergence velocity, especially during the final iterations.

A novel bio-inspired optimization technique, called social spider optimization (SSO) algorithm, was proposed in 2013 by Cuevas et al. [40]. In the SSO algorithm, the spiders are divided into two groups according to their gender, and the individuals emulate interaction with one another based on the biological laws of a cooperative colony through a communal web. Each spider is conducted by a set of different evolutionary operators, which mimic different cooperative behaviors that are typically found in the colony. Female spiders tend to present an attraction or dislike toward others. The attraction or dislike is commonly encoded as small vibrations that are critical for the collective coordination of all individuals. The vibrations for a particular spider are determined by the weight of other spiders and the distance between two individuals. Thus, a strong vibration is produced by a spider with good fitness or the neighbor spider of the specified one. Male spiders are divided into dominant and non-dominant populations according to their weights. The dominant male spiders are attracted to the closest female spider, whereas the non-dominant male spiders tend to concentrate upon the center of the male population [40], [41], [42]. In addition, mating, which is performed by dominant male spiders and their neighbor female spiders, is an important operation in SSO. Mating operation is assigned by the roulette method, which allows the individuals to exchange information and increase population diversity [40]. A comprehensive set of 19 functions were tested to examine the performance of the SSO algorithm. All the retrieval results showed that SSO algorithm achieved higher accuracy than those obtained by both PSO and artificial bee colony (ABC) algorithms [40]. Cuevas et al. [43] proposed a novel swarm algorithm, called SSO-C, based on the original SSO to solve constrained optimization tasks. For these problems, a penalty function introduces a tendency term into the original objective function to penalize the constraint violations, thereby solving a constrained task as an unconstrained one. A feasibility criterion was applied to bias the new individuals toward feasible regions. Eight well-defined constrained benchmark functions were tested to assess the performance of SSO-C algorithm, and the retrieval results were compared with those obtained by PSO, ABC and firefly method, which demonstrated the superiority of the SSO-C algorithm. To date, SSO algorithms have been successfully applied to the detection of energy theft [44] and the training of artificial neural networks [45], [46].

However, to the best knowledge of the authors, reports have not yet been conducted about the application of SSO algorithm for solving problems of inverse estimation tasks in radiation or coupled radiation–conduction heat transfer. Thus, this study aims to introduce the SSO algorithm to solve the inverse transient radiation and coupled radiation–conduction problems in an absorbing, scattering, and emitting parallel slab with diffuse and gray boundaries. Five improved SSO (ISSO) algorithms are proposed to accelerate convergence efficiency and search ability, thereby improving the computational performance of the original SSO algorithm. Furthermore, both the original SSO and ISSO algorithms are applied to estimate the multi-parameters in transient radiation and coupled radiation–conduction problems simultaneously.

The remainder of this work is organized as follows. In Section 2, the theoretical overview of the SSO algorithm is introduced thoroughly, and five ISSO algorithms are proposed to improve the search ability of the spider population and accelerate the convergence velocity. Section 3 describes the TRTE in radiation problems and the additional energy equation in conduction problems. The inverse estimation results are also presented in this section. First, the distribution of the radiative source term in radiation problems is estimated by SSO algorithms, and the comparison between SSO and ISSO algorithms is discussed. Afterwards, the sensitivities of the reflectance with respect to the extinction coefficient, scattering albedo, and scattering asymmetry factor in radiation problems are analyzed. Subsequently, these three parameters are simultaneously reconstructed. Finally, the sensitivities of the dimensionless temperature on the boundaries with respect to the conduction–radiation parameter, scattering albedo, and boundary emissivity in coupled radiation–conduction problems are analyzed. Thereafter, these three parameters are simultaneously estimated. The main conclusions and perspectives are presented in Section 4.

Section snippets

SSO algorithm

SSO algorithm is a novel bio-inspired swarm algorithm for solving optimization problems based on the cooperative behavior of social spiders [40]. In SSO algorithm, the spider population is divided into two search groups according to their genders (female and male), and their positions are updated at each iteration according to different evolutionary operators. Meanwhile, the communal web allows the spiders to communicate with each one another to transmit important information, which accelerates

Results and discussion

The SSO algorithms including the original SSO and ISSO models are applied to estimate the radiative source term and radiative properties of the medium in inverse transient radiative and coupled radiative–conductive heat transfer problems. Since measurement error is inevitable in the practical researches, the random standard deviation is added into the radiative signals obtained by the direct problem to show the influence of measurement errors on the simulations, which is expressed as:Ymea=Yexa+

Conclusions

The novel bio-inspired swarm algorithm SSO is introduced to solve the estimation problems of inverse transient radiation and coupled radiation–conduction for the first time. Based on the original SSO algorithm, five ISSO algorithms are developed to accelerate the convergence velocity and the global search ability of the inverse model. Furthermore, the SSO algorithms are applied to solve the inverse estimation problems in a 1D participating medium. The sensitivity analysis of the measured

Acknowledgements

The support of this work by the National Natural Science Foundation of China (No. 51576053, 51476043) and the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (No. 51421063) are gratefully acknowledged. A very special acknowledgement is made to the editors and referees who make important comments to improve this paper.

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