Abstract
Due to the fact that vastly different variables and constraints are simultaneously considered, truss layout optimization is a typical difficult constrained mixed-integer nonlinear program. Moreover, the computational cost of truss analysis is often quite expensive. In this paper, a novel fitness estimation based particle swarm optimization algorithm with an adaptive penalty function approach (FEPSO-AP) is proposed to handle this problem. FEPSO-AP adopts a special fitness estimate strategy to evaluate the similar particles in the current population, with the purpose to reduce the computational cost. Further more, a laconic adaptive penalty function is employed by FEPSO-AP, which can handle multiple constraints effectively by making good use of historical iteration information. Four benchmark examples with fixed topologies and up to 44 design dimensions were studied to verify the generality and efficiency of the proposed algorithm. Numerical results of the present work compared with results of other state-of-the-art hybrid algorithms shown in the literature demonstrate that the convergence rate and the solution quality of FEPSO-AP are essentially competitive.
1. Introduction
As a typical real world project, truss structural analysis is considered to be computationally expensive [1]. Moreover, truss layout optimization, which solves truss sizing variables (e.g., cross-sectional areas of elements) and shape variables (e.g., coordinates of nodes) simultaneously, is known as a typical multimodal and highly nonlinear problem [2]. The various differences (e.g., physical nature, magnitude, continuity, etc.) between the two types of variables make the truss layout optimization problem a difficult task [3]. Hence, in the past decades, researchers mainly focused on solving truss layout optimization via multilevel methods [4].
Vanderplaats and Moses [5] proposed an alternating gradient method to decompose truss layout optimization into a number of subproblems, each of which optimized a subset of the design variables. The subproblems are solved iteratively until a converged optimal solution is found. Zhou [6] used a similar two-level approximation concept to optimize the cross-sectional areas of the members and the coordinates of the joints. Gil and Andreu [7] used fully stressed design method and conjugate gradient method to optimize the sizing and shape parameters of bridges, respectively.
Due to the strong coupling between the variables, the search efficiency of multilevel methods is often limited [8]. Aiming for addressing this issue and also benefiting from recent rapid advances in computational power, single-level methods which optimize all design variables simultaneously are becoming popular and competitive. Wang et al. [9] presented an optimality criteria (OC) algorithm for spatial truss layout optimization. Fourie and Groenwold [10] used new operators, namely, the elite velocity and the elite particle, in the standard particle swarm optimization (PSO) algorithm to optimize the truss layout, with the purpose of increasing the probability of migration to regions with high fitness. In fact, various metaheuristic algorithms including simulated annealing (SA) [11], genetic algorithm (GA) [12], charged system search (CSS) [13], and artificial bee colony algorithm (ABC) [14] have been introduced to address truss layout optimization problems.
More recently, researchers turn to hybridizing different techniques to further enhance the searching efficiency of metaheuristic algorithms. Lingyun et al. [18] proposed a niche hybrid genetic algorithm to solve the truss shape and sizing optimization in a simple and effective manner. Kaveh and Zolghadr [23] developed a hybridized CSS-BBBC algorithm with trap recognition capability for truss layout optimization. Zuo et al. [24] proposed a hybrid OC-GA approach for fast and global truss layout optimization; Kaveh and Javadi [25] used harmony search and ray optimizer to enhance the PSO algorithm to optimize truss layout under multiple frequency constraints; Liu and Ye [26] designed a genetic simulated annealing algorithm for domes layout optimization. Gholizadeh [17] proposed a hybridized cellular automata and PSO algorithm for truss layout optimization.
However, the common weakness of metaheuristic algorithms based structural optimization is that a huge number of structural analyses are required, which is quite time-consuming. In this paper, we propose a new hybridized algorithm, termed FEPSO-AP for truss layout optimization that aims to enhance the optimal efficiency by using the fitness estimations to partly substitute the computationally expensive fitness calculations. The finite element method (FEM) is adopted to evaluate the structural performance. Empirical results demonstrate that the proposed method is highly promising for truss layout optimization.
2. Statement of Truss Layout Optimization Problem
The main aim of truss layout optimization can be formulated as follows: where and denote the weight of truss structure and the maximum violated structural constraint; is the vector of all design variables; , , and represent the material density, length, and cross-sectional area of the th truss element, respectively.
Different types of constraints might be considered simultaneously depending on the problem to be solved. Four typical design constraints involved in this work can be stated by in which is the truss element stress constraint, while and stand for the actual largest stress of the th truss element and its stress limit, respectively; is the nodal displacement constraint, while and stand for the actual largest nodal displacement of the th node and its displacement limit, respectively; is the local buckling constraint, while stands for the Euler critical stress of the th element; is the structural natural frequency constraint, while and stand for the structural natural frequency and its frequency limit; , , and represent the number of structural elements, nodes, and constrained natural frequencies, respectively.
3. FEPSO-AP Algorithm
This section describes the fitness estimation based PSO algorithm with an adaptive penalty function approach (FEPSO-AP) developed in this research. As FEPSO-AP integrates PSO, FE, and AP, this section recalls the basic concept of PSO algorithm, fitness estimation strategy, and adaptive constraint handling approach. Finally, a framework of FEPSO-AP algorithm is presented.
3.1. The PSO Algorithm
As one of the most popular metaheuristic algorithms, PSO has found a wide application in real world projects for its structural concision and searching efficiency [17]. In a standard PSO [27], it is assumed that each of the particles has a position and a certain velocity. The position of particle represents a candidate solution to the optimization problem, and the velocity of particle determines the particle’s movement. Hence, the flying of particles can be considered as the swarm searching of design domain.
If the particle flies from its current position to the next position, its velocity and position are updated by in which and indicate the velocity and position of the th particle at iteration ; and represent the historical best position of the th particle and the global best position of all particles till iteration , respectively. stands for the inertia weight; and are cognitive and social parameters, respectively. and are diagonal matrixes, of which the diagonal elements are uniformly distributed random numbers among the range of .
3.2. The Fitness Estimation Strategy
Sun et al. [28] proposed a novel fitness estimation strategy for PSO to solve computationally expensive problems.
According to (3), the positions of any two arbitrary particles selected from the swarm can be formulated by in which , , , and . It can be found that the above two equations are both related to the global best position . Hence, (4) can be abbreviated to
Use a virtual position to present the value of (5). Supposing the influences of all coefficients are similar, Figure 1 describes the geometrical relationship among the virtual position and all subitems contained by (5). As shown in Figure 1, the fitness of virtual position can be estimated by either the fitness of , , , and or the fitness of , , , and . Hence, if the current fitness value and the historical best fitness of every particle at iteration and are obtained already, the explicit relationship between the fitness of particle and at iteration can be expressed in the following formulations: in which , , , , , , , and represent the distance between the virtual position and the real positions , , , , , , , and , respectively.
3.3. Adaptive Constraint Handling Approach
As mentioned in Section 2, truss layout optimization problem is often taken as a multiple constrained optimization problem. Coello Coello [29] pointed out that the constraint handling procedure plays an important role in the exploration and exploitation of constrained optimization problems. Similar to other metaheuristic algorithms, FEPSO is designed for unconstrained problems. Hence, it is necessary to incorporate constraint handling approach into FEPSO to solve truss layout optimization problems.
In this work, we use penalty function methods (PEM) to transfer a constrained problem into an unconstrained problem by adding the influence of violated constraints to the initial fitness function. A pseudoobjective function is stated as follows: where and are positive penalty factor and penalty function. The common difficulty existing in standard PEM is to set the constant penalty factor properly. Runarsson and Yao [30] pointed out that if the penalty factor turns out to be too large, the searching landscape would be quite rough, and thus qualified exploitation and exploration of the design domain are hard to be achieved; on the contrary, if the penalty factor is too small, it would be quite possible to lose feasible solutions.
As summarized in [29], the common target of adaptive penalty factors is to make a good balance between the objective function and the constraints violation. In this work, we developed a laconic adaptive penalty factor as follows: in which is the ratio of feasible solutions in current population. It can be concluded that if is too small, the obtained unfeasible solutions will increase the value of , and thus, the algorithm would be pushed to explore more feasible solutions. On the other hand, if the feasible solutions congregate in the population, the value of will approach zero, and thus a detailed exploitation would be performed.
3.4. The Framework of FEPSO-AP Algorithm
The penalty function of truss layout optimization problem is determined by in which is the structural design vector, is the violated constraints, and and are defined by (1).
According to (8)–(10), the pseudoobjective function can be abbreviated to
To transfer the formulated truss layout optimization problem into an unconstrained optimization problem, the adaptive penalty function, the fitness estimation strategy, and the PSO algorithm are assembled as follows.
Step 1. Set the termination condition of optimization and the initial values of , , and . Set the iteration times .
Step 2. Generate the initial positions and the initial velocities of particles population randomly, .
Step 3. Calculate the fitness values of all particles and update every particle’s historical best position and the global best position by
Step 4. Update the velocity and the position of every particle based on (3). Set the iteration times .
Step 5. Calculate the distances between every pair of particles.Select particle randomly and calculate its fitness value based on (13).Choose a particle nearby particle and calculate the virtual position by using the historical information of particle based on (5).If does not coincide with , , , , , , , or , estimate the fitness value of particle by (6); otherwise, calculate the fitness value of particle based on (13).Repeat till the fitness values of all particles are evaluated.
Step 6. Update every particle’s historical best position and the global best position . If is obtained by estimation, recalculate the corresponding particle’s fitness value by (13), and rechoose the global best position . Repeat this step till a nonestimated value of is obtained.
Step 7. Repeat Steps 4–6 till the termination condition is reached. Output the global best position and its objective value.
4. Benchmark Examples
The following four benchmark examples have been used to demonstrate the generality and efficiency of the FEPSO-AP algorithm:(i)a planar 15-bar truss subjected to a single load condition and stress constraints,(ii)a spatial 25-bar truss subjected to a single load condition under stress and displacement constraints,(iii)a planar 37-bar truss subjected to multiple frequency constraints,(iv)a planar 47-bar truss subjected to three load conditions under stress and local buckling constraints.
Programs of FEPSO-AP algorithm and structural finite element method (FEM) algorithm are developed by using MATLAB R2013a. A personal computer with a Pentium E5700 processor and 2 GB memory under the Microsoft Windows 7 operating system has been used to run the optimization software.
For all benchmarks examined in this study, the FEPSO-AP algorithm parameters are set as the usual constants of standard PSO which are obtained by [31]: , , and . According to the design dimensions of four benchmarks, the population sizes are set as 46, 26, 38, and 88, while the maximum numbers of FEM analyses are set as 4000, 4500, 8000, and 20000, respectively.
Twenty-five independent runs are performed with the best one being selected for each problem.
4.1. Planar 15-Bar Truss
The original geometry of planar 15-bar truss is shown in Figure 2(a). Two nodes (ID: 1 and 5) are totally fixed and the -coordinates of other two nodes (ID: 4 and 8) are fixed as well. The single loading condition is listed in Table 1.
(a) Geometry and element definitions of the planar 15-bar truss
(b) Best solution of the planar 15-bar truss
All design variables are classified into 23 groups: sizing variables: , ; shape variables: ; ; ; ; ; ; ; .
Material parameters and design constraints are listed in Table 2.
Figure 2(b) shows the optimum design found by this work, of which two nodes (ID: 4 and 8) are highly overlapped. Table 3 compares the best design found by this work with those reported in the literature. It can be seen that the results achieved by the proposed algorithm are quite close to the best results reported in the literature.
4.2. Spatial 25-Bar Truss
The original geometry of spatial 25-bar truss is shown in Figure 3(a). Two nodes (ID: 1 and 2) are totally fixed and the -coordinates of four nodes (ID: 7, 8, 9. and 10) are fixed as well. The loading condition is listed in Table 4.
(a) Geometry and element definitions of the spatial 25-bar truss
(b) Best solution of the spatial 25-bar truss
To ensure the structural symmetries, all design variables are classified into 13 groups: sizing variables: ; ; ; ; ; ; ; ; shape variables: ; ; ; ; .
Material parameters and design constraints are listed in Table 5.
Figure 3(b) shows the optimum design found by this work. Table 6 compares the best design found by this work with those reported in the literature. It can be concluded that by using the proposed algorithm it is possible to achieve the best feasible results at a low computational cost.
4.3. Planar 37-Bar Truss
The original geometry of planar 37-bar truss is shown in Figure 4(a). Eleven nodes (ID: 1, 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20) are totally fixed and the -coordinates of all other nine nodes (ID: 3, 5, 7, 9, 11, 13, 15, 17, and 19) are fixed as well. A nonstructural mass of 10 kg is attached to all free nodes on the lower chord.
(a) Geometry and element definitions of the planar 37-bar truss
(b) Best solution of the planar 37-bar truss
To ensure the structural symmetric about the -axis, all design variables are classified into 19 groups: sizing variables: ; ; ; ; ; ; ; ; ; ; ; ; ; ; shape variables: ; ; ; ; ; ; ; ; .
Material parameters and design constraints are listed in Table 7.
Figure 4(b) shows the optimum design explored by this work. Table 8 compares the best design presented by this work with those reported in the literature. It can be seen that the results achieved by the proposed algorithm are even better than the best results reported in the literature.
4.4. Planar 47-Bar Truss
The original geometry of planar 47-bar truss is shown in Figure 5(a). Four nodes (ID: 15, 16, 17, and 22) are totally fixed and the -coordinates of other two nodes (ID: 1 and 2) are fixed as well. The loading conditions are listed in Table 9.
(a) Geometry and element definitions of the planar 47-bar truss
(b) Best solution of the planar 47-bar truss
To ensure the structural symmetric about the -axis, all design variables are classified into 44 groups: sizing variables: ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; shape variables: ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; .
Material parameters and design constraints are listed in Table 10.
Figure 5(b) shows the optimum design identified by this work. Table 11 compares the best design of this work with those reported in the literature. It can be seen that by using the proposed algorithm it is possible to achieve better results at a lower computational cost.
5. Conclusion
In this work, a new hybrid PSO algorithm is proposed to solve a quite challenging task in truss optimization area: truss layout optimization with multiple constraints.
Two computational techniques are adopted to further enhance the performance of PSO algorithm. In the first fitness estimation strategy, the evaluation of particles is partly substituted by the estimation of similar particles, with the purpose to reduce the computational cost of real world optimization problem. In the second adaptive penalty function approach, the iteration information is merged into the penalty function to find a good balance between the exploration and exploitation of the constrained design domain. The resulted algorithm is termed as FEPSO-AP.
Four benchmark truss layout optimization problems, subject to nodal displacement constraints, element stress constraints, natural frequency constraints, and local buckling constraints, are used to verify the performance of FEPSO-AP. Numerical results demonstrate that three out of four benchmarks, to which the FEPSO-AP based optimization is applied, delivered the best feasible designs to the author’s knowledge. Moreover, the convergence rate of the FEPSO-AP algorithm is quite competitive comparing to other state-of-the-art hybrid algorithms published in the former literatures.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.