Improved Genetic Algorithm with Two-Level Approximation for Truss Optimization by Using Discrete Shape Variables

Abstract

This paper presents an Improved Genetic Algorithm with Two-Level Approximation (IGATA) to minimize truss weight by simultaneously optimizing size, shape, and topology variables. On the basis of a previously presented truss sizing/topology optimization method based on two-level approximation and genetic algorithm (GA), a new method for adding shape variables is presented, in which the nodal positions are corresponding to a set of coordinate lists. A uniform optimization model including size/shape/topology variables is established. First, a first-level approximate problem is constructed to transform the original implicit problem to an explicit problem. To solve this explicit problem which involves size/shape/topology variables, GA is used to optimize individuals which include discrete topology variables and shape variables. When calculating the fitness value of each member in the current generation, a second-level approximation method is used to optimize the continuous size variables. With the introduction of shape variables, the original optimization algorithm was improved in individual coding strategy as well as GA execution techniques. Meanwhile, the update strategy of the first-level approximation problem was also improved. The results of numerical examples show that the proposed method is effective in dealing with the three kinds of design variables simultaneously, and the required computational cost for structural analysis is quite small.

Academic Editor:P. Beckers

School of Astronautics, Beihang University, XueYuan Road No. 37, HaiDian District, Beijing 100191, China

Received 25 September 2014; Revised 10 April 2015; Accepted 17 April 2015; 16 September 2015

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The optimal design of a truss structure has been an active research topic for many years. Important progress has been made in both optimality criteria and solution techniques. As is well known, the optimal shape design of a truss structure depends not only on its topology but also on the element cross-sectional areas. This inherent coupling of structural shape, topology, and element sections explicitly indicates that the truss shape or topology or sizing optimization should not be performed independently. To date, most researchers focus on the subject of truss shape and sizing optimization [1-3] or topology and sizing optimization [4, 5], while relatively little literature is available on truss shape, topology, and sizing simultaneous optimization [6, 7]. The main obstacle is that shape and topology and sizing variables are fundamentally different physical representations. Combining these three types of variables may entail considerable mathematical difficulties and, sometimes, lead to ill-conditioning problem because their changes are of widely different orders of magnitude.

GA has been widely applied in truss topology optimization, especially for mixed variable problems. Based on various given problems, many specific GA methods have been proposed [8, 9]. However, they are still not completely satisfactory owing to their high computational cost and unstable reliability, especially for large scale structures [10, 11]. It is, therefore, apparent that the efficiency and reliability of GA retain large space to be improved further.

To improve the efficiency of truss sizing/topology optimization, Dong and Huang [12] proposed a GA with a two-level approximation (GATA), which obtains an optimal solution by alternating topology optimization and size optimization. As the structural analyses are used for building a series of approximate problems and the GA is conducted based on the approximate functions, the computational efficiency is greatly improved and the number of structural analyses can be reduced to the order of tens. Later, Li et al. [5] improved the GATA to enhance its exploitation capabilities and convergence stability. However, the shape variables are not involved in their research.

In this paper, based on the truss topology and sizing optimization system using GATA [5], a series of techniques were proposed to implement the shape, topology, and sizing optimization simultaneously in a single procedure. Firstly, a new optimization model is established, in which the truss nodal coordinates are taken as shape variables. To avoid calculating sensitivity of shape variables, discrete variables are used by adding the length of the individual chromosome. Therefore, sizing variables are continuous and shape/topology variables are discrete. To solve this problem, a first-level approximate problem is improved for the change of truss shape. Then, GA is used to optimize the individuals which include discrete 0/1 topology variables representing the deletion or retention of each bar and the integer-valued shape variables corresponding to nodal coordinates. Within each GA generation, a nesting strategy is applied in calculating the fitness value of each member [5, 12]. That is, for each member, a second-level approximation method is used to optimize the continuous size variables [5, 12]. In terms of GA, hybrid gene coding strategy is introduced, as well as the improvement of genetic operators. The controlled mutation of shape variables is considered in the generation of initial population. The uniform crossover is implemented for discrete topology variables and shape variables independently. Meanwhile, the first-level approximation problem update strategy was also improved in this paper. The proposed method is examined with typical truss structures and is shown to be quite effective and reliable.

This paper is organized as follows. In Section 2, we describe the optimization formulation for truss sizing/shape/topology optimization. In Section 3, we describe the optimization method GATA and in Section 4 the details of improvements for the optimization method are stated. In Section 5, we present our numerical examples and the algorithm performance and conclusion remarks are given in Sections 6 and 7, respectively.

2. Problem Formulation

The truss sizing/shape/topology optimization problem is formulated in (1). Here, three kinds of design variables are defined as follows.

(1) Sizing Variables . [figure omitted; refer to PDF] is the size variable vector, with [figure omitted; refer to PDF] [figure omitted; refer to PDF] denoting the cross-sectional area of bar members in [figure omitted; refer to PDF] th group and [figure omitted; refer to PDF] denoting the number of groups.

(2) Shape Variables . [figure omitted; refer to PDF] is the shape variable vector, and [figure omitted; refer to PDF] is the number of shape variables; [figure omitted; refer to PDF] denotes the identifier number within the possible coordinates set [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is the number of possible coordinates of [figure omitted; refer to PDF] th shape variable.

(3) Topology Variables . [figure omitted; refer to PDF] is the topology variable vector. If [figure omitted; refer to PDF] , members in [figure omitted; refer to PDF] th group are removed, and [figure omitted; refer to PDF] is set to a very small value [figure omitted; refer to PDF] , which is generally calculated as [figure omitted; refer to PDF] multiplied by the initial value of [figure omitted; refer to PDF] ; if [figure omitted; refer to PDF] , members in [figure omitted; refer to PDF] th group are retained, and [figure omitted; refer to PDF] is optimized between the upper bound [figure omitted; refer to PDF] and the lower bound [figure omitted; refer to PDF] : [figure omitted; refer to PDF]

where [figure omitted; refer to PDF] is the total weight of the truss structure and [figure omitted; refer to PDF] denotes the weight of [figure omitted; refer to PDF] th group. [figure omitted; refer to PDF] represents [figure omitted; refer to PDF] th constraint in the model, which could be constraints of element stresses, node displacements, mode frequency, or buckling factor. [figure omitted; refer to PDF] denotes the total number of constraints, and [figure omitted; refer to PDF] is the number of frequency or bulking constraints. If some bar members are removed, the corresponding constraints are eliminated, such as the stress constraints of the removed members. Thus, [figure omitted; refer to PDF] indicates whether the respective constraint is eliminated; if [figure omitted; refer to PDF] , [figure omitted; refer to PDF] th constraint is eliminated; otherwise if [figure omitted; refer to PDF] , [figure omitted; refer to PDF] th constraint is retained.

To facilitate describing the optimization model, a truss structure is taken as an example, which is shown in Figure 1. [figure omitted; refer to PDF] coordinate of node 1 could be 100, 120, 140, and 160, while [figure omitted; refer to PDF] coordinate of node 2 could be 200, 240, and 280. It is required that the coordinate variation of node 1 and node 2 is independent and that node 1 should be symmetric with node 3 along the dotted line, which means that there are two independent shape variables. In addition, the cross-sectional dimension of each bar element is required to be optimized independently and each bar element is allowed to be deleted or retained, which means that there are 7 size variables and 7 topology variables.

Figure 1: Seven-bar truss and node coordinates.

[figure omitted; refer to PDF]

3. Optimization Method

3.1. The First-Level Approximate Problem

To solve problem (1) which is always implicit, a first-level problem is constructed to transform [figure omitted; refer to PDF] and [figure omitted; refer to PDF] into a sequence of nonlinear explicit approximate functions. In [figure omitted; refer to PDF] th stage, the approximate explicit problem can be stated as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are upper and lower bounds of size variable [figure omitted; refer to PDF] at [figure omitted; refer to PDF] th stage; [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the moving limits of [figure omitted; refer to PDF] at [figure omitted; refer to PDF] th stage; [figure omitted; refer to PDF] are [figure omitted; refer to PDF] th approximate constraint function at [figure omitted; refer to PDF] th stage, which is constructed as follows. First, structural and sensitivity analysis are implemented at the point [figure omitted; refer to PDF] to obtain the constraint response. Second, the results of structural and sensitivity analysis are used to construct a branched multipoint approximate (BMP) function ((5)-(8)) [5, 12]: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is [figure omitted; refer to PDF] th known point, [figure omitted; refer to PDF] is the number of points to be counted, and [figure omitted; refer to PDF] . When the number of known points is larger than [figure omitted; refer to PDF] (always set as 5), only the last [figure omitted; refer to PDF] points are counted; [figure omitted; refer to PDF] is a weighting function, which is defined in (7)-(8); [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the adaptive parameter controlling the nonlinearity of [figure omitted; refer to PDF] , which are determined by solving the least squares parameter estimation in (9). When [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . For more details of BMP function, please see the work by Dong and Huang (2004) [12]. Though problem (2) is explicit, it involves topology and shape variables which cannot be directly solved by mathematical programming method. Thus a GA is implemented for explicit mixed variables problem (2).

3.2. GA to Deal with Mixed Variables Problem

GA is used to generate and operate on sequences of mixed variables vector [figure omitted; refer to PDF] representing the truss shape and topology, in which [figure omitted; refer to PDF] is 0/1 variables and [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) is integer-valued variable. Based on the optimum vector [figure omitted; refer to PDF] obtained in the last iteration, the GA generates an initial population randomly, in which the vector [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) represents [figure omitted; refer to PDF] th individual in [figure omitted; refer to PDF] th generation at [figure omitted; refer to PDF] th iteration of the first-level approximate problem. Then, for every individual in the current generation, the optimal size variables vector [figure omitted; refer to PDF] is obtained by solving a second-level approximation problem, which will be described later in Section 3.3. To reduce the structural analyses, the objective value [figure omitted; refer to PDF] is calculated accurately with analytic expressions, and the constraint value [figure omitted; refer to PDF] is calculated with approximate functions ((5)-(9)). Then, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are used to calculate the fitness of individual [figure omitted; refer to PDF] with penalty function method (10). For more details of penalty functions, please see work by Li et al. (2014) [5]. Consider [figure omitted; refer to PDF]

After the fitness value of all the members in the initial generation is calculated, the genetic selection, crossover, and mutation operators work on the vector [figure omitted; refer to PDF] in sequence based on the individual fitness value [figure omitted; refer to PDF] to generate the next generation ( [figure omitted; refer to PDF] ). The different genetic operations on 0/1 variables vectors [figure omitted; refer to PDF] and integer-valued variables vectors [figure omitted; refer to PDF] will be described in Section 4. When the maximum generation ( [figure omitted; refer to PDF] ) is reached, the optimum vectors [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are obtained for the next iteration ( [figure omitted; refer to PDF] ) of the first-level approximate problem.

3.3. The Second-Level Approximate Problem

After constructing first-level approximate problem (2) and implementing GA to generate sequences of vector [figure omitted; refer to PDF] , original problem (1) is transformed to an explicit problem with continuous size variables only. To improve the computational efficiency, a second-level approximate problem is constructed using linear Taylor expansions of reciprocal design variables [5, 12]. In [figure omitted; refer to PDF] th step, the second-level approximate problem is stated in [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the approximate objective value and [figure omitted; refer to PDF] is the approximate value of [figure omitted; refer to PDF] th constraint in [figure omitted; refer to PDF] th step; [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are move limits of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are upper and lower bounds of [figure omitted; refer to PDF] in [figure omitted; refer to PDF] th step. After constructing the second-level approximate problem, a dual method and a BFGS are used to seek the optimal size variable [figure omitted; refer to PDF] [5, 12].

4. Improvements in GATA for Adding Shape Variables

To facilitate describing the improvements for adding shape variables in GATA, the truss structure in Figure 1 is also taken as an example.

4.1. Definition of Shape Variables and Variable Link

In problem (1), [figure omitted; refer to PDF] is the shape variable vector; [figure omitted; refer to PDF] denotes the identifier number of the possible coordinates. Each shape variable is defined with an array. As shown in Figure 2, [figure omitted; refer to PDF] represents the identifier number of shape variables; [figure omitted; refer to PDF] is the identifier number of the nodes to be moved; [figure omitted; refer to PDF] means the direction of coordinate, which could be 1 or 2 or 3, corresponding to [figure omitted; refer to PDF] - or [figure omitted; refer to PDF] - or [figure omitted; refer to PDF] -axis coordinate, respectively; [figure omitted; refer to PDF] is the number of possible discrete coordinate values of node [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] denotes the identifier number of node coordinates of the initial truss structure; [figure omitted; refer to PDF] denotes the discrete coordinate set of node [figure omitted; refer to PDF] , or variable [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] (or [figure omitted; refer to PDF] ). The shape variables can be linked with each other; that is, some node coordinates could vary with a given relation, such as symmetric variation. The definition of shape variable link relation is explained in Figure 3. [figure omitted; refer to PDF] represents the identifier number of shape variables; [figure omitted; refer to PDF] is the identifier number of the nodes that is expected to link; [figure omitted; refer to PDF] denotes the direction of coordinate which is expected to link, [figure omitted; refer to PDF] or 2 or 3, corresponding to [figure omitted; refer to PDF] - or [figure omitted; refer to PDF] - or [figure omitted; refer to PDF] -axis coordinate, respectively; [figure omitted; refer to PDF] is defined as a moving scaling factor, which means that the linked coordinate value is [figure omitted; refer to PDF] when the coordinate value of shape variable [figure omitted; refer to PDF] is [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] for symmetric nodes. According to the shape variable definition rules described above, the shape variables of node 1 and node 2 could be defined as follows:

: [figure omitted; refer to PDF] ,

: [figure omitted; refer to PDF] .

Figure 2: The definition of shape variables.

[figure omitted; refer to PDF]

Figure 3: The link method of shape variables.

[figure omitted; refer to PDF]

The node 3 is symmetric with node 1 along the dotted line in Figure 1; thus, a sentence should be defined to describe the shape variable link relationship; that is,

: [figure omitted; refer to PDF] .

The definitions of shape variables and variable link are further explained in Figures 2 and 3, respectively.

4.2. GA Execution Process

4.2.1. Hybrid Coding Strategy of Shape and Topology Variables

In GATA, discrete variables are optimized through GA. After introducing the discrete shape variables, the string of genes should include the information of both topology variables and shape variables. Decimal coding is adopted for nodal positions, while the topology variables keep using binary format. The gene of each individual could be written as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] represent the code of shape variables and topology variables, respectively. For instance, there are 2 shape variables and 7 topology variables in the truss of Figure 1; the gene of an individual is 1-3-1-1-1-1-1-0-1, which means the first shape variable taking the 1st coordinate in [figure omitted; refer to PDF] and the second shape variable taking the 3rd coordinate in [figure omitted; refer to PDF] . The corresponding truss configuration is shown in Figure 4.

Figure 4: Example of individual gene code.

[figure omitted; refer to PDF]

4.2.2. Generation of the Initial Population

The generation mechanism of the initial population is updated for involving shape variables. At the first/initial calling of GA, the initial population of the designs is generated randomly. Once the optimal members of the population have been obtained, the initial population of the next generation is generated according to the elite of former generations of the GA. That is to say, from the second calling of the GA, the initial population consists of three parts: (1) there are the optimal individuals of the former generations; (2) members which are generated according to the optimal individuals of the last generation; that is, [figure omitted; refer to PDF] [figure omitted; refer to PDF] sequentially mutate under control with a low probability (Section 4.2.4), while [figure omitted; refer to PDF] will approach 0 with a greater probability if the corresponding optimal size variable [figure omitted; refer to PDF] is small; (3) the mutation of [figure omitted; refer to PDF] is the same as that in (2), while [figure omitted; refer to PDF] mutate randomly with a given low probability. The mutation control technique of shape variables will be explained in Section 4.2.4.

According to our calculation experience, the population size and maximum evolutional generation should exceed twice the total design variables. If it is more than 100, then it will take 100.

4.2.3. Roulette-Wheel Selection

Roulette-wheel selection is used to select a father design and a mother design from the parent generation, which is easy to be executed. Suggesting that the population size is [figure omitted; refer to PDF] , the fitness value of [figure omitted; refer to PDF] th individual in [figure omitted; refer to PDF] th generation is [figure omitted; refer to PDF] ; then the probability of individual [figure omitted; refer to PDF] to be selected in the next generation is [figure omitted; refer to PDF]

It can be seen from (12) that the individual of higher fitness value has greater probability to be selected. The fitness value of each individual is obtained using (10), as described in Section 3.2.

4.2.4. Uniform Crossover

Uniform crossover is popularly applied in the GA since it could produce better individuals and has lower probability to break good individuals. Since decimal coding is adopted for nodal coordinates, while the topology variables keep using binary format, crossover operator could not be carried on between these two kinds of code. Uniform crossover which operates gene by gene is implemented to the two areas independently. Before crossover, two individuals are selected randomly as mother and father chromosomes. Then, for each gene, a random value [figure omitted; refer to PDF] within 0~1 is generated. Let [figure omitted; refer to PDF] value of gene from the mother and let [figure omitted; refer to PDF] value of gene from the father. Let [figure omitted; refer to PDF] value of gene from the first child and let [figure omitted; refer to PDF] value of gene from the second child. For 0/1 topology genes and integer-valued shape genes, [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the crossover probability. Repeat this process until a new population is generated with [figure omitted; refer to PDF] individuals.

4.2.5. Controlled Uniform Mutation of Shape Variables

Uniform mutation and controlled uniform mutation are implemented for 0/1 topology genes and integer-valued genes, respectively. For each gene, a random number [figure omitted; refer to PDF] between zero and one is generated. If [figure omitted; refer to PDF] (mutating probability), the gene is mutated. For 0/1 valued topology genes, the gene is mutated to its allelomorph ( [figure omitted; refer to PDF] ). For an integer-valued coordinate gene, a controlled mutation technique is implemented to limit the mutation range, which could decrease the numerical instability induced by the large change of coordinates and improve the accuracy of the first-level approximation functions.

Two parameters are included in the control mutation technique, which are mutation probability [figure omitted; refer to PDF] and move limit [figure omitted; refer to PDF] . Mutation operation is implemented to each point of shape gene sequentially with [figure omitted; refer to PDF] . First, if a particular point needs to mutate, let us assume that the number of coordinate positions with respect to this shape variable is [figure omitted; refer to PDF] and the present identifier number is [figure omitted; refer to PDF] , and then the upper limit [figure omitted; refer to PDF] and lower limit [figure omitted; refer to PDF] of allowable mutation range are obtained as [figure omitted; refer to PDF] Note that [figure omitted; refer to PDF] denotes the maximum integer not larger than [figure omitted; refer to PDF] .

Then, an integer between [figure omitted; refer to PDF] and [figure omitted; refer to PDF] will be generated as the mutation result. Normally, [figure omitted; refer to PDF] 0.001~0.5 and [figure omitted; refer to PDF] 0.3~0.5.

4.3. Update Strategy of the First-Level Approximation Problem

In [figure omitted; refer to PDF] th iteration process of GATA for truss shape and topology optimization, the results of the structural and sensitivity analysis at [figure omitted; refer to PDF] are used to construct the first-level approximation problem using the multipoint approximation function. After introducing the shape variables, the truss shape [figure omitted; refer to PDF] might be different from that in the last iteration. Therefore, it is necessary to update the first-level approximation problem, so as to make it correspond to the present shape. The update strategy of the first-level approximation problem is then modified as follows. If the shape code of the optimal individual is inconsistent with that of the last iteration, a new first-level approximation problem will be built, and the number of known points [figure omitted; refer to PDF] will be set as 1; else the first-level approximation problem is consistent with the last iteration and increases the number of known points [figure omitted; refer to PDF] . The update strategy of the first-level approximate problem in the whole optimization process is emphasized in the algorithm flowchart (Figure 5).

Figure 5: The flowchart of the present approach.

[figure omitted; refer to PDF]

4.4. Algorithm Flowchart

The flowchart of the IGATA (Improved Genetic Algorithm with Two-Level Approximation) for truss size/shape/topology optimization is shown in Figure 5. After getting the optimal [figure omitted; refer to PDF] from the GA, a convergence criterion in (15) is used to determine whether the first-level approximate problem is terminated. Here, [figure omitted; refer to PDF] is size variables convergence control parameter, [figure omitted; refer to PDF] is weight convergence control parameter, [figure omitted; refer to PDF] is the constraints control parameter, and [figure omitted; refer to PDF] is the maximum iterative number for first-level approximate problem. The computational cost of IGATA is low because the first-level approximate techniques reduce the number of structural analyses significantly and the second-level approximate techniques reduce the number of the design variables significantly [5, 12]: [figure omitted; refer to PDF]

5. Numerical Examples

5.1. Ten-Bar Truss

The ten-bar truss has been studied by Rajan [6], as shown in Figure 6. The unit of length is inch. Node 6 is the original point. The length of bars 1, 2, 3, 4, 5, and 6 is 360 in. Nodes 4 and 6 are separately applied to a force of 100000 lb. Young's modulus is [figure omitted; refer to PDF] Psi and material density is 0.1 lb/ [figure omitted; refer to PDF] . The section area of each bar is taken as independent variable, which is originally 10 in.² and is permitted to vary between 1 in.² and 34 in.² . [figure omitted; refer to PDF] coordinates of nodes 1, 3, and 5 are taken as independent shape variables. The moveable range of [figure omitted; refer to PDF] coordinate is 180 in. to 1000 in. Discrete coordinates are preset for the shape variables, as shown in Table 1. The shape variables and link relation are defined as definition. Therefore, there are 10 size variables, 10 topology variables, and 3 shape variables in all. The stress of each bar should not exceed ±25,000 Psi.

Table 1: Shape variables and coordinates identifier number.

S.V. ID	Coord.	Position number and corresponding coordinates
1	2	3	4	5	6	7	8
1	[figure omitted; refer to PDF]	360	610	687	760	787	810	860	1000
2	[figure omitted; refer to PDF]	300	330	360	450	540	554	560
3	[figure omitted; refer to PDF]	180	187	190	200	210	230	260	360

Note: S.V. are shape variables.

Figure 6: Ten-bar truss structure.

[figure omitted; refer to PDF]

The parameters of GA are set as follows: population size 30, evolution generations 35, crossover probability 0.8, and mutation probability 0.05. The optimized solution of the shape, topology, cross-sectional areas, structural weight, and constraint obtained by the present approach is listed in Table 2, for comparison with [6]. It is seen from Table 2 that the critical constraint is very close to the boundary and the optimal weight of this paper is 3173 lb, which is lower than the result of [6] by 81 lb. The optimized shape and topology configuration are contrasted in Figure 7. The iteration history is shown in Figure 8. It is seen that the optimized solution is obtained after only 4 iterations. This example demonstrated the validity and efficiency of the proposed method.

Table 2: Comparison of optimized design of ten-bar planar truss.

Variable	Initial design	Present paper	Reference [6]
[figure omitted; refer to PDF]	360	180	186.5
[figure omitted; refer to PDF]	360	330	554.5
[figure omitted; refer to PDF]	360	687	786.9
[figure omitted; refer to PDF]	10	21.6	9.9
[figure omitted; refer to PDF]	10	1.0	9.4
[figure omitted; refer to PDF]	10	12.6	11.5
[figure omitted; refer to PDF]	10	12.6	1.5
[figure omitted; refer to PDF]	10	4.4	0
[figure omitted; refer to PDF]	10	17.1	12.0
[figure omitted; refer to PDF]	10	0	11.5
[figure omitted; refer to PDF]	10	2.6	3.6
[figure omitted; refer to PDF]	10	0	0
[figure omitted; refer to PDF]	10	1.0	10.4
Struc. analyses		20	-
Weight (lb)		3173.0	3254.0
Critical constraint		2.1 × 10-4	-

Figure 7: The optimized shape and topology of ten-bar truss structure.

(a) Optimal shape and topology in this paper

[figure omitted; refer to PDF]

(b) Optimal shape and topology in [6]

[figure omitted; refer to PDF]

Figure 8: Iteration history of ten-bar truss.

[figure omitted; refer to PDF]

5.2. Twelve-Bar Truss

A twelve-bar truss has been studied by Zhang et al. [13], as shown in Figure 9. The unit of length is mm. The structural symmetry should be kept in the design process. Young's modulus is [figure omitted; refer to PDF] Pa and material density is 1 kg/mm³ . The section area of each bar is taken as independent variable, which is originally 10 mm² , and is permitted to vary between 1 mm² and 100 mm² . [figure omitted; refer to PDF] and [figure omitted; refer to PDF] coordinates of nodes 2 and 5 are taken as independent shape variables. The moveable range of [figure omitted; refer to PDF] coordinate is 0 mm to 50 mm, and the moveable range of [figure omitted; refer to PDF] coordinate is 0 to [figure omitted; refer to PDF] . Discrete coordinates are preset for the shape variables, as shown in Table 3. The shape variables and link relation are defined as definition. Therefore, there are 12 size variables, 12 topology variables, and 4 shape variables in all. The stress of each bar should not exceed ±450 Pa.

Table 3: Shape variables and coordinates identifier number of 12-bar planar truss.

S.V. ID	Coord.	Position number and corresponding coordinates
1	2	3	4	5	6	7
1	[figure omitted; refer to PDF]	5	10	15	20	25
2	[figure omitted; refer to PDF]	30	35	40	45
3	[figure omitted; refer to PDF]	2	5	10	15	20	25	30
4	[figure omitted; refer to PDF]	2	5	10	15	20	25	30

Figure 9: Twelve-bar truss structure.

[figure omitted; refer to PDF]

The parameters of GA are set as follows: population size 50, evolution generations 50, crossover probability 0.9, and mutation probability 0.05. The optimized solution of the shape, topology, cross-sectional areas, structural weight, and constraint obtained by the present approach is listed in Table 4, for comparison with [13]. The optimized shape and topology configuration are contrasted in Figure 10. The iteration history is shown in Figure 11. It is seen that the final structural weight is 1023 kg, which is lower than the result in [13] by 109 kg, and the critical constraint is very close to the boundary. The optimized solution is obtained after only 4 iterations. This example demonstrated the validity and efficiency of the proposed method.

Table 4: Comparison of optimized design of twelve-bar planar truss.

Variable	Initial design	Present paper	Reference [13]
[figure omitted; refer to PDF]	15	10	31.18
[figure omitted; refer to PDF]	30	35	38.30
[figure omitted; refer to PDF]	15	10	9.21
[figure omitted; refer to PDF]	15	5	7.28
[figure omitted; refer to PDF]	10	4.11	1.00
[figure omitted; refer to PDF]	10	15.90	14.50
[figure omitted; refer to PDF]	10	1.13	1.98
[figure omitted; refer to PDF]	10	14.73	10.60
[figure omitted; refer to PDF]	10	1.17	2.10
[figure omitted; refer to PDF]	10	12.88	10.80
[figure omitted; refer to PDF]	10	0	1.46
[figure omitted; refer to PDF]	10	1.00	1.00
[figure omitted; refer to PDF]	10	3.23	5.73
[figure omitted; refer to PDF]	10	1.74	2.51
[figure omitted; refer to PDF]	10	2.59	1.00
[figure omitted; refer to PDF]	10	0	1.00
Struc. analyses		17	-
Weight (kg)		1023.3	1132.6
Critical constraint		2.1 × 10-4	-

Figure 10: The optimized shape and topology of twelve-bar truss structure.

(a) Optimal shape and topology in this paper

[figure omitted; refer to PDF]

(b) Optimal shape and topology in [13]

[figure omitted; refer to PDF]

Figure 11: Iteration history of twelve-bar truss.

[figure omitted; refer to PDF]

6. Algorithm Performance

Consider the example of ten-bar truss in Section 5, with population size and maximum generations ( [figure omitted; refer to PDF] ) set from 10 to 100, respectively, which is shown in Figure 12, while other parameters remain as given before. At each parameter set point, 100 independent runs of IGATA are executed. Since there are 100-parameter set points, IGATA is executed in a total of 10,000 times. For each parameter set point, the average weight is shown in Figure 13.

Figure 12: GA parameters set point.

[figure omitted; refer to PDF]

Figure 13: Average weight at each set point.

[figure omitted; refer to PDF]

It can be seen from Figure 13 that the minimum weight is 2800 lb, which is less than the weight of the initial structure by 32.6%. To describe the efficiency of the IGATA involving size/shape/topology variables, we counted the number of the results that are lower than 3254 lb, which is the optimal result in [6]. It can be seen that 8 results with lower weight are obtained within the 100-parameter set points.

As compared with the IGATA only including size and topology variables [5], the algorithm performance in this paper is not so satisfactory. To test the reason for this situation, continuous shape variables instead of discrete variables were used in the hybrid coding strategy in GA. The results of repeated tests show that the algorithm performance does not improve obviously. Thus the main reason may not lie in the continuity of shape variables but lie in the quality of the first-level approximation function induced by the shape variables. When executing GA in [figure omitted; refer to PDF] th iteration process of IGATA for truss shape and topology optimization, the objective and constraint approximation functions of the optimal individual from the last iteration are used, which do not change along with the structure shape, although controlled mutation has been implemented.

To improve the accuracy and efficiency of IGATA, we will use continuous shape variables and add sensitivity information of shape variables in the first-level approximation problem in the subsequent work.

7. Conclusion

In this paper, aiming at simultaneous consideration of sizing, shape, and topology optimization of truss structures, a design method IGATA is presented, which is based on the truss sizing and topology optimization method GATA. The shape variables are involved by using GA and are considered as discrete to avoid the sensitivity calculation, through which the computational cost is decreased significantly. A comprehensive model is established for involving the three kinds of design of variables, in which the shape variables are corresponding to a set of discrete node coordinates. GA is used to solve the first-level approximate problem which involves sizing/shape/topology variables. When calculating the fitness value of each member in the current generation, a second-level approximation method is used to optimize the continuous size variables. The definition, link, and code of the shape variables are presented, and the crossover and mutation of the decimal/binary mix-coding population are realized. The update strategy of the first-level approximation problem is also improved for the cases when the truss shapes are different from the neighbor iterations, so as to ensure that the truss shape is corresponding with the approximation problem. The results of numerical example demonstrated the validity of the method. Moreover, truss optimization problem with sizing/shape/topology variables can be treated effectively with the proposed method.

Nomenclature

[figure omitted; refer to PDF] :

Size variable vector

[figure omitted; refer to PDF] :

Cross-sectional area of bar members in [figure omitted; refer to PDF] th group

[figure omitted; refer to PDF] :

Shape variable vector

[figure omitted; refer to PDF] :

Identifier number within the possible coordinates set [figure omitted; refer to PDF]

[figure omitted; refer to PDF] :

Topology variable vector

[figure omitted; refer to PDF] :

Total weight of the truss structure

[figure omitted; refer to PDF] :

The weight of [figure omitted; refer to PDF] th group

[figure omitted; refer to PDF] :

[figure omitted; refer to PDF] th constraint

[figure omitted; refer to PDF] :

The total number of constraints

[figure omitted; refer to PDF] :

The number of frequency constraints.

Acknowledgment

This research work is supported by the National Natural Science Foundation of China (Grant no. 11102009), which the authors gratefully acknowledge.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

[1] L. Wei, T. Tang, X. Xie, W. Shen, "Truss optimization on shape and sizing with frequency constraints based on parallel genetic algorithm," Structural and Multidisciplinary Optimization , vol. 43, no. 5, pp. 665-682, 2011.

[2] D. Wang, W. H. Zhang, J. S. Jiang, "Truss optimization on shape and sizing with frequency constraints," AIAA Journal , vol. 42, no. 3, pp. 622-630, 2004.

[3] O. Sergeyev, Z. Mroz, "Sensitivity analysis and optimal design of 3D frame structures for stress and frequency constraints," Computers and Structures , vol. 75, no. 2, pp. 167-185, 2000.

[4] R. Su, X. Wang, L. Gui, Z. Fan, "Multi-objective topology and sizing optimization of truss structures based on adaptive multi-island search strategy," Structural and Multidisciplinary Optimization , vol. 43, no. 2, pp. 275-286, 2011.

[5] D. Li, S. Chen, H. Huang, "Improved genetic algorithm with two-level approximation for truss topology optimization," Structural and Multidisciplinary Optimization , vol. 49, no. 5, pp. 795-814, 2014.

[6] S. D. Rajan, "Sizing, shape, and topology design optimization of trusses using genetic algorithm," Journal of Structural Engineering , vol. 121, no. 10, pp. 1480-1487, 1995.

[7] R. J. Balling, R. R. Briggs, K. Gillman, "Multiple optimum size/shape/topology designs for skeletal structures using a genetic algorithm," Journal of Structural Engineering , vol. 132, no. 7, pp. 1158-1165, 2006.

[8] P. Hajela, E. Lee, "Genetic algorithms in truss topological optimization," International Journal of Solids and Structures , vol. 32, no. 22, pp. 3341-3357, 1995.

[9] H. Kawamura, H. Ohmori, N. Kito, "Truss topology optimization by a modified genetic algorithm," Structural and Multidisciplinary Optimization , vol. 23, no. 6, pp. 467-472, 2002.

[10] W. Tang, L. Tong, Y. Gu, "Improved genetic algorithm for design optimization of truss structures with sizing, shape and topology variables," International Journal for Numerical Methods in Engineering , vol. 62, no. 13, pp. 1737-1762, 2005.

[11] K. Sawada, A. Matsuo, H. Shimizu, "Randomized line search techniques in combined GA for discrete sizing optimization of truss structures," Structural and Multidisciplinary Optimization , vol. 44, no. 3, pp. 337-350, 2011.

[12] Y. Dong, H. Huang, "Truss topology optimization by using multi-point approximation and GA," Chinese Journal of Computational Mechanics , vol. 21, no. 6, pp. 746-751, 2004.

[13] Z. Zhang, W. Yao, L. Zhou, "Study on size and shape collaborative optimization mehtod of truss strucure," Advances in Aeronautical Science and Engineering , vol. 3, no. 2, pp. 138-143, 2012.