改进磷虾群算法及其在结构优化中的应用

doi:10.13229/j.cnki.jdxbgxb.20221349

Abstract

Abstract:

In this paper， the standard krill algorithm （KH） has the disadvantages of slow convergence speed， insufficient calculation accuracy and easy to fall into local optimal solution for complex problems， an improved krill algorithm （SDEKH） which combines improved differential evolution operator and S-type adaptive inertia weight is proposed in this paper. Through a variety of standard test functions to compare and test a variety of intelligent algorithms such as SDEKH and KH， the excellent performance of SDEKH is verified， and SDEKH is used to optimize the truss structure， and the optimization results of SDEKH are compared with other methods to verify that the optimization efficiency and accuracy are improved， which provides a more efficient and accurate method for engineering structure optimization design.

Key words: structural engineering, improved krill algorithm, differential evolution operator, inertia weight, structural optimization

CLC Number:

TU323.4

Feng-guo JIANG,Yu-ming ZHOU,Li-li BAI,Shuang LIANG. Improved krill algorithm and its application in structural optimization[J].Journal of Jilin University(Engineering and Technology Edition), 2024, 54(8): 2256-2266.

Figures/Tables 19

Fig.1

Table 1

Standard test function"

序号	函数名称	函数表达式
1	Ackley	$F 1 (x) = - 20 e x p (- 0.2 1 n ∑ i = 1 n x i 2) - e x p (1 n ∑ i = 1 n c o s 2 π x i)) + 20 + e$
2	Rastrigin	$F 2 (x) = 10 ? n + ∑ i = 1 n (x i 2 - 10 ? c o s (2 π x i))$
3	Schwefel 1	$F 3 (x) = ∑ i = 1 n (∑ j = 1 i x j) 2$
4	Schwefel 2	$F 4 (x) = m a x {x i, 1 ≤ i ≤ n}$
5	Schwefel	$F 5 (x) = ∑ i = 1 n - x i s i n (x i) + 418.982 ? 9 × 5$
6	Step	$F 6 (x) = ∑ i = 1 n ([x i + 0.5]) 2$
7	Penalty	$F 7 (x) = π n (10 ? s i n (π y 1) + ∑ i = 1 n u (x i, 10,100,4) + (y n - 1) 2 + ∑ i = 1 n - 1 (y i - 1) 2 (1 + 10 s i n 2 (π y i + 1)))$
8	Rosebrock	$F 8 (x) = ∑ i = 1 n - 1 100 x i + 1 - x i 2) 2 + (x i - 1) 2]$
9	Griewank	$F 9 (x) = 1 4 ? 000 ∑ i = 1 n x i 2 - ∏ i n c o s (x i i) + 1$
10	Sphere	$F 10 (x) = ∑ i = 1 n x i 2$
11	Quartic	$F 11 (x) = ∑ i = 1 n i x i 4 + r a n d$

Table 1

Fig. 2

Fig. 3

Fig. 4

Table 2

Test function test results comparison"

函数	SDEKH			KH			WOA	PSO	DE
函数	最优值	平均值	标准差	最优值	平均值	标准差	最优值	最优值	最优值
$F 1$	2.472 1E^-08	2.236 2E^-04	3.938 9E^-04	0.050 2	2.006 4	0.885 3	0.008 5	9.967 1	8.856 1
$F 2$	5.066 3E^-08	1.248 6E^-04	4.475 1E^-04	11.923 5	60.154 9	43.544 5	0.001 6	21.043 8	16.772 3
$F 3$	3.775 4E^-34	2.288 1E^-25	1.226 8E^-24	1.174 7E^-04	15.936 7	26.862 6	1.413 1E⁺⁰⁵	1.959 1E⁺⁰⁴	1.965 8E⁺⁰³
$F 4$	1.118 1E^-04	9.776 1E^-04	8.023 1E^-04	10.939 2	14.968 7	2.438 4	88.461 5	25.527 6	35.786 2
$F 5$	-1.031 1E⁺⁰⁴	-6.854 2E⁺⁰³	1.500 7	-4.132 1E⁺⁰³	-1.723 2E⁺⁰³	2.736 4	-8.988 1E⁺⁰³	-4.354 2E⁺⁰³	-3.346 1E⁺⁰³
$F 6$	1.414 2	2.533 7	0.551 3	4.596 4E⁺⁰²	1.590 1E⁺⁰³	6.278 3	2.374 2	1.936 2E⁺⁰³	3.263 4E⁺⁰²
$F 7$	1.038 1E^-12	6.059 2E^-05	1.753 3E^-04	7.918 4	2.100 5E⁺⁰⁴	4.980 3E⁺⁰⁴	0.216 7	49.246 1	2.165 8
$F 8$	8.713 4	8.890 2	0.071 2	2.422 3E⁺⁰⁴	2.163 5E⁺⁰⁵	1.402 1E⁺⁰⁵	28.841 1	4.514 2E⁺⁰⁵	1.221 4E⁺⁰⁵
$F 9$	0.013 9	0.226 4	0.139 3	7.454 6	16.077 6	5.403 9	0.694 7	16.439 9	3.563 9
$F 10$	8.665 6E^-29	4.628 5E^-23	2.242 2E^-22	51.025 4	4.324 9e⁺⁰²	3.508 1E⁺⁰²	0.000 4	2.262 9E⁺⁰³	5.226 2E⁺⁰²
$F 11$	2.019 5E^-05	7.139 4E^-04	0.000 7	0.007 1	0.016 3	0.007 2	0.078 3	0.787 8	0.112 3

Table 2

Fig. 5

Fig. 6

Table 3

Fig. 7

Table 4

Fig. 8

Fig. 9

Table 5

Fig.10

Fig. 11

Fig. 12

Fig.13

Table 6

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参数	数值
拉压屈服应力均值/MPa	275.8
屈服应力变异系数	0.15
材料弹性模量/MPa	2.06×10⁵
材料密度/（kg·m^-3）	2.7×10³
荷载的均值/kN	89
变异系数	0.15

元件编号	元件横截面面积/cm²
元件编号	SDEKH	KH	WOA	对偶规划法^［14］
1，7	19.073 2	19.123 5	19.105 9	19.06
2，8	13.548 9	13.555 6	13.620 4	13.56
3，9，12，13	11.361 2	11.361 4	11.360 4	11.37
4，10	11.503 8	11.506 3	11.511 8	11.66
5，11	3.401 2	3.455 6	3.384 7	3.638
6	8.768 6	8.863 2	8.857 2	8.729
结构总质量/kg	136.818 4	137.194 6	137.0481	137.6

结构设计变量	杆件编号	结构设计变量	杆件编号
1	1~4	9	37~40
2	5~12	10	41~48
3	13~16	11	49~52
4	17~18	12	53~54
5	19~22	13	55~58
6	23~30	14	59~66
7	31~34	15	67~70
8	35~36	16	71~72

Improved krill algorithm and its application in structural optimization

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