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Comparison of two multi-objective optimization methods for composite radiation shielding materials
Applied Radiation and Isotopes 159 (2020) 109061
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Applied Radiation and Isotopes journal homepage: http://www.elsevier.com/locate/apradiso
Comparison of two multi-objective optimization methods for composite radiation shielding materials Yao Cai a, b, *, Rui Hao a, Shaojie Yu a, Chang Wang a, Guang Hu b a b
China Ship Development and Design Center, Wuhan, 430064, China School of Nuclear Science and Technology, Xi’an Jiaotong University, Xi’an, 710049, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Multi-objective Optimization Nuclear Shielding material
Combining the SCE algorithm (Shuffled Complex Evolution), MOEA/D algorithm (Multi-Objective Evolutionary Algorithm based on Decomposition), MCNP program and several prediction models, two multi-objective opti mization methods (priori method and posteriori method) for radiation shielding material, which considering the shielding, mass, volume, mechanical and thermal properties are established. The material is in the form of resin matrix composite. The shielding performance of the material is simulated by MCNP program. The mechanical property and thermal property are calculated by some widely used prediction models. Several materials are optimized by the two methods respectively, and comparisons among the materials are made. The results show that, both the two methods could achieve synergistic optimization of shielding, mass, volume, mechanical properties and thermal properties of material. Differently, the priori method only obtains one solution corre sponding to its weight values, while the posteriori method obtains the whole Pareto-optimal set. These two methods have their own advantages and disadvantages, which should be selected according to the actual situation.
1. Introduction Radiation shielding material is an important component of the nu clear facilities. In general, the shielding of neutrons and gamma rays need materials contain both heavy and light elements. Resin matrix composites have become an important type of shielding materials because of its strong designability, easy processing and properties of various components (Nambiar and Yeow, 2012). In addition, some other properties are also quite important for the shielding materials. Gener ally, the high-efficiency shielding material for nuclear radiation should be as small as possible in mass and volume under the condition of meeting the shielding requirements, while the mechanical and thermal properties of the material as good as possible. Obviously, the above properties are conflicting and contradictory, and it is difficult for traditional materials to meet these requirements at the same time. These years, optimization techniques using genetic al gorithms (Cai et al., 2018a), differential evolution (DiJulio et al., 2016), linear programming (Newman and Asadi–Zeydabadi, 2008), etc. have gradually applied to improve the material designing, and several shielding material optimization methods (Ashayer et al., 2012; Tunes et al., 2017) have been built. However, the material properties
considered in the previous works are always insufficient. Most of them just considered the shielding performance, while the mechanical and thermal properties have not been considered. Thus, further research is needed. Different from the single objective optimization problems, the optimal solution of multi-objective optimization problems is not unique, and there is a Pareto-optimal set. In this solution set, each solution at least has one better objective, while the other objectives are not inferior to the other solutions. It means that there is no absolute good or bad solutions for all the objectives, and the essence of solving multi-objective optimization problems is to select one or more solutions from the solu tion set according to the domain knowledge and personal preferences. Based on the manner of giving preference information, the multiobjective optimization methods can be divided into: 1) priori method (traditional multi-objective optimization method); 2) posteriori method (evolutionary multi-objective optimization method). Thus, there are also two multi-objective optimization methods for composite shielding material optimization. It is necessary to discuss the two methods and have a comparison between them. This study exactly addresses this problem. First, the characterization of material properties are presented, and
* Corresponding author. China Ship Development and Design Center, Wuhan, 430064, China. E-mail address: [email protected] (Y. Cai). https://doi.org/10.1016/j.apradiso.2020.109061 Received 9 October 2019; Received in revised form 4 January 2020; Accepted 24 January 2020 Available online 14 February 2020 0969-8043/© 2020 Elsevier Ltd. All rights reserved.
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2.1.2. Mechanical property of the material Elastic modulus is the most significant mechanical properties of composite material. It describes the difficulty to deform the materials, and should be kept as large as possible. The common used Kerner model (Kerner, 1956) as follows was adopted. �X ���X � ki ϕi ϕi (1) kc ¼ 3ki þ 4μm 3ki þ 4μm n P
μc ¼ μ
i¼2 mP n i¼2
Fig. 1. Schematic of the calculation model used in the optimization.
νc ¼
the two multi-objective optimization methods to optimize the shielding material are studied (Section 2). Second, several materials are optimized by the two methods respectively, and comparisons among them are made (Section 3). At last, the relationship between the solutions ob tained by the two methods and the application of the two methods are discussed (Section 4).
ϕi μi ð7 5νm Þμm þð8 10νm Þμi
þ 15ð1ϕmνm Þ
ϕi μm ð7 5νm Þμm þð8 10νm Þμi
þ 15ð1ϕmνm Þ
3kc 2μc 6kc þ 2μc
(2)
(3) (4)
Ec ¼ 2μc ð1 þ νc Þ
where, ki is the bulk modulus of component i; μi is the shear modulus of component i; ϕi is the volume fraction of component i; νi is the Poisson ratio of component i; Ei is the elastic modulus of component i; subscripts c, m, p represent the composite material, matrix material and rein forcement material respectively.
2. Methodology
2.1.3. Thermal properties of the material Thermal properties mainly include thermal expansion coefficient and thermal conductivity. For shielding materials, the thermal con ductivity should be as large as possible to reduce the radial temperature gradient of the shielding. The thermal expansion coefficient reflects the degree of thermal deformation of materials, and should be kept as small as possible.
2.1. Characterization of the material properties As mentioned in Section 1, the material properties considered in this study including shielding performance, mass, volume, mechanical properties and thermal properties. Among them, the mechanical prop erties are represented by the elastic modulus of the material, and the thermal properties are represented by the thermal expansion coefficient (CTE) and thermal conductivity of the material. It should be noted that, the composite radiation shielding material researched in this study is a kind of particle reinforced material. It is assumed that the reinforced particles are spherical (quite small) and uniformly dispersed in the matrix.
2.1.3.1. Thermal expansion coefficient of the material. The thermal expansion coefficient describes the change of material length caused by the change of unit temperature. The common used Schapery model (Elomari et al., 1998) as follows was adopted. � � � 1 kc 1 kp � αc ¼ αp þ αm αp � (5) 1 km 1 kp
2.1.1. Shielding performance, mass and volume of the material In this study, the shielding performance of materials is calculated by Monte Carlo program MCNP (with the ENDF/B-VI cross section set). Since thousands of transport calculations are needed during the opti mization, the spherical shell model as shown in Fig. 1 is used to reduce the calculation time (Cai et al., 2018b). Of which, an isotropic point source is placed at the center position, the thickness of shielding mate rial is set at 40 cm (about 10 free paths of neutrons in such materials), and the count surface is set 2 m away from the source (distance protection). The shielding performance is represented by the dose equivalent caused by transmitted neutrons and gamma rays. The smaller the value, the better the shielding performance. The F2 tally card was used with the DE and DF cards to assess the dose equivalentrate transmitted. For this, the NCRP-38 (Rossi, 1971) neutron flux-to-dose rate conversion factors and the 1977 ANSI/ANS (Battat, 1977) photon flux-to-dose rate con version factors were used. Obviously, the mass and volume of the material are both hoped to be as small as possible. The mass property is simple to describe with for mula m ¼ ρV, but the volume property is described slightly different due to the fixed thickness of shielding material used in the optimization. In our previous study (Cai et al., 2018a), it is found that the pro portion of secondary gamma rays in the total dose equivalent (Hng/Ht, the self-absorption capacity) can be used as a sub-objective to charac terize the volume property indirectly and effectively. The smaller the ratio, the smaller volume of material needed to achieve the same shielding requirement.
kc ¼ km þ
ϕp 1 kp km
þ km ϕþm4μ
(6)
3 m
where, αi is the thermal expansion coefficient of component i. 2.1.3.2. Thermal conductivity of the material. Thermal conductivity re flects the ability of materials to conduct heat. The common used Brug geman model (Lin et al., 2010) as follows was adopted. � �1=3 λp λc λm 1 ϕp ¼ (7) λp λm λc where, λi is the thermal conductivity of component i. It can be seen, the mechanical and thermal properties of the com posites can be obtained according to formulas (1) - (7) as long as the basic parameters and volume fractions of the components are known. That is to say, the above properties of the material are all directly related to its component X, which provides the precondition of material optimization. 2.2. Multi-objective optimization of the material To design a compact, lightweight shielding material with good me chanical and thermal properties, it is necessary to consider the effects of dose equivalent, mass, volume, mechanical properties and thermal properties comprehensively. For this multi-objective optimization 2
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(12)
L�X � U
where, N is the number of components in the composite material; X¼(x1, x2, …, xN) is the variable vector, mass components matrix in the com posite material; L is the component vector of minimum; U is the component vector of maximum; F(X) is the total fitness function value; c1 ; c2 ; c3 ; c4 ; c5 and c6 are the weight coefficient respectively; HT(X) is the total dose equivalent, Sv; Hn(X) is the dose equivalent of neutron, Sv; Hg(X) is the dose equivalent of gamma rays, Sv; Hng(X) is the dose equivalent of secondary gamma rays, Sv; M is the mass of material, kg; fE is the elastic modulus of the composites, Gpa; fk is the thermal expansion coefficient of the composites, μm⋅m 1⋅K 1; fλ is the thermal conductivity of the composite, W⋅m 1⋅K 1; H0T ; M0 ; f 0E ; f 0λ and f 0λ are the reference values to make the function dimensionless. In formula (8), the first term reflects the shielding performance, the second term reflects the mass, the third term reflects the volume, the fourth term reflects the mechanical properties, and the last two terms reflect the thermal performance. Different from the previous study (Cai et al., 2018a), the existing materials are no longer used as reference materials to reduce the effect of artificial factors. Instead, c1 ¼ 1; c2 ¼ c3 ¼ c4 ¼ c5 ¼ c6 ¼ 0, are pre-set, and a sub-optimal solution is obtained by optimizing 5 genera tions, which is used as reference materials for material optimization. The algorithm of SCE (the Shuffled Complex Evolution algorithm) is an efficient choice for the shielding optimization after evaluating 14 well known metaheuristic algorithms (Cai et al., 2018c). It has excellent global search capabilities and optimization efficiency. Thus, the algo rithm of SCE is adopted in this study. The steps in the optimization are shown in Fig. 2.
Fig. 2. Flow chart of the optimization.
problem, the priori method (traditional multi-objective optimization method) transforms it into a single-objective problem by linear weighting, and solves it using single-objective optimization technology then. The posteriori method (evolutionary multi-objective optimization method) obtains a Pareto-optimal set first, and selects the desired so lution from the set then according to personal preferences.
2.2.2. The evolutionary multi-objective optimization method Different from the traditional multi-objective optimization method, no prior informations (such as weight coefficients and dimensionless parameters) are needed in the evolutionary multi-objective optimization method. And the optimization problems can be presented as follows:
2.2.1. The traditional multi-objective optimization method For the multi-objective optimization problem researched in this study, the objective function can be written as follows: FðXÞ ¼ c1
HT M Hng f0 fk f0 þ c2 þ c3 þ c4 E þ c5 0 þ c6 λ 0 M0 HT fE fλ HT fk
N X
s:t:
where, f1 ¼ HT , f2 ¼ M, f3 ¼ Hng =HT , f4 ¼ 1=fE , f5 ¼ fk , f6 ¼ 1=fλ . The meanings of the parameters in the formula are same with those in sec tion 2.2.1. Over the past decades, many scholars have proposed various evolu tionary multi-objective optimization algorithms. The common used ones are SPEA-II, PESA-II, NSGA-II, MOPSO and MOEA/D. After evaluating these algorithms with some classical test functions, it is found that the algorithm of MOEA/D (Multi-Objective Evolutionary Algorithm based on Decomposition) has better convergence and diversity than the others. Thus, it is adopted to perform the evolutionary multi-objective
(9)
The optimization problems can be presented as follows: (10)
FðXÞ N X
s:t:
(11)
xi ¼ 1
(15)
L�X � U
where
min
(14)
xi ¼ 1 i¼1
(8)
HT ðXÞ ¼ Hn ðXÞ þ Hg ðXÞ
(13)
minFðxÞ ¼ ðf1 ðxÞ; f2 ðxÞ; ⋯; f6 ðxÞÞT
i¼1
Table 1 Material components optimized by the traditional multi-objective optimization method. The first column provides the names of the optimized material, where the symbol “FS” represents “fission energy spectrum of 235U”, and the Arabic numerals indicate the number of objects considered in the optimization. Material FS1 FS2 FS3 FS4 FS5 FS6
Weight coefficient
Material composition /wt%
c1
c2
c3
c4
c5
c6
PE
B4C
W
Fe
Cu
Pb
1 1 1 1 1 1
0 1 1 1.5 2 2.5
0 0 1 1 1 1
0 0 0 1 1 1
0 0 0 0 1 1
0 0 0 0 0 1
2.37 12.66 14.30 6.41 3.92 3.48
0.05 1.59 2.00 21.65 31.16 33.93
96.86 54.20 77.12 45.38 57.13 54.71
0.16 0.46 1.68 2.59 0.72 0.36
0.48 30.71 3.53 23.78 6.66 7.36
0.09 0.38 1.38 0.19 0.41 0.15
3
Density /g⋅cm 3 13.10 4.93 4.86 4.84 4.92 4.80
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Fig. 3. Comparison among shielding materials by the MCNP simulation.
optimization of composite radiation material follows.
components were obtained (Table 1). It can be seen, when the shielding performance is merely considered, the material mainly consist of W, and the density is quite large. After considering the mass of the material, the density decreases sharply. Because the density of the material optimized at this time is large enough, the ability of restraining and self-absorption of secondary gamma rays is quite good, the density does not change much after futher put the volume term into consideration. Later, because the elastic modulus, thermal expansion coefficient and thermal conductivity of the material are positively correlated with the volume fraction of the rein forced material, in order to control the density of the material, the weight of the mass term is increased with each additional factor in the subsequent material optimization. The results show that, except FS1, the densities of materials obtained by other optimizations are simply with a slight difference. To compare the shielding performance of various materials intui tively, 108 particles were simulated by the MCNP programm (Fig. 3). The results show that the shielding performance of FS1 is much better than the others, the performance of FS2 and FS3 are basically the same, FS4, FS5 and FS6 are slightly worse than the former, but generally similar. Besides shielding properties, the mechanical and thermal properties of materials are also very important in practical applications. Fig. 4 shows a comparison of the mechanical and thermal properties of the optimized materials. It can be seen that, with the increase of factors considered in the optimization, the comprehensive properties of
3. Simulation and results As a typical case, the fission energy spectrum of 235U (beam intensity is 1010 fission/s) which mixed neutrons and gamma rays was adopted as the radiation source in this study. It release 2.407 neutrons (Watt fission energy spectrum) and 7.77 gamma rays every event. An empirical for mula for the prompt gamma rays spectrum was employed (Schaeffer, 1973): 8 6:6 0:1 < Eγ � 0:6 MeV < 0:6 < Eγ � 1:5MeV 20:2 exp ð 1:78Eγ Þ NðEγ Þ ¼ : 7:2 exp ð 1:09Eγ Þ 1:5 < Eγ � 10:5MeV (16) To compare the two multi-objective optimization methods, com posites composed of polyethylene (PE), B4C, W, Fe, Cu and Pb were optimized by them respectively. 3.1. Optimization by the traditional multi-objective optimization method To find the optimal shielding material, the domain constraints of each variable are set at 0 to 1. The parameters of the algorithm are selected according to previous studies (Duan et al., 1993), and set at m ¼ 2nþ1, p ¼ 3, s ¼ pm, q ¼ nþ1, α ¼ 1, β ¼ 2nþ1. By adjusting the weight coefficients and running the program, a series of material 4
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Fig. 4. Mechanical and thermal properties of materials designed for Table 2 Performance parameters of the materials optimized for Material FS1 FS2 FS3 FS4 FS5 FS6 Ref
T /cm
ρ
51.20 59.90 60.50 61.40 62.80 63.40 58.74
13.10 4.93 4.86 4.84 4.92 4.80 6.14
/g⋅cm
1 2 3 4 5 … 100
U.
235
U. The last line is the parameters of the reference material used in the optimization process.
3
V /m3
M /ton
fE /GPa
fk /μm⋅m
0.96 1.43 1.46 1.52 1.61 1.65 1.36
1.25 7.03 7.11 7.36 7.93 7.93 8.33
4.18 1.62 1.36 4.17 7.24 8.47 2.49
68.9 145.0 163.0 65.7 41.4 36.4 101.0
Material composition /wt%
F
PE
B4C
W
Fe
Cu
Pb
3.77E-2 3.77E-2 3.34E-2 3.34E-2 2.66E-2
4.03E-1 4.03E-1 2.67E-1 2.67E-1 2.46E-1
5.50E-1 5.50E-1 5.89E-1 5.89E-1 7.24E-1
9.22E-3 9.22E-3 5.89E-6 5.89E-6 8.87E-6
5.50E-6 5.50E-6 1.11E-1 1.11E-1 8.87E-6
5.50E-6 5.50E-6 5.89E-6 5.89E-6 4.38E-3
4.62 4.62 4.68 4.68 4.71
7.38E-2
7.72E-6
7.72E-1
7.72E-6
1.49E-1
4.73E-3
7.82
1
⋅K
1
fλ /W⋅m 1⋅K 10.30 1.49 1.08 8.22 15.50 17.80 3.49
1
f 3.83 7.21 8.61 3.68 3.12 3.02 5.00
materials gradually improved, and FS6 basically achieves the synergistic optimization of various properties. To further compare the materials visually and conveniently, a comprehensive evaluation index is established. Since the purpose of this study is to make the material as compact and lightweight as possible under the condition of meeting the shielding requirements, while the mechanical and thermal properties of the material as good as possible. Thus, set the protective requirement as 2.5 μSv⋅h 1 (at the detection location), and the material performance comprehensive evaluation function can be written as follows.
Table 3 The ranked 100 Pareto-optimal solutions for 6-objective optimization. Just a part of them are listed considering the length of article. No.
235
f¼
5
M V fE0 fk fλ0 þ þ þ þ M0 V0 fE fk0 fλ
(17)
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3.2. Optimization by the evolutionary multi-objective optimization method
Table 4 The compositions and total objective function values of materials optimized by the two methods. Material FS2 M2 FS3 M3 FS4 M4 FS5 M5 FS6 M6
Material composition /wt%
To compare the optimization results with those in Section 3.1, the numbers of optimization objective are set at 2–6 respectively, and the domain constraints of each variable are set at 0 to 1 as well. The internal and external populations are both set at 100, the max generation is set at 120, the other parameters are selected according to previous studies (Zhang and Li, 2007), and set at CR ¼ 1.0, δ ¼ 0.9, nr ¼ 2, F ¼ 0.5, T ¼ 20, η ¼ 20, pm ¼ 1/n. By running the program, 100 Pareto-optimal solutions are obtained for each optimization. Then, a question arises, which one would be chosen from so many solutions? To answer this question and facilitate comparison with the results in Section 3.1, the 100 solutions are sorted by Formula (8) (parameters are selected same as those in Section 3.1). The results of six-objective optimization situation are shown in Table 3. Then, choose the top one in each case as the required solution respectively, named M2-M6, and list them in Table 4. It can be seen, there is just a little difference between the two methods in terms of the total objective function value. Furthermore, the shielding performance of each material is simu lated by MCNP program, and the results are shown in Fig. 5. It can be seen that the shielding performance of the materials optimized by the two methods is similar. Fig. 6 shows a comparison of the mechanical and thermal properties of the materials optimized by the two methods. It can be seen that, with the increase of the objective numbers, the material ultimately achieves
F
PE
B4C
W
Fe
Cu
Pb
12.66 14.30 14.30 14.30 6.41 7.63 3.92 4.37 3.48 3.77
1.59 0.95 2.00 0.70 21.65 21.80 31.16 30.30 33.93 40.30
54.20 84.70 77.12 84.30 45.38 67.90 57.13 61.40 54.71 55.00
0.46 0.00 1.68 0.45 2.59 0.00 0.72 0.00 0.36 0.92
30.71 0.00 3.53 0.16 23.78 2.61 6.66 3.92 7.36 0.00
0.38 0.00 1.38 0.00 0.19 0.03 0.41 0.01 0.15 0.00
1.70 1.64 1.69 1.68 3.11 3.09 3.99 4.02 4.59 4.62
The meanings of each parameter in the formula are consistent with those in formula (8), and the spherical geometry is used as well. The smaller of f, the better comprehensive performance of the material. Table 2 shows the performance parameters and evaluation indicators of the materials optimized. It can be seen, the material FS6 exactly has the best comprehensive properties, it realizes compactness and light weight, and has excellent mechanical and thermal properties simulta neously. It means that the multi-objective optimization method could exactly achieve the synergistic optimization of multi-performance on the premise of ensuring the shielding performance of materials.
Fig. 5. Comparison among shielding materials optimized by the two methods. 6
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Fig. 6. Mechanical and thermal properties of materials optimized by the two methods. Table 5 Performance parameters of the materials optimized by the two method. Material M2 M3 M4 M5 M6 FS6
T /cm
ρ
59.94 59.91 60.97 62.33 64.46 63.40
5.04 5.05 4.88 4.95 4.36 4.80
/g⋅cm
3
V /m3
M /ton
fE /GPa
fk /μm⋅m
1.43 1.43 1.49 1.58 1.73 1.65
7.20 7.21 7.29 7.83 7.52 7.93
1.28 1.27 3.34 6.39 8.63 8.47
172.00 172.00 79.40 45.90 36.00 36.40
the synergistic optimization of elastic modulus, thermal expansion co efficient and thermal conductivity, and the mechanical and thermal properties of the materials obtained by the two methods are compara tively consistent. Samely, the comprehensive evaluation index shown in formula (16) is used to present the performance of each material visually (Table 5). It can be seen that the performance of M6 and FS6 is similar, both of them have achieved compactness and lightweight under the condition of meeting the shielding requirements, and with excellent mechanical and thermal properties simultaneously.
1
⋅K
1
fλ /W⋅m 1⋅K 0.97 0.96 5.23 13.30 16.40 17.80
1
f 9.17 9.22 4.17 3.21 3.03 3.02
methods have both obtained their optimal solutions. The difference is that, the traditional multi-objective optimization method only obtains one solution corresponding to its weight values, while the evolutionary multi-objective optimization method obtains the whole Pareto-optimal set. From the point of optimization results, the evolutionary multiobjective optimization method should be a more advanced method. It is not affected by the prior informations (such as weight coefficients and dimensionless parameters), and could obtain the whole Pareto-optimal set at one time. Researchers only need to select the solution according to their own requirements afterwards. However, the evolutionary multi-objective optimization method needs much more computation time than the traditional multi-objective optimization method. If the researcher has clear requirements on the properties of the materials to be developed and has clear reference materials (sufficient prior information), the traditional multi-objective optimization method could obtain a satisfactory solution in a rela tively short time. That is to say, the two methods both have their own advantages and
4. Discussion To further illustrate the relationship between the solutions obtained by the two methods, the solutions obtained in two-objective and threeobjective cases are compared respectively (Fig. 7). It can be seen, the solutions obtained by the traditional multi-objective optimization method are exactly on the Pareto frontier which is obtained by the evolutionary multi-objective optimization method. It means the two 7
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thermal properties of material. Differently, the traditional multiobjective optimization method only obtains one solution correspond ing to its weight values, while the evolutionary multi-objective optimi zation method obtains the whole Pareto-optimal set. The later method needs no prior informations, while the former method is less computa tional. The two methods both have their own advantages and disad vantages, researchers should choose them according to their specific situations in practical application. It may provides a new method for the research and development of new multifunctional and high performance composite materials. Declaration of competing interest The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative in terest that represents a conflict of interest in connection with the work submitted. Acknowledgments This research was supported by the National Natural Science Foun dation of China (Grant No. 11975182). References Ashayer, S., Askari, M., Afarideh, H., 2012. Optimal per cent by weight of elements in diagnostic quality radiation shielding materials. Radiat. Protect. Dosim. 149, 268–288. Battat, M., 1977. ANS-6.1. 1 Working Group,“American National Standard Neutron and Gamma-Ray Flux-to-Dose Rate Factors,”. American Nuclear Society, LaGrange Park, Illinois, USA. ANSI/ANS-6.1. 1-1977. Cai, Y., Hu, H., Lu, S., Jia, Q., 2018a. Optimization of radiation shielding material aiming at compactness, lightweight, and low activation for a vehicle-mounted acceleratordriven D-T neutron source. Appl. Radiat. Isot. 135, 147–154. Cai, Y., Hu, H., Pan, Z., Hu, G., Zhang, T., 2018b. A method to optimize the shield compact and lightweight combining the structure with components together by genetic algorithm and MCNP code. Appl. Radiat. Isot. 139, 169–174. Cai, Y., Hu, H., Pan, Z., Sun, W., Yan, M., 2018c. Metaheuristic optimization in shielding design for neutrons and gamma rays reducing dose equivalent as much as possible. Ann. Nucl. Energy 120, 27–34. DiJulio, D.D., Bjorgvinsdottir, H., Zendler, C., Bentley, P.M., 2016. Population-based metaheuristic optimization in neutron optics and shielding design. Nucl. Instrum. Methods Phys. Res. Sect. A Accel. Spectrom. Detect. Assoc. Equip. 835, 157–162. Duan, Q.Y., Gupta, V.K., Sorooshian, S., 1993. Shuffled Complex Evolution Approach for Effective and Efficient Global Minimization. Plenum Press. Elomari, S., Skibo, M.D., Sundarrajan, A., Richards, H., 1998. Thermal expansion behavior of particulate metal-matrix composites. Compos. Sci. Technol. 58, 369–376. Kerner, E.H., 1956. The elastic and thermo-elastic properties of composite media. Proc. Phys. Soc. B 69, 808. Lin, F., Bhatia, G.S., Ford, J.D., 2010. Thermal conductivities of powder-filled epoxy resins. J. Appl. Polym. Sci. 49, 1901–1908. Nambiar, S., Yeow, J.T., 2012. Polymer-composite materials for radiation protection. ACS Appl. Mater. Interfaces 4, 5717–5726. Newman, F., Asadi–Zeydabadi, M., 2008. An optimization model and solution for radiation shielding design of radiotherapy treatment vaults. Med. Phys. 35, 171. Rossi, H.H., 1971. NCRP Scientific Committee 4 on Heavy Particles, Protection against Neutron Radiation. National Council on Radiation Protection and Measurements. Schaeffer, N.M., 1973. Reactor Shielding for Nuclear Engineers. US Atomic Energy Commission Office of Information Services. Tunes, M.A., de Oliveira, C.R.E., Schon, C.G., 2017. Multi-objective optimization of a compact pressurized water nuclear reactor computational model for biological shielding design using innovative materials. Nucl. Eng. Des. 313, 20–28. Zhang, Q., Li, H., 2007. MOEA/D: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans. Evol. Comput. 11, 712–731.
Fig. 7. Comparison of the solutions obtained by the two methods.
disadvantages, and complement each other somehow. Researchers should choose the method according to their specific situations in practical application. 5. Conclusions Two methods optimizing the shielding material compact, light weight, good mechanical and thermal properties for nuclear radiation were established combing SCE algorithm, MOEA/D algorithm, MCNP programm and several prediction models. The factors needed to be considered and the steps to reach the optimization objective were pre sented. Several materials were optimized by the two methods, and comparisons among the materials were made. The results show that, the performance of materials optimized by the two methods are similar. Both the two methods could achieve synergistic optimization of shielding performance, mass, volume, mechanical properties and
8
Contents lists available at ScienceDirect
Applied Radiation and Isotopes journal homepage: http://www.elsevier.com/locate/apradiso
Comparison of two multi-objective optimization methods for composite radiation shielding materials Yao Cai a, b, *, Rui Hao a, Shaojie Yu a, Chang Wang a, Guang Hu b a b
China Ship Development and Design Center, Wuhan, 430064, China School of Nuclear Science and Technology, Xi’an Jiaotong University, Xi’an, 710049, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Multi-objective Optimization Nuclear Shielding material
Combining the SCE algorithm (Shuffled Complex Evolution), MOEA/D algorithm (Multi-Objective Evolutionary Algorithm based on Decomposition), MCNP program and several prediction models, two multi-objective opti mization methods (priori method and posteriori method) for radiation shielding material, which considering the shielding, mass, volume, mechanical and thermal properties are established. The material is in the form of resin matrix composite. The shielding performance of the material is simulated by MCNP program. The mechanical property and thermal property are calculated by some widely used prediction models. Several materials are optimized by the two methods respectively, and comparisons among the materials are made. The results show that, both the two methods could achieve synergistic optimization of shielding, mass, volume, mechanical properties and thermal properties of material. Differently, the priori method only obtains one solution corre sponding to its weight values, while the posteriori method obtains the whole Pareto-optimal set. These two methods have their own advantages and disadvantages, which should be selected according to the actual situation.
1. Introduction Radiation shielding material is an important component of the nu clear facilities. In general, the shielding of neutrons and gamma rays need materials contain both heavy and light elements. Resin matrix composites have become an important type of shielding materials because of its strong designability, easy processing and properties of various components (Nambiar and Yeow, 2012). In addition, some other properties are also quite important for the shielding materials. Gener ally, the high-efficiency shielding material for nuclear radiation should be as small as possible in mass and volume under the condition of meeting the shielding requirements, while the mechanical and thermal properties of the material as good as possible. Obviously, the above properties are conflicting and contradictory, and it is difficult for traditional materials to meet these requirements at the same time. These years, optimization techniques using genetic al gorithms (Cai et al., 2018a), differential evolution (DiJulio et al., 2016), linear programming (Newman and Asadi–Zeydabadi, 2008), etc. have gradually applied to improve the material designing, and several shielding material optimization methods (Ashayer et al., 2012; Tunes et al., 2017) have been built. However, the material properties
considered in the previous works are always insufficient. Most of them just considered the shielding performance, while the mechanical and thermal properties have not been considered. Thus, further research is needed. Different from the single objective optimization problems, the optimal solution of multi-objective optimization problems is not unique, and there is a Pareto-optimal set. In this solution set, each solution at least has one better objective, while the other objectives are not inferior to the other solutions. It means that there is no absolute good or bad solutions for all the objectives, and the essence of solving multi-objective optimization problems is to select one or more solutions from the solu tion set according to the domain knowledge and personal preferences. Based on the manner of giving preference information, the multiobjective optimization methods can be divided into: 1) priori method (traditional multi-objective optimization method); 2) posteriori method (evolutionary multi-objective optimization method). Thus, there are also two multi-objective optimization methods for composite shielding material optimization. It is necessary to discuss the two methods and have a comparison between them. This study exactly addresses this problem. First, the characterization of material properties are presented, and
* Corresponding author. China Ship Development and Design Center, Wuhan, 430064, China. E-mail address: [email protected] (Y. Cai). https://doi.org/10.1016/j.apradiso.2020.109061 Received 9 October 2019; Received in revised form 4 January 2020; Accepted 24 January 2020 Available online 14 February 2020 0969-8043/© 2020 Elsevier Ltd. All rights reserved.
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2.1.2. Mechanical property of the material Elastic modulus is the most significant mechanical properties of composite material. It describes the difficulty to deform the materials, and should be kept as large as possible. The common used Kerner model (Kerner, 1956) as follows was adopted. �X ���X � ki ϕi ϕi (1) kc ¼ 3ki þ 4μm 3ki þ 4μm n P
μc ¼ μ
i¼2 mP n i¼2
Fig. 1. Schematic of the calculation model used in the optimization.
νc ¼
the two multi-objective optimization methods to optimize the shielding material are studied (Section 2). Second, several materials are optimized by the two methods respectively, and comparisons among them are made (Section 3). At last, the relationship between the solutions ob tained by the two methods and the application of the two methods are discussed (Section 4).
ϕi μi ð7 5νm Þμm þð8 10νm Þμi
þ 15ð1ϕmνm Þ
ϕi μm ð7 5νm Þμm þð8 10νm Þμi
þ 15ð1ϕmνm Þ
3kc 2μc 6kc þ 2μc
(2)
(3) (4)
Ec ¼ 2μc ð1 þ νc Þ
where, ki is the bulk modulus of component i; μi is the shear modulus of component i; ϕi is the volume fraction of component i; νi is the Poisson ratio of component i; Ei is the elastic modulus of component i; subscripts c, m, p represent the composite material, matrix material and rein forcement material respectively.
2. Methodology
2.1.3. Thermal properties of the material Thermal properties mainly include thermal expansion coefficient and thermal conductivity. For shielding materials, the thermal con ductivity should be as large as possible to reduce the radial temperature gradient of the shielding. The thermal expansion coefficient reflects the degree of thermal deformation of materials, and should be kept as small as possible.
2.1. Characterization of the material properties As mentioned in Section 1, the material properties considered in this study including shielding performance, mass, volume, mechanical properties and thermal properties. Among them, the mechanical prop erties are represented by the elastic modulus of the material, and the thermal properties are represented by the thermal expansion coefficient (CTE) and thermal conductivity of the material. It should be noted that, the composite radiation shielding material researched in this study is a kind of particle reinforced material. It is assumed that the reinforced particles are spherical (quite small) and uniformly dispersed in the matrix.
2.1.3.1. Thermal expansion coefficient of the material. The thermal expansion coefficient describes the change of material length caused by the change of unit temperature. The common used Schapery model (Elomari et al., 1998) as follows was adopted. � � � 1 kc 1 kp � αc ¼ αp þ αm αp � (5) 1 km 1 kp
2.1.1. Shielding performance, mass and volume of the material In this study, the shielding performance of materials is calculated by Monte Carlo program MCNP (with the ENDF/B-VI cross section set). Since thousands of transport calculations are needed during the opti mization, the spherical shell model as shown in Fig. 1 is used to reduce the calculation time (Cai et al., 2018b). Of which, an isotropic point source is placed at the center position, the thickness of shielding mate rial is set at 40 cm (about 10 free paths of neutrons in such materials), and the count surface is set 2 m away from the source (distance protection). The shielding performance is represented by the dose equivalent caused by transmitted neutrons and gamma rays. The smaller the value, the better the shielding performance. The F2 tally card was used with the DE and DF cards to assess the dose equivalentrate transmitted. For this, the NCRP-38 (Rossi, 1971) neutron flux-to-dose rate conversion factors and the 1977 ANSI/ANS (Battat, 1977) photon flux-to-dose rate con version factors were used. Obviously, the mass and volume of the material are both hoped to be as small as possible. The mass property is simple to describe with for mula m ¼ ρV, but the volume property is described slightly different due to the fixed thickness of shielding material used in the optimization. In our previous study (Cai et al., 2018a), it is found that the pro portion of secondary gamma rays in the total dose equivalent (Hng/Ht, the self-absorption capacity) can be used as a sub-objective to charac terize the volume property indirectly and effectively. The smaller the ratio, the smaller volume of material needed to achieve the same shielding requirement.
kc ¼ km þ
ϕp 1 kp km
þ km ϕþm4μ
(6)
3 m
where, αi is the thermal expansion coefficient of component i. 2.1.3.2. Thermal conductivity of the material. Thermal conductivity re flects the ability of materials to conduct heat. The common used Brug geman model (Lin et al., 2010) as follows was adopted. � �1=3 λp λc λm 1 ϕp ¼ (7) λp λm λc where, λi is the thermal conductivity of component i. It can be seen, the mechanical and thermal properties of the com posites can be obtained according to formulas (1) - (7) as long as the basic parameters and volume fractions of the components are known. That is to say, the above properties of the material are all directly related to its component X, which provides the precondition of material optimization. 2.2. Multi-objective optimization of the material To design a compact, lightweight shielding material with good me chanical and thermal properties, it is necessary to consider the effects of dose equivalent, mass, volume, mechanical properties and thermal properties comprehensively. For this multi-objective optimization 2
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(12)
L�X � U
where, N is the number of components in the composite material; X¼(x1, x2, …, xN) is the variable vector, mass components matrix in the com posite material; L is the component vector of minimum; U is the component vector of maximum; F(X) is the total fitness function value; c1 ; c2 ; c3 ; c4 ; c5 and c6 are the weight coefficient respectively; HT(X) is the total dose equivalent, Sv; Hn(X) is the dose equivalent of neutron, Sv; Hg(X) is the dose equivalent of gamma rays, Sv; Hng(X) is the dose equivalent of secondary gamma rays, Sv; M is the mass of material, kg; fE is the elastic modulus of the composites, Gpa; fk is the thermal expansion coefficient of the composites, μm⋅m 1⋅K 1; fλ is the thermal conductivity of the composite, W⋅m 1⋅K 1; H0T ; M0 ; f 0E ; f 0λ and f 0λ are the reference values to make the function dimensionless. In formula (8), the first term reflects the shielding performance, the second term reflects the mass, the third term reflects the volume, the fourth term reflects the mechanical properties, and the last two terms reflect the thermal performance. Different from the previous study (Cai et al., 2018a), the existing materials are no longer used as reference materials to reduce the effect of artificial factors. Instead, c1 ¼ 1; c2 ¼ c3 ¼ c4 ¼ c5 ¼ c6 ¼ 0, are pre-set, and a sub-optimal solution is obtained by optimizing 5 genera tions, which is used as reference materials for material optimization. The algorithm of SCE (the Shuffled Complex Evolution algorithm) is an efficient choice for the shielding optimization after evaluating 14 well known metaheuristic algorithms (Cai et al., 2018c). It has excellent global search capabilities and optimization efficiency. Thus, the algo rithm of SCE is adopted in this study. The steps in the optimization are shown in Fig. 2.
Fig. 2. Flow chart of the optimization.
problem, the priori method (traditional multi-objective optimization method) transforms it into a single-objective problem by linear weighting, and solves it using single-objective optimization technology then. The posteriori method (evolutionary multi-objective optimization method) obtains a Pareto-optimal set first, and selects the desired so lution from the set then according to personal preferences.
2.2.2. The evolutionary multi-objective optimization method Different from the traditional multi-objective optimization method, no prior informations (such as weight coefficients and dimensionless parameters) are needed in the evolutionary multi-objective optimization method. And the optimization problems can be presented as follows:
2.2.1. The traditional multi-objective optimization method For the multi-objective optimization problem researched in this study, the objective function can be written as follows: FðXÞ ¼ c1
HT M Hng f0 fk f0 þ c2 þ c3 þ c4 E þ c5 0 þ c6 λ 0 M0 HT fE fλ HT fk
N X
s:t:
where, f1 ¼ HT , f2 ¼ M, f3 ¼ Hng =HT , f4 ¼ 1=fE , f5 ¼ fk , f6 ¼ 1=fλ . The meanings of the parameters in the formula are same with those in sec tion 2.2.1. Over the past decades, many scholars have proposed various evolu tionary multi-objective optimization algorithms. The common used ones are SPEA-II, PESA-II, NSGA-II, MOPSO and MOEA/D. After evaluating these algorithms with some classical test functions, it is found that the algorithm of MOEA/D (Multi-Objective Evolutionary Algorithm based on Decomposition) has better convergence and diversity than the others. Thus, it is adopted to perform the evolutionary multi-objective
(9)
The optimization problems can be presented as follows: (10)
FðXÞ N X
s:t:
(11)
xi ¼ 1
(15)
L�X � U
where
min
(14)
xi ¼ 1 i¼1
(8)
HT ðXÞ ¼ Hn ðXÞ þ Hg ðXÞ
(13)
minFðxÞ ¼ ðf1 ðxÞ; f2 ðxÞ; ⋯; f6 ðxÞÞT
i¼1
Table 1 Material components optimized by the traditional multi-objective optimization method. The first column provides the names of the optimized material, where the symbol “FS” represents “fission energy spectrum of 235U”, and the Arabic numerals indicate the number of objects considered in the optimization. Material FS1 FS2 FS3 FS4 FS5 FS6
Weight coefficient
Material composition /wt%
c1
c2
c3
c4
c5
c6
PE
B4C
W
Fe
Cu
Pb
1 1 1 1 1 1
0 1 1 1.5 2 2.5
0 0 1 1 1 1
0 0 0 1 1 1
0 0 0 0 1 1
0 0 0 0 0 1
2.37 12.66 14.30 6.41 3.92 3.48
0.05 1.59 2.00 21.65 31.16 33.93
96.86 54.20 77.12 45.38 57.13 54.71
0.16 0.46 1.68 2.59 0.72 0.36
0.48 30.71 3.53 23.78 6.66 7.36
0.09 0.38 1.38 0.19 0.41 0.15
3
Density /g⋅cm 3 13.10 4.93 4.86 4.84 4.92 4.80
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Fig. 3. Comparison among shielding materials by the MCNP simulation.
optimization of composite radiation material follows.
components were obtained (Table 1). It can be seen, when the shielding performance is merely considered, the material mainly consist of W, and the density is quite large. After considering the mass of the material, the density decreases sharply. Because the density of the material optimized at this time is large enough, the ability of restraining and self-absorption of secondary gamma rays is quite good, the density does not change much after futher put the volume term into consideration. Later, because the elastic modulus, thermal expansion coefficient and thermal conductivity of the material are positively correlated with the volume fraction of the rein forced material, in order to control the density of the material, the weight of the mass term is increased with each additional factor in the subsequent material optimization. The results show that, except FS1, the densities of materials obtained by other optimizations are simply with a slight difference. To compare the shielding performance of various materials intui tively, 108 particles were simulated by the MCNP programm (Fig. 3). The results show that the shielding performance of FS1 is much better than the others, the performance of FS2 and FS3 are basically the same, FS4, FS5 and FS6 are slightly worse than the former, but generally similar. Besides shielding properties, the mechanical and thermal properties of materials are also very important in practical applications. Fig. 4 shows a comparison of the mechanical and thermal properties of the optimized materials. It can be seen that, with the increase of factors considered in the optimization, the comprehensive properties of
3. Simulation and results As a typical case, the fission energy spectrum of 235U (beam intensity is 1010 fission/s) which mixed neutrons and gamma rays was adopted as the radiation source in this study. It release 2.407 neutrons (Watt fission energy spectrum) and 7.77 gamma rays every event. An empirical for mula for the prompt gamma rays spectrum was employed (Schaeffer, 1973): 8 6:6 0:1 < Eγ � 0:6 MeV < 0:6 < Eγ � 1:5MeV 20:2 exp ð 1:78Eγ Þ NðEγ Þ ¼ : 7:2 exp ð 1:09Eγ Þ 1:5 < Eγ � 10:5MeV (16) To compare the two multi-objective optimization methods, com posites composed of polyethylene (PE), B4C, W, Fe, Cu and Pb were optimized by them respectively. 3.1. Optimization by the traditional multi-objective optimization method To find the optimal shielding material, the domain constraints of each variable are set at 0 to 1. The parameters of the algorithm are selected according to previous studies (Duan et al., 1993), and set at m ¼ 2nþ1, p ¼ 3, s ¼ pm, q ¼ nþ1, α ¼ 1, β ¼ 2nþ1. By adjusting the weight coefficients and running the program, a series of material 4
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Fig. 4. Mechanical and thermal properties of materials designed for Table 2 Performance parameters of the materials optimized for Material FS1 FS2 FS3 FS4 FS5 FS6 Ref
T /cm
ρ
51.20 59.90 60.50 61.40 62.80 63.40 58.74
13.10 4.93 4.86 4.84 4.92 4.80 6.14
/g⋅cm
1 2 3 4 5 … 100
U.
235
U. The last line is the parameters of the reference material used in the optimization process.
3
V /m3
M /ton
fE /GPa
fk /μm⋅m
0.96 1.43 1.46 1.52 1.61 1.65 1.36
1.25 7.03 7.11 7.36 7.93 7.93 8.33
4.18 1.62 1.36 4.17 7.24 8.47 2.49
68.9 145.0 163.0 65.7 41.4 36.4 101.0
Material composition /wt%
F
PE
B4C
W
Fe
Cu
Pb
3.77E-2 3.77E-2 3.34E-2 3.34E-2 2.66E-2
4.03E-1 4.03E-1 2.67E-1 2.67E-1 2.46E-1
5.50E-1 5.50E-1 5.89E-1 5.89E-1 7.24E-1
9.22E-3 9.22E-3 5.89E-6 5.89E-6 8.87E-6
5.50E-6 5.50E-6 1.11E-1 1.11E-1 8.87E-6
5.50E-6 5.50E-6 5.89E-6 5.89E-6 4.38E-3
4.62 4.62 4.68 4.68 4.71
7.38E-2
7.72E-6
7.72E-1
7.72E-6
1.49E-1
4.73E-3
7.82
1
⋅K
1
fλ /W⋅m 1⋅K 10.30 1.49 1.08 8.22 15.50 17.80 3.49
1
f 3.83 7.21 8.61 3.68 3.12 3.02 5.00
materials gradually improved, and FS6 basically achieves the synergistic optimization of various properties. To further compare the materials visually and conveniently, a comprehensive evaluation index is established. Since the purpose of this study is to make the material as compact and lightweight as possible under the condition of meeting the shielding requirements, while the mechanical and thermal properties of the material as good as possible. Thus, set the protective requirement as 2.5 μSv⋅h 1 (at the detection location), and the material performance comprehensive evaluation function can be written as follows.
Table 3 The ranked 100 Pareto-optimal solutions for 6-objective optimization. Just a part of them are listed considering the length of article. No.
235
f¼
5
M V fE0 fk fλ0 þ þ þ þ M0 V0 fE fk0 fλ
(17)
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3.2. Optimization by the evolutionary multi-objective optimization method
Table 4 The compositions and total objective function values of materials optimized by the two methods. Material FS2 M2 FS3 M3 FS4 M4 FS5 M5 FS6 M6
Material composition /wt%
To compare the optimization results with those in Section 3.1, the numbers of optimization objective are set at 2–6 respectively, and the domain constraints of each variable are set at 0 to 1 as well. The internal and external populations are both set at 100, the max generation is set at 120, the other parameters are selected according to previous studies (Zhang and Li, 2007), and set at CR ¼ 1.0, δ ¼ 0.9, nr ¼ 2, F ¼ 0.5, T ¼ 20, η ¼ 20, pm ¼ 1/n. By running the program, 100 Pareto-optimal solutions are obtained for each optimization. Then, a question arises, which one would be chosen from so many solutions? To answer this question and facilitate comparison with the results in Section 3.1, the 100 solutions are sorted by Formula (8) (parameters are selected same as those in Section 3.1). The results of six-objective optimization situation are shown in Table 3. Then, choose the top one in each case as the required solution respectively, named M2-M6, and list them in Table 4. It can be seen, there is just a little difference between the two methods in terms of the total objective function value. Furthermore, the shielding performance of each material is simu lated by MCNP program, and the results are shown in Fig. 5. It can be seen that the shielding performance of the materials optimized by the two methods is similar. Fig. 6 shows a comparison of the mechanical and thermal properties of the materials optimized by the two methods. It can be seen that, with the increase of the objective numbers, the material ultimately achieves
F
PE
B4C
W
Fe
Cu
Pb
12.66 14.30 14.30 14.30 6.41 7.63 3.92 4.37 3.48 3.77
1.59 0.95 2.00 0.70 21.65 21.80 31.16 30.30 33.93 40.30
54.20 84.70 77.12 84.30 45.38 67.90 57.13 61.40 54.71 55.00
0.46 0.00 1.68 0.45 2.59 0.00 0.72 0.00 0.36 0.92
30.71 0.00 3.53 0.16 23.78 2.61 6.66 3.92 7.36 0.00
0.38 0.00 1.38 0.00 0.19 0.03 0.41 0.01 0.15 0.00
1.70 1.64 1.69 1.68 3.11 3.09 3.99 4.02 4.59 4.62
The meanings of each parameter in the formula are consistent with those in formula (8), and the spherical geometry is used as well. The smaller of f, the better comprehensive performance of the material. Table 2 shows the performance parameters and evaluation indicators of the materials optimized. It can be seen, the material FS6 exactly has the best comprehensive properties, it realizes compactness and light weight, and has excellent mechanical and thermal properties simulta neously. It means that the multi-objective optimization method could exactly achieve the synergistic optimization of multi-performance on the premise of ensuring the shielding performance of materials.
Fig. 5. Comparison among shielding materials optimized by the two methods. 6
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Fig. 6. Mechanical and thermal properties of materials optimized by the two methods. Table 5 Performance parameters of the materials optimized by the two method. Material M2 M3 M4 M5 M6 FS6
T /cm
ρ
59.94 59.91 60.97 62.33 64.46 63.40
5.04 5.05 4.88 4.95 4.36 4.80
/g⋅cm
3
V /m3
M /ton
fE /GPa
fk /μm⋅m
1.43 1.43 1.49 1.58 1.73 1.65
7.20 7.21 7.29 7.83 7.52 7.93
1.28 1.27 3.34 6.39 8.63 8.47
172.00 172.00 79.40 45.90 36.00 36.40
the synergistic optimization of elastic modulus, thermal expansion co efficient and thermal conductivity, and the mechanical and thermal properties of the materials obtained by the two methods are compara tively consistent. Samely, the comprehensive evaluation index shown in formula (16) is used to present the performance of each material visually (Table 5). It can be seen that the performance of M6 and FS6 is similar, both of them have achieved compactness and lightweight under the condition of meeting the shielding requirements, and with excellent mechanical and thermal properties simultaneously.
1
⋅K
1
fλ /W⋅m 1⋅K 0.97 0.96 5.23 13.30 16.40 17.80
1
f 9.17 9.22 4.17 3.21 3.03 3.02
methods have both obtained their optimal solutions. The difference is that, the traditional multi-objective optimization method only obtains one solution corresponding to its weight values, while the evolutionary multi-objective optimization method obtains the whole Pareto-optimal set. From the point of optimization results, the evolutionary multiobjective optimization method should be a more advanced method. It is not affected by the prior informations (such as weight coefficients and dimensionless parameters), and could obtain the whole Pareto-optimal set at one time. Researchers only need to select the solution according to their own requirements afterwards. However, the evolutionary multi-objective optimization method needs much more computation time than the traditional multi-objective optimization method. If the researcher has clear requirements on the properties of the materials to be developed and has clear reference materials (sufficient prior information), the traditional multi-objective optimization method could obtain a satisfactory solution in a rela tively short time. That is to say, the two methods both have their own advantages and
4. Discussion To further illustrate the relationship between the solutions obtained by the two methods, the solutions obtained in two-objective and threeobjective cases are compared respectively (Fig. 7). It can be seen, the solutions obtained by the traditional multi-objective optimization method are exactly on the Pareto frontier which is obtained by the evolutionary multi-objective optimization method. It means the two 7
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thermal properties of material. Differently, the traditional multiobjective optimization method only obtains one solution correspond ing to its weight values, while the evolutionary multi-objective optimi zation method obtains the whole Pareto-optimal set. The later method needs no prior informations, while the former method is less computa tional. The two methods both have their own advantages and disad vantages, researchers should choose them according to their specific situations in practical application. It may provides a new method for the research and development of new multifunctional and high performance composite materials. Declaration of competing interest The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative in terest that represents a conflict of interest in connection with the work submitted. Acknowledgments This research was supported by the National Natural Science Foun dation of China (Grant No. 11975182). References Ashayer, S., Askari, M., Afarideh, H., 2012. Optimal per cent by weight of elements in diagnostic quality radiation shielding materials. Radiat. Protect. Dosim. 149, 268–288. Battat, M., 1977. ANS-6.1. 1 Working Group,“American National Standard Neutron and Gamma-Ray Flux-to-Dose Rate Factors,”. American Nuclear Society, LaGrange Park, Illinois, USA. ANSI/ANS-6.1. 1-1977. Cai, Y., Hu, H., Lu, S., Jia, Q., 2018a. Optimization of radiation shielding material aiming at compactness, lightweight, and low activation for a vehicle-mounted acceleratordriven D-T neutron source. Appl. Radiat. Isot. 135, 147–154. Cai, Y., Hu, H., Pan, Z., Hu, G., Zhang, T., 2018b. A method to optimize the shield compact and lightweight combining the structure with components together by genetic algorithm and MCNP code. Appl. Radiat. Isot. 139, 169–174. Cai, Y., Hu, H., Pan, Z., Sun, W., Yan, M., 2018c. Metaheuristic optimization in shielding design for neutrons and gamma rays reducing dose equivalent as much as possible. Ann. Nucl. Energy 120, 27–34. DiJulio, D.D., Bjorgvinsdottir, H., Zendler, C., Bentley, P.M., 2016. Population-based metaheuristic optimization in neutron optics and shielding design. Nucl. Instrum. Methods Phys. Res. Sect. A Accel. Spectrom. Detect. Assoc. Equip. 835, 157–162. Duan, Q.Y., Gupta, V.K., Sorooshian, S., 1993. Shuffled Complex Evolution Approach for Effective and Efficient Global Minimization. Plenum Press. Elomari, S., Skibo, M.D., Sundarrajan, A., Richards, H., 1998. Thermal expansion behavior of particulate metal-matrix composites. Compos. Sci. Technol. 58, 369–376. Kerner, E.H., 1956. The elastic and thermo-elastic properties of composite media. Proc. Phys. Soc. B 69, 808. Lin, F., Bhatia, G.S., Ford, J.D., 2010. Thermal conductivities of powder-filled epoxy resins. J. Appl. Polym. Sci. 49, 1901–1908. Nambiar, S., Yeow, J.T., 2012. Polymer-composite materials for radiation protection. ACS Appl. Mater. Interfaces 4, 5717–5726. Newman, F., Asadi–Zeydabadi, M., 2008. An optimization model and solution for radiation shielding design of radiotherapy treatment vaults. Med. Phys. 35, 171. Rossi, H.H., 1971. NCRP Scientific Committee 4 on Heavy Particles, Protection against Neutron Radiation. National Council on Radiation Protection and Measurements. Schaeffer, N.M., 1973. Reactor Shielding for Nuclear Engineers. US Atomic Energy Commission Office of Information Services. Tunes, M.A., de Oliveira, C.R.E., Schon, C.G., 2017. Multi-objective optimization of a compact pressurized water nuclear reactor computational model for biological shielding design using innovative materials. Nucl. Eng. Des. 313, 20–28. Zhang, Q., Li, H., 2007. MOEA/D: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans. Evol. Comput. 11, 712–731.
Fig. 7. Comparison of the solutions obtained by the two methods.
disadvantages, and complement each other somehow. Researchers should choose the method according to their specific situations in practical application. 5. Conclusions Two methods optimizing the shielding material compact, light weight, good mechanical and thermal properties for nuclear radiation were established combing SCE algorithm, MOEA/D algorithm, MCNP programm and several prediction models. The factors needed to be considered and the steps to reach the optimization objective were pre sented. Several materials were optimized by the two methods, and comparisons among the materials were made. The results show that, the performance of materials optimized by the two methods are similar. Both the two methods could achieve synergistic optimization of shielding performance, mass, volume, mechanical properties and
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