A Hybrid Method Combining Improved Binary Particle Swarm Optimization with BESO for Topology Optimization Yin Zhang, Liang Gao, Hao Li
A Hybrid Method Combining Improved Binary Particle Swarm Optimization with BESO for Topology Optimization 1
Yin Zhang, 2Liang Gao, 3Hao Li 1. The State Key Laboratory of Digital Manufacturing Equipment and Technology, School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan, Hubei, China The State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, Hunan, China
[email protected] *2. The State Key Laboratory of Digital Manufacturing Equipment and Technology, School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan, Hubei, China,
[email protected] 3. The State Key Laboratory of Digital Manufacturing Equipment and Technology, School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan, Hubei, China,
[email protected]
Abstract This paper presents a new approach for topology optimization by combining improved binary particle swarm optimization (BPSO) with bi-directional evolutionary structural optimization (BESO). A dual coding method is employed to represent individuals and population for finite element analysis. Three operators of BPSO update, communication and mutation are developed for the improved BPSO. Several 2D and 3D examples of minimum mean compliance problems are performed to demonstrate the efficiency of this method and some of them are compared with the SIMP and the BESO methods.
Keywords: Topology Optimization, Improved Binary Particle Swarm Optimization, Bi-Directional Evolutionary Structural Optimization, Dual Coding Method, Finite Element Analysis
1. Introduction Topology optimization for continuum structures aims at finding the optimum distribution of materials within the design domain under given constraints and loads to reduce structural weight and improve structural performance. One of the most typical methods for topology optimization is the evolutionary structural optimization (ESO) method [1]. The basic concept of ESO is that by gradually removing inefficient elements, the structure evolves towards an optimum. But it may lead to a non-optimal solution, for once an element has been removed it cannot be recovered. The bi-directional evolutionary structural optimization (BESO) method, which is an extension of ESO, was proposed to solve this problem [2]. In BESO, elements are allowed to be added and deleted simultaneously. The elemental removal in ESO/BESO is controlled by the evolutionary ratio (ER), which determines the target volume of each iteration and governs the efficiency. At the end of each iteration, solid elements with sensitivities below the defined level will be removed while void
International Journal of Advancements in Computing Technology(IJACT) Volume5, Number1,January 2013 doi: 10.4156/ijact.vol5.issue1.44
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A Hybrid Method Combining Improved Binary Particle Swarm Optimization with BESO for Topology Optimization Yin Zhang, Liang Gao, Hao Li
elements with sensitivities above the level will be added. The procedure will be repeated until it reaches the objective volume and convergents to an optimum. The result of ESO/BESO provides a clear topology without grey area for it only deals with discrete design variables, making it easy to manufacture. However, due to the heuristic feature, sometimes the original ESO/BESO methods fail to achieve an optimal solution [3]. Huang and Xie [4][5] presented a modified BESO method to deal with the mesh-dependency and non-convergent solutions problem. Particle swarm optimization (PSO)[6] is a population-based algorithm inspired by the behavior of group animals, the update formula can be expressed as follow (1)
,
(2) where
and
are the acceleration constant coefficients.
evenly distributed between 0 and 1.
,
and
particle and the global best position of all the particles.
and
are two random numbers
are the best previous position of the ith is the inertia weight [7].
Binary particle swarm optimization (BPSO) [8] extends the basic concept of PSO, but the particles move in the search space restricted to 0 or 1 on each dimension. In BPSO, the updating Eq. (1) remains the same while the particle changes the value of its bit string as follow
(3)
where
0
1
(4)
is the sigmoid function to transform the particle’s velocity into binary value 0 or 1
and rand( ) is a random number uniformly distributed in [0, 1] Literatures using BPSO for topology has been found in recent years [9][10]. Tseng et al.[9] proposed an enhanced binary particle swarm optimization (EBPSO) based on pure binary bit-string frameworks to deal with structural topology problems with two enhancement strategies, namely the stress-based and pair-switched strategies. Luh et al.[10] proposed a modified BPSO algorithm adopting the concept of genotype-phenotype and multiple diverse topology structures could be obtained through it. But due to the stochastic nature of BPSO, the structural discontinuity may occur, making the structure incapable of carrying and transferring loads. Although literatures using BPSO for topology optimization [9][10] have performed structural connectivity analysis to deal with such a problem, the computational costs are so enormous that these two methods have only dealt with small-scale problems and not been extended to large-scale problems especially 3D structures. Our approach proposes the first study to combine the features of BPSO with BESO by utilizing a dual coding method. The improved BPSO operators consisting of BPSO update, communication and
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A Hybrid Method Combining Improved Binary Particle Swarm Optimization with BESO for Topology Optimization Yin Zhang, Liang Gao, Hao Li
mutation will be applied locally on the elements and the treatments of elements (delete or add) are based on their fitness, reducing the computational cost significantly. A penalization mechanism is developed to deal with the connectivity problem. Several 2D and 3D examples are presented to demonstrate the efficiency of the proposed method. To realize the bi-directional characteristic, a filter scheme proposed by Huang and Xie [4] is used to extrapolate sensitivities to void elements.
2. Sensitivity calculation and filter scheme 2.1. Problem formulation Usually in the BESO method, obtaining the overall structural stiffness with volume constraint can be stated as follow [11]
Minimize Subject to:
∗
∑
0 1
Where K and u are the global stiffness matrix of the structure and the displacement vector. prescribed total structural volume and elements.
(5)
∗
is the
is the volume of a single element. N is the number of
is a binary design variable which denotes the density of an individual element(‘1’
denotes solid elements while a small value of
e.g. 0.001 represents void elements).
2.2. Sensitivity calculation and filter scheme The elemental sensitivity represents the efficiency of one element and the element removal/addition is based on the ranking of them. In order to obtain mesh-independency and convergent solutions, the ), elemental sensitivities needs to be processed with the filter scheme [4]. The elemental sensitivities need to be history-averaged to stabilize the evolutionary procedure and improve the accuracy of the elemental sensitivity [4][11] .
nodal sensitivity numbers (
3. Implementation of the proposed method 3.1. Coding and population size The proposed method applies a dual coding method. As it is shown in Figure 1, on the whole structure, the code ‘1’ represents solid element, while the code ‘0’ represents void element. Besides, elements are treated as individuals, having a binary string of a certain uniform length. The binary strings have no physical meaning but they are the basic for the individuals’ later performance. In the initialization step, binary strings with pure ‘1’-digits are assigned to solid elements, while for void elements, mixed strings with a certain low percentage of ‘1’-digits are assigned to them so that
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A Hybrid Method Combining Improved Binary Particle Swarm Optimization with BESO for Topology Optimization Yin Zhang, Liang Gao, Hao Li
they might have chances to be added into the design domain. Normally the length should be longer than 6 to balance between speed and accuracy.
Figure 1 A simple example of dual coding method Initially the whole population is divided into three groups according to the relative ranking of their sensitivities. The size of the superior group with highest sensitivities is set to be
, which
is constant during the process. The rest are divided into two groups. At the beginning before the first iteration, the size of the inferior group is set according to the evolutionary ratio (ER), later it is determined by the sum of ER and the progress. And the maximum size of the inferior group is 1
.
Since BESO allows both adding and deleting elements simultaneously, we use another updating equation which can be expressed as follow (6)
,
where
,
and
have similar meaning with
,
and
.
3.2. Improved BPSO operators The BPSO update operator performs over the superior and inferior group to help the efficient elements have more ‘1’-digit while inefficient ones have more ‘0’-digits by using Eq. (1) and (6). But it does not perform over the intermediate group to reduce the chance of removing or adding moderate-effective elements. As a result, no sudden change of the whole structure will occur. The communication operator is performed over the intermediate group to keep population diversity. The probability to choose a partner from the same group is
, which is larger than 0.5 because large
makes the society of that group stable. Take the ith element in the intermediate group as an example, if the partner comes from the same group, they will exchange some segments of their binary strings; otherwise some segments of the partner’s binary string will be inherited by the ith particle while the partner’s binary string remains unchanged. The mutation operator is an auxiliary measure to overcome the oscillating nature of the BPSO and
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A Hybrid Method Combining Improved Binary Particle Swarm Optimization with BESO for Topology Optimization Yin Zhang, Liang Gao, Hao Li
get out of local optimum. It tends to manifold ‘1’-digits and ‘0’-digits in the binary strings of individuals in the superior and inferior group by changing the value of one digit in the binary strings .
directly. The probability for one digit to mutate is set to be
3.3. Selection rules and convergence criterion The selection rules are based on the number of ‘1’-digits (fitness) in the binary strings. At the end of each iteration, elements with pure ‘0’-digits will be removed while others remain in the design domain to build up a new design. And elements, which were void in the last iteration and have current binary strings with a certain percentage of ‘1’-digits, will be added into the design domain. The percentage is set to be 50% in this paper. The optimization process will repeated until both the objective volume V ∗ and the following convergence criterion are satisfied [11] |∑
|
(7)
∑
where k is the current iteration number and
is the allowable convergence error. N is an integral is set to be 0.001 and N is selected as 5.
number depending on the convergence speed. In this paper,
3.4. Controlling parameters A penalization mechanism is introduced to deal with the connectivity problem by pushing the probability parameters like
and
gradually to their maxima with the development of the
process. For example, the penalization mechanism for ,
where
,
and
,
,
can be stated as follow (8)
,
are the minimum and maximum values for
. Prog is the progress
indicator of the optimization process. Pen is a penalty factor controlling the development of Similar penalization mechanism can be applied to
.
. But for the inertia weight ω, to balance
between global search and local search[12], the self-adaptive inertia weight can be expressed as ∗
(9)
In this paper, the evolutionary ratio (ER) indicates the search region, which is different from the ER in conventional BESO, thus the target volume of each iteration is not strictly controlled. To sum up, the proposed method uses BESO to control global search region based on the elemental sensitivities while utilizes the improved BPSO to perform local search. It tends to favor BESO when the volume approaches the object volume
∗
. In later iterations the elemental treatment will more
and more directly guided by the sensitivities so that the proposed method will be less indeterminate.
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A Hybrid Method Combining Improved Binary Particle Swarm Optimization with BESO for Topology Optimization Yin Zhang, Liang Gao, Hao Li
4. Optimization procedure of the proposed method
Figure 2. Flow chart of the hybrid method The steps of the proposed method are shown by a flowchart given in Figure 2. Note that when the current volume satisfies the volume constraint the intermediate group will disappear and the individuals will be divided into only two groups (superior and inferior group), thus the communication operator will be skipped.
5. Numerical examples and discussions The parameters setting for all examples are listed in Table 1. These parameters were determined according to literatures [13][14] and through numerical experiments after multiple simulation runs. Table 1. Parameter setting of the proposed method Parameter
Value
ωmax
1.2
ωmin
0.8
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A Hybrid Method Combining Improved Binary Particle Swarm Optimization with BESO for Topology Optimization Yin Zhang, Liang Gao, Hao Li
Vmax
4
Vmin
-4
C1
1.4962
C2
1.4962
Binary string length
10
Max iteration number of improved BPSO
50
ER
5%
Rc,min
0.6
Rc,max
1
Rm,min
0.5
Rm,max
1
Pen
3
5.1. Cantilever Example 5.1 presents a cantilever beam. The design domain is divided into 60×30, namely 1800 four-node quadrilateral elements. The Young’s modulus is 1MPa and the Poisson’s ratio is 0.3. The volume fraction is 0.5 with a filter radius of 3mm. Six runs were performed to show the efficiency of the hybrid method with the results of BESO and SIMP with the same model[15][16]. The boundary conditions, loading and final structural layouts of the proposed method, SIMP and BESO are displayed in Figure 3. We can see that all the topologies are quite similar except that the topology derived from SIMP has blurry boundary because the density in SIMP is treated as continuous design variables [17], adding some difficulties to distinguish and manufacture. Since the proposed method directly deals with discrete design variables (
or 1) with the penalty exponent p=3, it will not
suffer from such a problem.
Figure 3. Cantilever problem and final structural layouts The results of the six runs and BESO are shown in Table 2. It can be seen that the computation efforts of the proposed method and BESO are similar, and the proposed method can obtain better results. Both of these two methods are based on the simple concept of slowly removing inefficient elements, but unlike BESO, the proposed method tends to remove inefficient elements with a certain probability while the searching direction is still guided by its sensitivity. Through random local search it is likely to get better solutions and get out of local optimum. Besides, one can see that the mean
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A Hybrid Method Combining Improved Binary Particle Swarm Optimization with BESO for Topology Optimization Yin Zhang, Liang Gao, Hao Li
compliance of the SIMP method is higher than that of the other two. This is caused by the over-estimated strain energy of intermediate density elements in the topology [4]. Table 2. Comparison of computational effort and mean compliance nth run
Total iterations
Mean compliance (N·mm)
1
51
31.3895
2
47
31.4272
3
41
31.3368
4
48
31.3940
5
50
31.5725
6
47
31.4255
Average
47.67
31.4243
Soft-kill BESO
46
31.7690
SIMP
50
36.4234
The evolutionary histories of the volume and mean compliance for the first run are given in Figure 4 One can see that the volume gradually decreases while the mean compliance increases with occasional sudden jumps caused by a significant change in the structure with some bars breaking up in the intermediate design. But soon the mean compliance returns to normal state and the procedure develops towards the right direction. When the volume reached the objective volume, it keeps constant until the mean compliance satisfies the convergence criterion.
Figure 4. Evolutionary histories for the first run in example 5.1
5.2. Initial guess design To verify that the hybrid method has the ability to optimize problems in both directions, this example presents a minimum mean compliance problem for a cantilever from initial guess design shown in Figure 5 with the initial guess design. The whole structure is discretized into 80×50, totally
402
A Hybrid Method Combining Improved Binary Particle Swarm Optimization with BESO for Topology Optimization Yin Zhang, Liang Gao, Hao Li
4000 plane stress four-node quadrilateral elements, where parameters are: ER=5%, E=206MPa, ν=0.3. The remaining volume is 50% of the design domain with a filter radius of 3mm. The penalty exponent in this example is also set to be 3.
Figure 5. Dimensions of the design domain, loading, boundary conditions and initial design Table 3 gives the total iterations, mean compliance for six runs and the average values of them. It indicates that starting from an initial guess design can save computation time for large-scale problems when an initial guess design is not far from the optimum solution. However, if an initial guess design is far different from optimum it may demand more number of iterations to reach the optimum. Table 3. Records of six runs and their average values nth run
Total iterations
Mean compliance (N·mm)
1
32
922.0978
2
28
922.7601
3
26
923.8096
4
38
919.2018
5
28
922.4605
6
31
921.5840
Average
30.5
921.9856
The final topologies of the six runs are displayed in Figure 6. One can see that the average iteration number is smaller than that from the initial full design, all the six topologies are quite similar and their mean compliances are very close, therefore the results seem to be stable. The filter scheme helps extrapolate the sensitivity numbers for void elements through filtering the sensitivity numbers of their neighboring solid elements [16]. Besides, since at early stages the size of the superior group is far larger than that of the inferior group, the individuals in the intermediate group has larger probability to perform inter-group communication operator with the superior group, which tends to help them have more ‘1’-digits in their binary strings. So the void elements may have the chance to be added into the design domain.
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A Hybrid Method Combining Improved Binary Particle Swarm Optimization with BESO for Topology Optimization Yin Zhang, Liang Gao, Hao Li
Figure 6. Results of example 5.2
5.3. 3D cantilever
Figure 7. 3D cantilever problem and final structural layout The proposed method can be easily extended to 3D problems. The design domain, boundary conditions, loading of the 3D cantilever and the final structural layout are shown in Figure 7. The structure is modeled using 32×10×20, totally 6400 eight-node brick elements. The object volume is 10% of the design domain with a filter radius of 1.8m. The Young’s Modulus is 206GPa and Poisson’s ratio is 0.3. The evolutionary histories of the mean compliance and volume are shown in Figure 8. Totally 40 iterations have been used to get a convergent solution and the minimum mean compliance is 41.2708N·m. It is difficult for common versions of BPSO to solve these problems, for with the same fine mesh the finite element analysis has to be performed over a number of individuals when updating particles, the problem size is so enormous. But in this proposed method, updating particles are performed locally over the individuals, in each iteration only one time finite element analysis is needed. This will reduce the computational costs significantly thus the proposed method could deal with large-scale problems.
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A Hybrid Method Combining Improved Binary Particle Swarm Optimization with BESO for Topology Optimization Yin Zhang, Liang Gao, Hao Li
Figure 8. Evolutionary histories of example 5.3
6. Conclusions This paper has proposed a hybrid method combining improved binary particle swarm optimization with BESO. The improved BPSO operators consisting of BPSO update, communication and mutation have been developed. The method performs the search for optimum in a semi-stochastic way while the search direction and space are strongly guided by the sensitivities. It deals with connectivity problems through a penalization mechanism on probability parameters. Several 2D and 3D examples have been performed to demonstrate the efficiency of the proposed method and its capability of adding and removing elements simultaneously like BESO. Comparing with SIMP and BESO, it is able to obtain better solution. Besides, it can be easily extended to large-scale optimization problems especially for 3D structures comparing with other versions of BPSO. Although this method has stochastic feature, through numerical experiments one can see that it is able to produce solutions converging to similar results.
Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 51175197) and the Open Research Fund Program of the State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body (No. 31115020).
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