Proceedings of 11th Asian Pacific Conference on Shell and Spatial Structures-APCS2015 May 14-16, 2015 Xi’an, China
Discrete firefly algorithm in the application of topology optimization of truss structures Qingjie HU*, Qingpeng LIa, Yue WUa *a
Key Lab of Structures Dynamic Behavior and Control of Ministry of Education, Harbin Institute of Technology Address: Room301, School of Civil Engineering, Harbin Institute of Technology, Harbin 150090, China Email:
[email protected]
Abstract As an advanced nature-inspired metaheuristic algorithm, firefly algorithm (FA) has been applied to solve combinatorial optimization problems in many fields of engineering, which is inspired by the social behavior of fireflies and the phenomenon of bioluminescent communication. Based on the basic principles of FA, the topology optimization of truss structure is discussed and discrete firefly algorithm (DFA) is used to solve the problem of truss structures in this paper. Firstly, the development of structural topology optimization methods and the basic principle of standard FA are introduced in details. Then, a DFA which is suitable for discrete variable optimization problems is formed by improving the original firefly locations and location updating formula. And then, using ground structure method as method of topological variation of trusses, applying the topology optimization mathematical model based on cross-section optimization, and combining with the variable elastic modulus technology and geometric construction analysis, an optimization method for the topological design of truss based on DFA is proposed. Finally, in order to show the feasibility of the topology optimization method in this paper, numerical examples of a plane truss and a spatial truss are conducted and compared with other optimization algorithms. Keywords: discrete firefly algorithm, truss structure, topology optimization design, discrete variable, ground structure method.
1. Introduction The topology optimization methods of truss structures mainly consist of analytical method and numerical method. The Michell theory studied by Prager et al. [1] and Rozvany et al. [2] is an analytical method, which cannot be directly used in practical engineering issues. However, it had a profound impact on structural topology optimization study. Numerical methods were raised after Dorn et al. [4] putting forward the ground structure method. Their optimization algorithms mainly include mathematical programming algorithm, optimality criteria algorithm and nature-inspired metaheuristic algorithms. The nature-inspired metaheuristic algorithms are the most popular optimization algorithms for two reasons. On one hand, they do not have analytical requirements. On the other hand, through appropriate measures they can assist researchers easily find the singular optimal studied by CHENG et al. [5] compared with mathematical programming algorithm and optimality criteria algorithm. Over the last few years, a large numbers of nature-inspired metaheuristic algorithms have been proposed and widely used, such as tabu search algorithm, simulated annealing algorithm, genetic algorithm, ant colony algorithm and PSO. The firefly algorithm developed by Yang [6] studied in this paper is one of the newest nature-inspired metaheuristic algorithms which is easier and more effective to find better solution than genetic algorithm and particle swarm optimization algorithm when solving continuous optimization problems, combinatorial optimization problems, constrained optimization combinatorial optimization problems, multi-objective optimization combinatorial optimization problems and dynamic programming. According to Fister et al.[7]， firefly algorithm has been applied in almost all areas of optimization, as well as engineering practice. Because the topology optimization problem of truss structures owns the characteristics of discrete and multi-peak, to some extent, it can be regard as a kind of combinatorial optimization problem. Hence it can also be solved by firefly algorithm. Based on this, the firefly algorithm is used to solve topology optimization problem of truss structures by discretization in this paper.
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2. Firefly algorithm (FA) 2.1. Standard FA The Firefly algorithm (FA) is a very recent nature-inspired metaheuristic algorithm developed by Yang which is inspired by the flashing behavior of fireflies. According to Yang [6], FA optimization has three idealized rules. (a) All fireflies are unisex, so that one firefly is attracted to other fireflies regardless of their sex. (b) Attractiveness is proportional to brightness, so for any two flashing fireflies, the less bright firefly will move towards the brighter one. Both attractiveness and brightness decrease as the distance between fireflies increases. If there is no firefly brighter than a particular firefly, that firefly will move randomly. (c) The brightness of a firefly is affected or determined by the landscape of the objective function. Based on these three rules, there are two essential components to FA: the variation of light intensity and the formulation of attractiveness. The latter is assumed to be determined by the brightness of the firefly, which in turn is related to the objective function of the problem under study and can be defined by:
I i f ( xi )
(1)
where f(xi ) is the objective function of the problem under study. As light intensity and attractiveness decrease and the distance from the source increases, the variation of light intensity and attractiveness should be a monotonically decreasing function. For example, the light intensity can be:
I ij (rij ) I i e
rij2
(2)
in which the light absorption coefficient γ is a parameter of the FA and rij is the distance between fireflies i and j at xi and xj, respectively, which can be defined as the Cartesian distance rij=||xi -xj||. Because a firefly’s attractiveness is proportional to the light intensity seen by other fireflies, it can be defined by:
ij (rij ) 0e
rij2
(3)
in which β0 is the attractiveness at r=0. Finally, the probability of a firefly i being attracted to another, more attractive (brighter) firefly j is determined by (4) where t is the generation number, εj is a random vector (e.g., the standard Gaussian random vector in which the mean is 0 and the standard deviation is 1) and α is the randomization parameter. The second term of Eq. (4) represents the attraction between the fireflies and the third term is the random movement. Eq. (4) can be called location update formula. A swarm of fireflies which contain n fireflies is generated in the preliminary stage of optimization process and a firefly whose location is a multidimensional vector containing multiple design variables corresponds to a candidate solution of optimization problem. Under the effect of attraction, new candidate solutions are chosen in the process of fireflies update their location constantly. 2.2. Discretization of FA To solve the topology optimization of truss structures, which is a discrete optimization problem, standard FA needs to be discretized to form a discrete firefly algorithm (DFA), because it only suits continuous optimization problems. Discretization of firefly algorithm can be divided into two parts as follow: 2.2.1. Discretization of initials location of fireflies Initial locations of fireflies, which consist of continuous real variables in standard firefly algorithm, need to be discretized so that they can be composed of discrete integer variables. The modified initial location firefly j can be: (5) Where xji is the nth element of firefly j, n is the number of optimization variables and the product round means integer conversion.
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Proceedings of 11th Asian Pacific Conference on Shell and Spatial Structures-APCS2015 May 14-16, 2015 Xi’an, China
2.2.2. Discretization of location updating formula As the second term and the third term of location update formula may be non-integers, these terms need to be discretized to ensure the updated locations of fireflies are integers. The modified location update formula can be:
x j (t 1) x j (t ) round (ij (rij )(xi (t ) x j (t ))) round (ε j )
(6)
The above two parts of discretization can ensure that the locations of each fireflies are discrete integer variables.
3. Topology optimization of truss structures The ground structure approach is followed in the proposed methodology. This scheme, initially proposed by Dorn et al., starts with a universal truss containing all (or almost all) possible member connections among all nodes in the structure. The topology optimization procedure is then employed to discard the unnecessary members. In other words, the algorithm chooses, from all possible members, the members that remain in the structure. Simultaneously, the algorithm performs the size optimization of the truss by changing the cross-sectional area of the remaining structural members. This optimization procedure seeks the minimum structural weight of the truss subjected to stress and displacement constraints. 3.1. Optimized mathematical model With the ground structure approach, topology optimization problem can be transformed into size optimization problem, and the design variables for topology and size are grouped into the vector x=(A1,A2,…,Am)T. Thus, the optimization problem can be posed as: find x ( A1, A2 ,, Am )T m
min W i Ali i
(7)
i1
st. g j ,l ( A) 0 ( j 1,, J ),(l 1,, L) Ai S
(i 1,, m)
Where Ai is the cross-sectional area of the ith bar, W is the structural weight, ρi is the specific weight of the ith bar material, li is the length of ith bar, gj,l is the jth constraint under lth load case, S=(S1,S2,…,Sn) is the discrete set of bar section, S1→0 means that the bar is deleted and n is the number of elements of S. In the optimization process, the bar section and structural topology are adjusted by changing the section number of each bars which represents a cross section arranged in the discrete set S in order from small to large. 3.2. Variable elastic modulus technology Researchers are apt to be confronted with singular optimums when using ground structure approach to solve topology optimizations. The singular optimums are caused by replacing the section of canceled bar with a smaller value for the purpose of topology change. They may cause great difficulties to bar removal or insertion during the process of optimization. Essentially, the measure using a smaller value to replace the section of canceled bar may cause the canceled bar whose stress is zero greater than the allowable stress. So it can set the stress of canceled bar to zero on purpose to avoid the singular optimums. Theoretically, there are two measures, setting the displacement of nodes connected to the canceled bar to zero or setting the element stiffness matrix of canceled bar to null matrix, to take the axial force of canceled bar to zero in the static analysis with finite element method. However, the displacement of nodes connected to many bars including canceled is difficult to be set to zero in practice because as long as there are retained bars connected, the displacement of nodes cannot be set to zero. So the measure of variable modulus of elasticity by which the element stiffness matrix of canceled bar is set to null matrix is the only feasible method to take the axial force of canceled bar to zero and to avoid the singular optimum. 3.3. Geometric construction analysis The geometric construction analysis implemented by checking the positive definiteness of the global stiffness matrix of structure for structure whose topology is generated randomly topology by the firefly algorithm is a necessary measure to disregard the presence of mechanisms. After geometric construction analysis, a static analysis is performed for structure and a penalty value for mechanisms to reduce unnecessary computation cost.
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There is a notable problem that in the process of geometric composition’s analysis, useless nodes will turn up without connected bar in the structure. There is no doubt that the useless nodes should be deleted. In this paper, these nodes are treated as fixed hinge bearing for the reason that if they were deleted directly, the singular global stiffness matrix of structure would be caused. 3.4. Truss topology optimization process A topology optimization method of truss structures based on the discrete firefly algorithm is put forward in this paper by applying the discrete firefly algorithm and combining it with the above optimized mathematical model and measures, whose flow chart shown in Figure 1.
Figure 1: Flow chart of topology optimization of truss structures based on discrete firefly algorithm
4. Numerical examples Standard test problems are useful for checking optimization algorithms. The benchmark examples given in this section have been widely used for this purpose. Due to the stochastic nature of the DFA, the final result can vary depending on the seed used for the random number generation. With the goal of providing a statistical basis, this paper presents the results of over 100 runs for each example. In this way, the average values and standard deviations are presented along with the optimal results. 4.1. 15-bar planar truss example The 15-bar planar truss is used widely in structural optimization to verify various nature-inspired metaheuristic algorithms. The topology and nodal numbering of a 15-bar planar truss structure is shown in Figure 2(unit: cm) and the form of the supports is fixed hinge bearing. Fifteen bars are categorized into 8 groups using symmetry: (1) A1 A9, (2) A2 A10, (3) A3 A11, (4) A4 A12, (5) A5 A13, (6) A6 A14, (7) A7 A15, (8) A8. The material
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Proceedings of 11th Asian Pacific Conference on Shell and Spatial Structures-APCS2015 May 14-16, 2015 Xi’an, China
density is 2.768×103 (kg/m3) and the modulus of elasticity is 6.987×104 (MPa). In this example, the structure is subjected to two load cases listed in Table 1. The members are subjected to the stress limits of 172.435 MPa. Maximum displacement limitation of 2.032cm is imposed on node 5 in y direction. The set of discrete cross-sectional areas is {6.452,9.677,22.581,32.258,45.161,70.968,88.871,103.226,129.032,161.29,193.548}cm2. The following set of parameters is used in this example: α = 1.5, β0 = 1, γ = 0.04, Gmax = 120, n = 30.
Figure 2: 15-bar plane truss structure
Case
Node 3 5 7 4 6 8
1
2
Px/ kN 0 0 0 0 0 0
Py/ kN -4.45×105 -4.45×105 -4.45×105 -4.45×105 -4.45×105 -4.45×105
Table 1: Load cases of 15-bar truss The best result over 100 runs achieved by the DFA has a weight of 396.66kg with a probability of 75%. The mean value achieved by the DFA is equal to 401.55kg and the coefficient of variation is 4.9%. The convergence of the topology optimization is shown in Figure 3 and the best topology is shown in Figure 4. 1000
Best Mean
950 900 850
Weight(kg)
800 750 700 650 600 550 500 450 400 350 0
20
40
60
80
100
120
Generations
Figure 3: Convergence for the topology optimization of 15-bar truss
Figure 4: Best topology for the topology optimization of 15-bar truss
Although, the best result presents in this paper is identical to the best design developed using GA studied by CHEN et al. which uses a population size P= 40, 3 populations and 40 generations, it performs better than GA for the reason that the GA get the best solution after 3000 analyses while it takes only 1590 analyses averagely for the DFA. This demonstrates the effectiveness of the DFA to deal with topology optimization of truss structures. 4.2. 25-bar space truss example The 25-bar space truss is also often used as a benchmark problem. The topology and nodal numbering of a 25-bar space truss structure is shown in Figure 5(unit: cm) and the form of the supports is fixed hinge bearing.
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Twenty-five bars are categorized into 8 groups using symmetry: (1) A1, (2) A2-A5, (3) A6-A9, (4) A10-A11, (5) A12-A13, (6) A14-A17, (7) A18-A21, (8) A22-A25,. The material density is considered as 2.768×103 kg/m3 and the modulus of elasticity is taken as 6.987×104MPa. The allowable strength of each bars are shown in Table 2. In this example, the structure is subjected to two load cases listed in Table 3. Maximum displacement limitation of 0.899cm is imposed on node 1 and node 2 in x and y directions. The set of discrete cross-sectional areas is {0.774,1.355,2.142,3.348,4.065,4.632,6.542,7.742,9.032,10.839,12.671,14.581,21.483,34.839,44.516,52. 903,60.258,65.226}cm2. The following set of parameters is used in this example: α = 1.5, β0 = 1, γ = 0.02, Gmax = 240, n = 30.
Figure 5: 25-bar space truss structure
Group
1
2
3
-242.044
-79.941
-119.36
275.896
275.89 6
275.89 6
4 -242.04 4 275.896
5
6
7
8
-242.044
-46.619
-46.619
-76.437
275.896
275.896
275.89 6
275.89 6
Table 2: Allowable stress
Case 1
2
Node 1 2 3 6 1 2
Px/ kN 44.5 0 2.225 2.225 0 0
Py/ kN 44.5 44.5 0 0 89.0 -89.0
Pz/ kN -22.25 -22.25 0 0 -22.25 -22.25
Table 3: Load cases of 25-bar space truss
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Proceedings of 11th Asian Pacific Conference on Shell and Spatial Structures-APCS2015 May 14-16, 2015 Xi’an, China
750
Best Mean
700 650
Weight(kg)
600 550 500 450 400 350 300 250 0
20
40
60
80
100
120
140
160
180
200
220
240
Generations
Figure 7: Best topology for the topology optimization of 25-bar space truss
Figure 6: Convergence history for the topology 25-bar space truss optimization problem
The best result over 100 runs achieved by the DFA has a weight of 256.91kg with a probability of 83%. The mean value achieved by the DFA is equal to 259.66kg and the coefficient of variation is 2.32%. Figure 6 shows the convergence of the topology optimization and the best topology is shown in Figure 7. It is 275.05kg, which has the same topology with the best result of DFA, for the RDQA studied by CHAI et al. that worse than DFA’s. This also demonstrates the effectiveness of the DFA to deal with topology optimization of space truss structures.
5. Conclusions This paper employs the DFA, which is modified by FA, in the topology optimization of truss structures. The results show that the approach is especially suited to combinatorial optimization problems, which is a typical scenario for such a problem. Furthermore, using ground structure method as method of topological variation of trusses, applying the topology optimization mathematical model based on cross-section optimization, and combining with the variable elastic modulus technology and geometric construction analysis, an optimization method for the topological design of truss structures based on DFA is proposed. The effectiveness of the DFA in solving the topology optimization of truss structures is demonstrated through the numerical examples of a plane truss and a spatial truss, the results of which are similar or even better than those reported in the literature, and with lower computational costs. These examples emphasize the capabilities of the proposed methodology in this field.
References [1] Prager W., A Note on Discredited Michell Structures. Computer Methods in Applied Mechanics and Engineering, 1974, 3(3), 349-355. [2] Rozvany G.I.N., Some Shortcomings in Michell's Truss Theory. Structural Optimization, 1997, 13(2-3), 203-204. [3] Rozvany G.I.N., Partial Relaxation of the Orthogonality Requirement for Classical Michell trusses. Structural Optimization, 1997, 13(4), 271-274. [4] Dorn WS, Gomory RE, and Greenberg HJ, Automatic Design of Optimal Structures. Journal de Mechanics, 1964, 3(1), 25-52. [5] CHENG Gengdong, On singular optima of structural topology optimization of trusses. Journal of Dalian University of Technology, 2000, 40(4), 379-383[in chinese]. [6] YANG Xin-she, Firefly Algorithms for Multimodal Optimization, in Proc of the 5th International Symposium on Stochastic Algorithms：Foundations and Applications, 2009, 169-178.
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