Corresponding relationships between bird foraging and PSO algorithm.
--> . In addition, the results indicate that optimizing the weights corresponding to the control points can improve the quality of designed shape. However, the MGA has the obvious drawback that the convergence velocity is relatively low.
The PSO algorithm is a kind of biologically inspired algorithm whose search process is similar to foraging of birds and it was proposed in 1995 by Eberhart and Kennedy . The physical model of PSO is very simple and the computational program is easy to be implemented, it also has strong robustness and achieves good performance on computational efficiency and accuracy. In addition, PSO algorithm can well balance the global and local search of particles, which enhance the global convergence of the algorithm. Farahmand et al.  investigated the inverse geometry design of two-dimensional radiative enclosures with diffuse gray surfaces based on the PSO and the retrieval results show that PSO algorithm obtains better performance in satisfying the design goal based in terms of computational accuracy and CPU time compared with MGA. However, the standard PSO algorithm also suffers from easily trapping into local optima in solving high dimensional problems. In order to strengthen the applicability of PSO, some improvements have been proposed and widely applied, including Stochastic PSO (SPSO), Differential Evolution PSO (DEPSO), Multi Phase PSO (MPPSO), and so on. However, to the authors’ best knowledge, there are few reports concerning about the applications of improved PSO algorithms for solving inverse geometry design problems of radiative enclosures.
In this chapter, the application of PSO algorithms in solving inverse geometry design problems of two-dimensional radiative enclosures filled with participating media is investigated. The design goal is to satisfy a uniform distribution of radiative heat flux on the designed surface. The discrete ordinate method (DOM) with a body-fitted coordinate system is used to solve the RTE. The standard PSO, SPSO and DEPSO algorithms are applied to optimize the locations of the control points, and Akima cubic interpolation is adopted to obtain the boundary geometry shape through these points. A typical inverse geometry design test is studied to demonstrate the good performance of PSO algorithms and the effects of corresponding parameters are also discussed.
The remainder of this chapter is organized as follows: the theoretical principles of PSO algorithms are introduced in Section 2. The feasibility of PSO algorithms by four famous benchmark functions is verified in Section 3. The inverse geometry design of two-dimensional radiative enclosures and the influences of the number of control points and the radiative properties of media on the inverse design results are investigated in Section 4. The main conclusions of the researches in this chapter are summarized in Section 5.
Bird individuals will communicate with each other to share their information about food when they are foraging, which can help birds find food faster. The advantages of cooperation of bird swarm are much greater than the disadvantages of competition among bird individuals. Based on the features of bird foraging behavior, Kennedy and Eberhart proposed PSO algorithm in 1995 . The solving process of PSO algorithm is similar to the foraging behavior of birds, and the corresponding relationships are shown in Table 1.
|Bird foraging behavior||PSO algorithm|
|Foraging domain of bird individual||Searching space of each particle|
|Flight speed of bird||Moving speed of particle|
|Location of bird||Location of particle, which represents a solution|
of optimization problems
|Location of food||The best solution of optimization problems|
There are two dominant parameters in the PSO algorithm, namely, the speed and the location of particles. The moving speed decides the direction and distance of the particles, and every location of particles can be considered as the potential a solution of optimization problems. PSO adopts a combination of local and global searches and shares the evolutionary information among particle individuals to find the optimal solution.
At the beginning of the optimization of PSO, every location and velocity of particles are randomly generated. During each iteration, there will be two extreme values. One is best location that an individual particle found so far, which is called local best location. Another is the best location that the whole particle swarm found so far, which is called global best location. The velocity and location of each particle are stochastically accelerated according to these two extremes and the evolutionary formula can be expressed as follows 
According to Eq. (1), we can find that the velocity of
The main procedure of PSO algorithm for solving optimization problems can be carried out according to the following steps:
The flowchart of the basic PSO algorithm is shown in Figure 2.
However, there are some obvious shortcomings in the basic PSO algorithm, such as slow convergence, easy to fall into local optimum, and even the velocities tend to be infinity in some occasions. Thus, many modifications have been proposed to overcome these drawbacks.
There are two important capabilities in PSO algorithm, namely exploration and exploitation of particles. Exploration is the phenomenon that particles leave the original orbit and search for new space. Exploitation is the phenomenon that particles look for better locations along the original track. In order to better take advantage of these two search way, Shi and Eberhart put forward the standard PSO algorithm on the basis of basic PSO in 1998 , in which an inertia weight coefficient
Comparing Eq. (1) with Eq. (3) we can find that basic PSO algorithm is a special circumstance of standard PSO algorithm that inertia weight is set as
In order to overcome the drawback that PSO algorithm converges too early and make sure to reach the goal of global convergence, Zeng and Cui proposed SPSO algorithm in 2004 [30, 31], in which a stopped changing particle is utilized to improve the global searching ability of particle swarms.
In SPSO algorithm, the inertia weight coefficient is set as
According to Eq. (5), it can be found that the local searching ability of SPSO is increased compared with standard PSO. However, the global searching ability is reduced significantly. In order to further strengthen the global searches of SPSO, the algorithm randomly generates a particle in the searching space whose location is
After above updates, the following criterions are executed:
Through the above analysis we can find that there is at least one particle’s location reaches the global best location at a particular iteration, which indicates that at least one particle is randomly generated at each iteration. Therefore, SPSO algorithm has been proved with strong global search ability.
Differential Evolution (DE) algorithm adopts simple differential operation among potential solutions to produce new candidate solution, which is a parallel, direct and stochastic searching technique. It was first proposed for solving Chebyshev polynomials and global optimization problems over continuous spaces by Storn and Price in 1995 . Taking a cue from DE, the mutation operation is introduced into PSO algorithm to overcome the drawback of trapping in local optima, which is called DEPSO.
In DEPSO algorithm, the differential evolution operator is introduced to increase the diversity of particle swarms which is defined as
However, the location obtained by mutation operation maybe a worse result which will cause a bad influence on the search of other particles. In order to make sure the rapidity and stability of DEPSO algorithm, the following judgment should be executed before the location is updated:
(1) If the fitness value of the new location is better than the fitness value of the earlier location, which demonstrates the mutation is successful. Then the location of
(2) If the new fitness value is worse than before, which indicates this mutation is failed. Then the location of
In addition, there is a significant difference between DEPSO and basic PSO that the velocities of particles are not limited in the searching process, which can increase the convergence rate.
In order to test the performance of the above PSO algorithms, four benchmark optimization functions are used for verification whose details are shown in Table 2. The parameters in PSO algorithms are set as follows: the number of particle swarm population is set as , the maximum velocity is set as , the inertia weight coefficient is chose according to Eq. (4). The acceleration constants are set as and in SPSO and DEPSO algorithms, whereas and in standard PSO algorithm, respectively. The objective functions are set as the same as the test functions and the iteration stops upon the fitness value reaches the maximum iteration number
Considering a radiative equilibrium problem in two-dimensional irregular enclosures filled with participating media whose schematic diagram is shown in Figure 4. The curve EF represents the design surface and the design purpose is to produce a uniform distribution of radiative heat flux on the design surface. The bottom surface AD is the heating surface which can considered as the radiative heat source and its temperature is fixed as
In order to optimize the geometry shape of the design surface to meet the specified requirement, the objective function (being equal to the fitness function in PSO algorithms) is defined as the square residuals between the estimated and average dimensionless radiative heat flux values which can be expressed as
To evaluate the optimization results, the relative error is defined as
The design surface is discretized into a series of control points, and Akima cubic interpolation is used to approximate the geometry shape of the surface in optimization process. Akima interpolation is formulated by a cubic polynomial between two control points. First, the prerequisite is introduced as 
and the function
At the endpoints, the value of the function
If the above equations are satisfied, then the cubic polynomial in the subinterval [, ] can be determined as 
|Case 1||(0.0, 1.0)||(0.2, 1.2)||(0.4, 1.5)||(0.6, 1.5)||(0.8, 1.2)||(1.0, 1.0)|
|Case 2||(0.0, 1.0)||(0.2, 1.5)||(0.4, 1.3)||(0.6, 1.3)||(0.8, 1.5)||(1.0, 1.0)|
|Case 3||(0.0, 1.0)||(0.28, 1.2)||(0.41, 1.4)||(0.67, 1.1)||(0.88, 1.3)||(1.0, 1.0)|
In order to test the performance of the Akima cubic interpolation, three interpolation cases are applied, in which six specified points are used as control points whose coordinates are shown in Table 4. The curves in Figure 5 indicate the Akima cubic interpolation can successfully applied for geometry shape fitting.
The boundary shape changes with the optimization of the design surface and the computational domain must be re-meshed, which greatly increase the solving difficulty of the inverse geometry design problems. In addition, the forward radiative heat transfer problem cannot be precisely solved by the normal numerical method. For the purpose of fitting the irregular boundary shape, the DOM with a body-fitted coordinate system is adopted to solve the RTE. For the participating media, the forward can be written as 
which is an integro-differential type, where
The Jacobian matrix is used for coordinate transformation to solve the radiative heat transfer in irregular enclosures
The spatially discretized RTE with a body-fitted coordinate system can be expressed as
The step scheme is applied to solve the above equations, and Eq. (22) can be expressed as
where the subscripts
Eq. (24) can be expressed in matrix form
Consider a non-radiative equilibrium problem in the two-dimensional irregular enclosure filled with participating media. The four boundaries are cold surfaces whose temperature is 0K and all the boundaries are assumed as blockbody. The temperature of media is set as
The inverse geometry design model in Section 4.1 is considered and the standard PSO algorithm is abbreviated as PSO if there is no special instruction. The initial shape of the enclosure is rectangular, whose size is set as . The absorption and scattering coefficients of media are set as m-1 and m-1, respectively. The parameters in the PSO algorithms are set as the same as the ones in test cases of Section 3. The stopping criteria are set as follows: (1) the objective function value is less than 10-7, (2) the iteration times reach a maximum of 1000 and (3) the fitness value remains unchanged in consequent 100 iterations. The initial and final geometry shape of the design surfaces and their dimensionless radiative heat flux distributions are shown in Figure 8(a) and 8(b), respectively. The curves demonstrate that a uniform distribution of radiative heat flux on the design surface is obtained by means of PSO algorithms. Figure 8(c) shows the relative error distributions of the dimensionless radiative heat flux on the design surface. The average relative errors are 0.0310%, 0.0234% and 0.0279%, and the maximum relative errors are 0.0728%, 0.0540% and 0.0807% for PSO, SPSO and DEPSO algorithms, respectively. Overall, the retrieved results clearly reveal that the specified requirement of producing a uniform radiative heat flux distribution on the design surface can be obtained by PSO algorithm.
|Fitness values||Average relative|
In view of the random characteristic of intelligent algorithms, all the tests are repeated 50 trials to compare the performance of PSO algorithms. Table 5 shows the comparison of the results obtained by PSO, SPSO and DEPSO algorithms. It can be found that all the PSO algorithms have reached the special design requirement and both SPSO and DEPSO achieve better performance than the initial PSO in terms of computational accuracy and efficiency.
In order to enhance the computational efficiency of inverse geometry design problems, the effects of corresponding parameters are investigated in this study. For the fact that the design surface is discretized into a series of control points in the inverse design process, the number of control point has a direct impact on the inverse geometry design results. The radiative physical parameters of media are kept as the same as the above typical example and the numbers of control point are set as
|Fitness values||Average relative|
The physical properties of the media have an important influence on the energy transfer and then affect the optimization results of radiative enclosures. The effects of the extinction coefficient and the scattering albedo on the inverse design results are studied here. The scattering albedo of media is fixed as , the extinction coefficients are set as , 3.0 and 5.0, respectively. The initial guessed and final optimized geometry shapes of the design surface are shown in Figure 10. It can be seen that three PSO algorithms achieve similar boundaries and the geometry shapes are significantly different for different extinction coefficients. The extinction coefficient of media is fixed as , the scattering albedos are set as , 0.5 and 0.9, respectively. The optimized geometry shapes of the design surface by means of SPSO are shown in Figure 11. It can be seen that the designed boundaries are very close to each other for different scattering albedos. Tables 7 and 8 list the detailed inverse design results for different extinction coefficients and scattering albedos, respectively. As shown, both the optimized geometry the retrieval results for different extinction coefficients are significantly different, and both the height of the final designed surface and the dimensionless radiative heat flux decrease with the extinction coefficient increases. However, the retrieval results for different scattering albedo are close. Therefore, the scattering albedo of the media has little influence on the inverse geometry design results when the extinction coefficient is determined.
radiative heat flux
relative error (%)
radiative heat flux
relative error (%)
The scattering of media will affect the original transfer direction of radiative heat or energy, so the scattering characteristic of media also has influence on the inverse geometry design of radiative enclosures. The extinction coefficient and scattering albedo of the media are set as and , respectively. The initial and final geometry shapes of the design surface by using SPSO algorithm for three kinds of scattering characteristics of media are shown in Figure 12. It can be found that the optimal geometry shapes of the design surface are close. Table 9 plots the inverse geometry design results for different scattering characteristics. As shown, the values of dimensionless radiative heat flux on the design surface and relative errors are significantly different. Therefore, the scattering characteristics of media mainly affect the radiative heat flux on the boundaries and have no obvious effect on the final geometry shape of inverse radiative design problems.
radiative heat flux
relative error (%)
In this chapter, the basic theoretical principles of PSO algorithm is introduced in detail and three kinds of PSO algorithms—standard PSO, SPSO and DEPSO—are applied for solving the inverse geometry design problem of a two-dimensional radiative enclosure filled with participating media. The design purpose is to produce a uniform distribution of radiative heat flux on the designed surface. The design surface is discretized into a series of control points and the Akima cubic interpolation is used to approximate the geometry shape of the boundary. The radiative heat transfer problem in the irregular enclosures is solved by the DOM with a body-fitted coordinate system. The pre-required radiative heat flux distribution is satisfied by optimizing the positions of control points based on the PSO algorithms. The retrieval results show that PSO algorithms can be successfully applied to solve inverse geometry design problems and SPSO achieves the best performance on computational time. Meanwhile, the scattering albedo and scattering characteristics of media have little effect on the geometry shape of the design surface. To improve the computational efficiency, the number of control points is recommended as
The supports of this work by the National Natural Science Foundation of China (Nos. 51476043 and 51576053), the Major National Scientific Instruments and Equipment Development Special Foundation of China (No. 51327803), 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 also made to the editors and referees who make important comments to improve this chapter.
Submitted: October 6th, 2015 Reviewed: February 2nd, 2016 Published: September 21st, 2016
© 2016 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.