Improved PSO algorithm for shape and sizing optimization of truss structure.

Li, Yancang ; Peng, Yang ; Zhou, Shujing 等

Introduction

The engineering structure optimization can improve the design quality, shorten the design cycle, and cut down the engineering cast (Cai et al. 2011; Gandomi, Yang 2011). The traditional structure optimization algorithms are mainly classified into the mathematical programming (MP) algorithms and the mechanics criterion (MC) algorithms. The former requires the complex analysis, and for multi-variable optimization problems, it needs large amount of computation, which limits its use in large complex engineering structure design optimization. Though the MC has clear concepts, it is not stable because it easily falls into premature convergence with the increase of iterations (Huang, Meng 2009; Sesok, Belevicius 2008; Wang et al. 2002b).

How to solve the problem and provide novel methods for the structures optimization are urgent questions for researchers and engineers. Up to date, lots of works have been done on this, by the attention and efforts of researchers in corresponding fields, many algorithms and models were proposed (Hadidi et al. 2011; Hagishita, Ohsaki 2009; Kaveh, Talatahari 2010, 2011, 2012; Kaveh et al. 2008; Sonmez 2011; Takada 2012; Talatahari et al. 2012; Wang, Ohmori 2012; Xia et al. 2013; Zhou, Gao 2012). In some degree, all these studies give rise to the structural optimization, but we still have a long way to go. This has signified the need for the related study.

According to this, a new method--the improved PSO algorithm--was introduced to solve the problem. As a population-based stochastic approach for solving continuous and discrete optimization problems, especially the complicated combinational optimization problems, the particle swarm optimization (PSO) algorithm, proposed by Reeves (1983), is a new swarm intelligence technique, inspired by the social behavior of bird flocking or fish schooling. In PSO, simple software agents, called particles, move in the search space of an optimization problem. The position of a particle represents a candidate solution to the optimization problem at hand. Each particle searches for better positions in the search space by changing its velocity according to rules originally inspired by behavioral models of bird flocking.

Within the field of computer graphics, the first antecedents of PSO can be traced back to the work of Reeves (1983), who proposed particle systems to model objects that are dynamic and cannot be easily represented by polygons or surfaces. Examples of such objects are fire, smoke, water, and clouds. In these systems, particles are independent of each other and their movements are governed by a set of rules. Then, Reynolds (1987) used a particle system to simulate the collective behavior of a flock of birds. In a similar kind of simulation, Heppner and Grenander (1990) included a roost that was attractive to the simulated birds. Both models inspired the set of rules that were later used in the original PSO algorithm. Then, PSO was introduced by Kennedy and Eberhart (1995). They pointed out that the rules which govern the movement of the particles in a problem's search space can also be seen as a model of human social behavior in which individuals adjust their beliefs and attitudes to conform with those of their peers. It has roots in the simulation of social behaviors, in particular the dynamic theory of social impact (Kennedy 2006; Nowak et al. 1990), using tools and ideas taken from computer graphics and social psychology research.

The PSO algorithm is simple and easy to understand, and it has the higher search efficiency with occupying fewer programming resources. With these advantages, PSO finds a number of applications in the areas of optimization, including the unconstrained, single-objective optimization problems and constrained problems, multi-objective optimization problems, and problems with dynamically changing landscapes. To date, there have been hundreds of publications reporting applications of PSO algorithms (Poli 2008; Zhang, Zhan 2009). But, like other population-based stochastic approaches, the biggest bottleneck of limiting its application is the premature convergence. By the attention and efforts of researchers in corresponding fields, PSO algorithm is improved and expanded based on the initial model (Bu et al. 2010; Li et al. 2009; Reng, Li 2008; Tang et al. 2009; Zhou et al. 2011). Although the improved PSO algorithm's performance is improved comparing with the basic algorithm, there still exists the drawback of easily to premature convergence (Shi, Eberhart 2001; Wang et al. 2002a; Xie et al. 2011). Here, we proposed an improved PSO based on chaos theory and metropolis criteria, and introduced it to the optimization of the shape of space truss. Engineering practice and comparison with the other algorithms showed its efficiency.

The rest of the paper is organized as follows. First, the attention was paid to the modification of the algorithm after the brief introduction of the basic knowledge of PSO. In the following part, application study and performance comparison with other algorithms on the structural optimization were introduced and the advantage of the improved algorithm was analyzed.

1. Improvement of basic PSO algorithm

1.1. Basic knowledge of PSO algorithm

The PSO algorithm is a global stochastic search algorithm by simulating the nature of birds' activities and swarm intelligence. Particles, with no size and quality, always maintain two performances expressed by vectors in the search process, one is the speed expressed by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the other is the position expressed by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The speed vector represents the search direction of the flight, and position vector is the location of the solution in the search process. Each particle can maintain its own history optimal position [H.sub.best] and global optimal position vector [G.sub.best] in flight process. The velocity and location updating rules are as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[x.sup.j.sub.i] = [x.sup.j.sub.i] + [v.sup.j.sub.i], (2)

where i is the number of a particle, n is the dimension of the problem solved, j is the number of routine generation, w is the inertia weight, [c.sub.1] and [c.sub.2] are acceleration coefficients, [rand.sup.j.sub.1] and [rand.sup.j.sub.2] are the random numbers in the range of [0, 1].

1.2. Improvement of initial population

The basic PSO has the shortcoming of premature convergence, which is related to the quality of the initial population. Here, the idea of random direction was employed to produce high-quality initial population to ensure that the algorithm can find the optimal solution and avoid stagnating while the normal particles are in the search midway. The procedure producing high-quality initial population is as follows: Step 1: Determining the initial search space according to the problem to be solved:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Step 2: Initializing of the number of an individual in the population j = 1 and selecting a random position [x.sub.j] in the search space as base position.

Step 3: Initializing array X (k) for storing population individuals and k is the total number of individuals in the population.

Step 4: Producing a random vector [[??].sub.j] in the given search space.

Step 5: Producing individual [x.sub.j+1] = [x.sub.j] + [[??].sub.j] in sequence.

Step 6: Comparing [x.sub.j+1] with [x.sub.j] to judge whether they are beyond the search space, if [x.sub.j+1] is better than [x.sub.j], [x.sub.j+1] is reserved, j = j + 1 and return to Step 5. Otherwise, return to Step 4 until the particle is generated which meets the terminate conditions.

1.3. Parameters improvement of PSO algorithm

Improvement of inertia parameter w

The process of birds searching for food in the nature is a continuous learning procedure. With the increase of searching time, their experience and the ability of acclimatization increases rapidly. It can be controlled by adjusting w. In Shi and Eberhart (2001), w was set to a constant and the abilities of the population study and environment sensitivity were ignored. Although then some scholars proposed linear w, they still ignore other constrains. Here, the accelerations of exploration experience and environmental suitability were considered to adjust parameters dynamically. w was defined as a nonlinear variable. In the early search phase, particles should fly with the lower speed around the initial position to avoid destroying the optimal value found. In the later search period, the particles should fly with rapid speed away from the initial position to explore the solution space as much as possible. We define w as follows:

w = ([w.sub.ini] - [w.sub.end]) x [([g.sub.max] - g/[g.sub.max]).sup.2] + [w.sub.end], (4)

where [w.sub.ini] and [w.sub.end] represent initial and final inertia weight value, respectively, g is the value of current iteration times and [g.sub.max] is the maximum iterating times.

Improvement of acceleration factors [c.sub.1] and [c.sub.2]

The velocity direction of individual is incessantly changing with the surrounding environment changing. Here, the fuzzy reasoning was employed to adjust the acceleration coefficients adaptively. In the early search process, the flight direction should be adjusted constantly for obtaining a high acceleration factor. In the middle search process, the acceleration factors should be moderate. In the later process, acceleration factor should be small because of the larger flight rate. The acceleration factors adjusted by the fuzzy reasoning can make proper adjustment according to the experience and environment which can help the algorithm to find the better solution. The acceleration factors improved by fuzzy reason membership functions are shown in the (5) equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where c is a constant between [2, 3], [w.sub.min] is the minimum inertia weight according to experts' experience and [w.sub.max] is the maximum inertia weight according to experts' experience.

1.4. Improvement based on metropolis criteria

As a conversion function in the simulated annealing algorithm, the metropolis criteria can ensure the algorithm to jump out of local optimal solutions and possibly find the global optimal solution. PSO algorithm only accepts better solution position but ignores the bad solution positions having potential to find the global optimal solution, which makes the algorithm easily fall into premature convergence. By employing the metropolis criteria, PSO algorithm can accept some bad solutions, which improves its performance. The steps of the improved PSO algorithm are as follows.

Step 1: Initialization. According to the given problem, determine the value of space dimension n, inertia weight [w.sub.ini], [w.sub.end], [w.sub.min] and [w.sub.max], acceleration factor c, the number of a routine population k and other parameters. Then, produce initial population according to the former section of "Improvement of initial population".

Step 2: Judge whether the algorithm should stop searching according to the fitness value fit(i). If the terminate condition is met, stop searching, otherwise, go to Step 3.

Step 3: Remember the history optimal position [H.sub.best] of every individual and global optimal position vector [G.sub.best] of all particles.

Step 4: Producing next new generation population.

Step 5: Calculating the fitness value fit[(j).sup.k+1] of the new population and comparing with their parent's fit(j), if fit[(j).sup.k+1] > fit[(j).sup.k], the new generation will replace their parents, or else calculate the acceptable probability of the new population according to the following metropolis criterion:

p(g) = exp - (fit(j) - fit(i)/g). (6)

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Step 6: Producing a random number rand (0,1), if P (g) [??]rand (0,1), replace their parents with the new generation, or else return to Step 2.

In order to validate the efficiency of the improved PSO we proposed, we employed it to optimize two typical functions with multiple extreme values and compared the results with the original PSO algorithm. The two chosen functions with different characteristics are the Girewank function and Rastrigrin function shown in Figures 1 and 2. The reason we chose the two functions is that both they have multiple local minima and their optimization quality is easier to examine. The parameters are as follows: [w.sub.ini] = 0.9; [w.sub.end] = 0.2; [w.sub.max] = 0.7; [w.sub.min] = 0.4; c = 2; [g.sub.max] = 30 and n = 20. Because the algorithms are stochastic, in order to overcome the shortage that taking results of different numerical experiments would show completely different results, the comparison was based on a number of numerical experiments.

The comparison results based on 20 numerical experiments were shown in Table 1 and the average convergence curves were shown in Figures 3 and 4.

Through the results above, we know that the improved PSO algorithm can find the optimal of both functions. It performs better than the original PSO algorithm. From the average convergence curves, we see that the improved algorithm is obviously faster than the basic algorithm in finding the optimal solution. In Figure 3, the improved algorithm needs 10 iterations, yet the original PSO needs to circulate 18 iterations to find the solution. From Figure 4, we observe that the original PSO algorithm is difficult to jump out of the multiple local peaks. The improved algorithm can easily jump out the local optimal solution and find the global optimal solution.

Through the above comparison, we know that the performance of the algorithm has been improved and it can be employed to solve the structure shape optimization problems.

2. Engineering practice

In order to verify the efficiency of the improved PSO in truss structure optimization design, we employed it to solve the problem of truss structure shape optimization shown in Wang et al. (2002a).

The initial shape of the selected 37-bar plane truss structure is shown in Figure 5. Suppose that the position of lower chord panel points keeps without change, the upper panel points can move in the vertical direction and the structure remains symmetrical in the optimization. The load on the bottom chord node is P = 10 kN. The vertical direction displacement of the 10th node is restricted to [0, 10] mm. The minimum cross-sectional area is 50 [mm.sup.2], elastic modulus of the material is E = 210 GPa, material density is [rho] = 7800 kg/[m.sup.3] and the allowable stresses of all bars are restricted to [-240, 240] Pa.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

Before the application of the improved PSO, we set up the following mathematical model: Objective function:

min W = [rho] [n.summation over (i=1)] [A.sub.i][l.sub.i] + [lambda]M. (7)

Stress constraint conditions:

[[sigma].sub.min] [less than or equal to] [FN.sub.i]/[A.sub.i] [less than or equal to] [[sigma].sub.max]. (8)

Displacement constraint condition:

[summation] [FN.sub.i] x [FL.sub.i] x [l.sub.i]/E x [A.sub.i] [less than or equal to] [D.sub.max]. (9)

The optimized node displacement constraint conditions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where W represents the structure quality; M is multiplied by a penalty; [lambda] is a punish factor, if the optimized variables satisfy the restriction condition, its value equals to zero, or else, it is 1. [A.sub.i] and [l.sub.i] (i = 1, 2, ..., n) represent the cross-sectional area and length of the ith bar, respectively. [FN.sub.i] is the calculated axial force on ith bar; [[sigma].sub.min] and [[sigma].sub.max] represent the minimum and maximum of stress of bar, respectively; [FL.sub.i] is an axial force in ith bar resulting from a positive unit load; E is the value of material elastic modulus; [D.sub.max] is the maximum displacement value; [Y.sup.min.sub.j] and [Y.sup.max.sub.j] represent the minimum and maximum displacement of the jth optimized node in the Y direction, respectively.

The improved PSO was used to solve this problem and the parameters selected are as follows: the particles number PN= 25, the maximum iteration times INT = 1000, the acceleration parameter c = 2 the inertial weight coefficients: [w.sub.ini] = 1.0; [w.sub.end] = 0.2; [w.sub.min] = 0.4, and [w.sub.max] = 0.8. Numerical simulation shows under these values of coefficients, the algorithm performs best. To keeping symmetry in the process of the truss structure optimization, the position optimization variables are

[[[A.sub.1], [A.sub.3], [A.sub.5], [A.sub.7], [A.sub.9], [A.sub.11], [A.sub.13], [A.sub.15], [A.sub.17], [A.sub.19], [A.sub.21], [A.sub.23], [A.sub.25], [A.sub.27], [A.sub.29], [A.sub.3], [A.sub.33], [A.sub.35], [A.sub.37], [Y.sub.3], [Y.sub.5], [Y.sub.7], [Y.sub.9], [Y.sub.11]].sup.T]

and the design variables are shown in Table 2. Combined with finite element method (FEM), the improved PSO was employed to the optimization and the programs were written in Matlab7.1 language.

The optimized results of structure quality are shown in Table 2 and convergence curve is shown in Figure 6. And the final optimization shape is shown in Figure 7.

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

In Figure 6, the convergence curve shows that in the early searching process, the objective function is around 1000 kg which is approximately 15 times of the optimal solution. And the iterations account for a little over one percent of the total, which means the random direction algorithm can obviously produce the high-quality initial population. The curve has no stagnation in the first 600 iterations, and this is made possible because of the dynamic adjustment by fuzzy system. In all search process, the curve shows concussion adjustment phenomenon, which is due to the function of the embedded metropolis criterion.

Then, the results were compared with the results obtained from Tang et al. (2009) and Wang et al. (2002a). The comparison was shown in Tables 3 and 4.

From Table 3, we know that the displacement of the tenth node is 9.93 mm in the Y direction, which meets the displacement constraint conditions. The total quality of the optimized structure is 69.27 kg, which is better than the outcome of Tang et al. (2009) and Wang et al. (2002a). The improved PSO we proposed has the better global search performance with the stronger ability to jump out the local optimal solution.

Figure 7 shows that the optimized truss shape is an arch that can decompose the vertical load to horizontal direction and reduce the stress concentration effect. This is more reasonable than the original trapezoid shape according to the mechanical knowledge. From Table 4, we see that both the mean and standard deviation of the stress obtained by our algorithm are smaller than the results of Tang et al. 2009). Although the mean stress we obtained is bigger than results of Wang et al. 2002a), the displacement of the latter is 2.23 mm and quality is 105.2 kg which are too conservative. The above analysis shows that improved PSO we proposed performs better than the other two algorithms. Engineering practice also verifies the applicability of the results we obtained.

Conclusions

It is theoretically and practically significant to study the algorithms for the structure optimization. In order to overcome the shortcoming of the premature convergence in the common algorithms, we proposed a novel PSO and introduced it to the shape optimization of a truss structure. The application and comparison with other algorithms show that the new approach using combination of stochastic direction algorithm, fuzzy system, and metropolis criterion has significant implementation advantages over the earlier techniques. This study provides a promising method for the structure optimization.

doi: 10.3846/13923730.2013.786754

Acknowledgements

This work was supported by the Natural Science Foundation of Hebei Province, China No. E2012402030) and Foundation of Hebei Educational Committee, China (ZD2010222).

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Yancang LI. Received the PhD degree in Project Management from Tianjin University, China in 2008. Currently he is are searcher at Hebei University of Engineering, China. His research interests include artificial intelligence and its application in the engineering practice.

Yang PENG. He is studying at Civil Engineering College, Hebei University of Engineering, majored in artificial intelligence and its application in the engineering practice.

Shujing ZHOU. He is a researcher at Hebei University of Engineering, China. His research interests include artificial intelligence and its application in the engineering practice.

Yancang LI, Yang PENG, Shujing ZHOU

College of Civil Engineering, Hebei University of Engineering, Handan, 056038, China

Received 11 Jan. 2013; accepted 25 Feb. 2013

Corresponding author: Yancang Li

E-mail: [email protected]

Table 1. Comparison of original PSO algorithm and the improved PSO
algorithm on Girewank function and Rastrigrin function

Function    Function name

[f.sub.1]   Girewank
[f.sub.2]   Rastrigrin

Function    Formulas

[f.sub.1]   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
[f.sub.2]   f(x) = [n.summation over (i=1)] [x.sup.2.sub.i] -
            10 cos(2[pi][x.sub.i] + 10]

                 Test outcome value

Function    Original PSO    Improved PSO

[f.sub.1]   0.3172          0.0000
[f.sub.2]   9.1205          0.0001

Table 2. Symmetric groups of bars and nodes

Symmetric bars    1, 2     3, 4     5, 6     7, 8    9, 10    11, 12

Optimized bars    1        3        5        7       9        11
Symmetric bars    21, 22   23, 24   25, 26   27      28, 29   30, 31
Optimized bars    21       23       25       27      29       31
Symmetric nodes   1, 20    2, 18    3, 19    4, 16   5, 17    6, 14
Optimized nodes                     3                5

Symmetric bars    13, 14   15, 16   17, 18   19, 20

Optimized bars    13       15       17       19
Symmetric bars    32, 33   34, 35   36, 37
Optimized bars    33       35       37
Symmetric nodes   7, 15    8, 12    9, 13    10, 11
Optimized nodes   7                 9        11

Table 3. Comparison of different algorithms on the 37-bar
plane truss optimization

                          Improved PSO      Results of Tang
Variable                  we proposed       et at. (2009)

[A.sub.1] ([mm.sup.2])    567.8             870.7
[A.sub.3] ([mm.sup.2])    158.7             51.9
[A.sub.5] ([mm.sup.2])    120.9             51.3
[A.sub.7] ([mm.sup.2])    760.1             818.6
[A.sub.9] ([mm.sup.2])    60.2              52.0
[A.sub.11] ([mm.sup.2])   56.2              50.1
[A.sub.13] ([mm.sup.2])   564.8             776.2
[A.sub.15] ([mm.sup.2])   87.5              50.3
[A.sub.17] ([mm.sup.2])   50.0              50.3
[A.sub.19] ([mm.sup.2])   565.4             754.0
[A.sub.21] ([mm.sup.2])   98.6              50.1
[A.sub.23] ([mm.sup.2])   67.8              50.9
[A.sub.25] ([mm.sup.2])   598.1             746.8

                          Results of Wang
Variable                  et at. (2002a)    Variable

[A.sub.1] ([mm.sup.2])    883.1             [A.sub.27] ([mm.sup.2])
[A.sub.3] ([mm.sup.2])    50.0              [A.sub.29] ([mm.sup.2])
[A.sub.5] ([mm.sup.2])    50.0              [A.sub.31] ([mm.sup.2])
[A.sub.7] ([mm.sup.2])    715.4             [A.sub.33] ([mm.sup.2])
[A.sub.9] ([mm.sup.2])    50.0              [A.sub.35] ([mm.sup.2])
[A.sub.11] ([mm.sup.2])   115.3             [A.sub.37] ([mm.sup.2])
[A.sub.13] ([mm.sup.2])   646.1             [Y.sub.3] ([mm.sup.2])
[A.sub.15] ([mm.sup.2])   50.0              [Y.sub.5] ([mm.sup.2])
[A.sub.17] ([mm.sup.2])   348.1             [Y.sub.7] ([mm.sup.2])
[A.sub.19] ([mm.sup.2])   553.8             [Y.sub.9] ([mm.sup.2])
[A.sub.21] ([mm.sup.2])   54.1              [Y.sub.11] ([mm.sup.2])
[A.sub.23] ([mm.sup.2])   50.0              Quality (kg)
[A.sub.25] ([mm.sup.2])   528.2             Node10 displacement (mm)

                          Improved PSO      Results of Tang
Variable                  we proposed       et at. (2009)

[A.sub.1] ([mm.sup.2])    205.4             50.5
[A.sub.3] ([mm.sup.2])    65.3              67.4
[A.sub.5] ([mm.sup.2])    50.0              50.5
[A.sub.7] ([mm.sup.2])    52.3              50.2
[A.sub.9] ([mm.sup.2])    51.2              51.3
[A.sub.11] ([mm.sup.2])   54.2              50.2
[A.sub.13] ([mm.sup.2])   496.5             508.2
[A.sub.15] ([mm.sup.2])   897.6             904.4
[A.sub.17] ([mm.sup.2])   1186.2            1178.1
[A.sub.19] ([mm.sup.2])   1370.4            1346.1
[A.sub.21] ([mm.sup.2])   1406.8            1363.4
[A.sub.23] ([mm.sup.2])   69.27             77.46
[A.sub.25] ([mm.sup.2])   -9.93             - 8.05

                          Results of Wang
Variable                  et at. (2002a)

[A.sub.1] ([mm.sup.2])    50.0
[A.sub.3] ([mm.sup.2])    183.7
[A.sub.5] ([mm.sup.2])    183.7
[A.sub.7] ([mm.sup.2])    194.0
[A.sub.9] ([mm.sup.2])    192.8
[A.sub.11] ([mm.sup.2])   187.4
[A.sub.13] ([mm.sup.2])   1021
[A.sub.15] ([mm.sup.2])   1718
[A.sub.17] ([mm.sup.2])   2269
[A.sub.19] ([mm.sup.2])   2669
[A.sub.21] ([mm.sup.2])   2734
[A.sub.23] ([mm.sup.2])   105.2
[A.sub.25] ([mm.sup.2])   - 2.23

Table 4. Comparison of stresses in all bars (MPa)

                                           Results of
Bar      Improved PSO    Results of Tang   Wang et al.
number   we proposed     et al. 2009)      2002a)        Bar number

1st      178.22          114.08            71.33         23rd
3rd      - 63.0          - 192.68          - 200         25th
5th      13.92           1.998             - 71.21       27th
7th      126.34          116.23            79.34         29th
9th      - 178.55        - 193.20          - 149.13      31st
11th     14.55           - 18.04           5.00          33rd
13th     163.12          119.05            81.78         35th
15th     - 120.53        - 186.76          - 209.96      37th
17th     29.55           - 0.61            9.37          Mean
19th     157.48          119.89            87.44         Standard
21st     - 112.88        - 199.13          - 240.00      deviation

                         Results of        Results of
Bar      Improved PSO    Tang et al.       Wang et al.
number   we proposed     2009)             2002a)

1st      - 32.24         - 83.55           - 43.32
3rd      148.66          122.79            86.74
5th      - 31.49         - 62.82           - 118.87
7th      - 21.01         3.38              5.94
9th      - 27.44         4.51              5.94
11th     2.60            6.35              - 7.22
13th     14.54           - 6.85            - 5.76
15th     31.31           - 7.39            1.09
17th     78.47           82.59             76.76
19th     64.83           75.43             75.44
21st