基于投影优选法的结构静态位移多目标优化方法

doi:10.13229/j.cnki.jdxbgxb.20230333

摘要/Abstract

摘要：

本文提出了一种基于投影优选法的结构静态位移多目标优化方法。投影优选法是根据参数灵敏度向量在特定的多维子空间的投影角度和投影长度来对参数进行排序的参数选取方法。其中，本文采用的结构静态灵敏度的解析计算方法是结合Epsilon算法和改进的纽曼级数建立的。该多目标优化方法每步迭代都可按照给定的参数个数要求来选取合理的参数组合并得到对应参数的修改量，通过多次迭代计算得到优化解。该方法特别适用于限定参数个数的多目标结构优化问题。在算例中，使用该方法对一个桁架结构进行多目标优化。算例中还用随机选取的参数组与本文方法选定的参数组做优化结果对比。对比结果表明：应用本文方法的优化结果更理想。算例进一步证明了本文优化方法具有很高的工程应用价值。

关键词: 固体力学, 灵敏度分析, 投影优选法, 最优修改量, Epsilon算法

Abstract:

In this paper， a multi-objective optimization method of structural static displacement based on projection priority selection method is proposed. Projection priority selection method is a parameter selection method that sorts the parameters according to the projection angle and projection length of the parameter sensitivity vector in a specific multidimensional subspace. The structural static sensitivity is calculated by the analytical method based on Epsilon algorithm and improved Newman series. In each iteration of the optimization method， the reasonable combination of parameters can be selected according to the given number of parameters and the modification of the corresponding parameters can be obtained. The optimal solution can be obtained through multiple iterations. This method is especially suitable for solving multi-objective structural optimization problems with limited number of parameters. In the example， the proposed method is used for multi-objective optimization of a truss structure. The optimization results were compared between the randomly selected parameter group and the parameter group selected by the method in this paper. The results show that the optimization results of the Projection priority selection method are more ideal. Examples further prove that the proposed optimization method has high engineering application value.

Key words: solid mechanics, sensitivity analysis, projection priority selection method, optimal amount of modification, Epsilon algorithm

中图分类号:

O342

麻凯,孙建航,闫森康,陶炎,汪文涛,郭桂凯. 基于投影优选法的结构静态位移多目标优化方法[J]. 吉林大学学报(工学版), 2025, 55(1): 74-83.

Kai MA,Jian-hang SUN,Sen-kang YAN,Yan TAO,Wen-tao WANG,Gui-kai GUO. Multi-objective optimization method of structural static displacement based on projection priority selection method[J]. Journal of Jilin University(Engineering and Technology Edition), 2025, 55(1): 74-83.

图/表 16

图1

图2

图3

表1

各节点位移"

序号	初始位移/ $m m$	目标位移/ $m m$	最优参数组/ $m m$	对照1组/ $m m$	对照2组/ $m m$	对照3组/ $m m$	对照4组/ $m m$
序号	初始位移/ $m m$	目标位移/ $m m$	$A 64, A 79, A 55 A 61, A 2, A 69, A 40$	$A 23 ~ A 29$	$A 34 ~ A 40$	$A 29, A 31, A 21 A 34, A 36, A 38, A 40$	$A 44 ~ A 50$
1	119.72	115	119.04	119.57	119.72	119.64	119.56
2	68.59	64	68.06	68.49	68.55	68.54	68.48
3	97.56	93	97.00	97.48	97.57	97.49	97.48
4	52.54	48	52.01	52.45	52.49	52.48	52.44
5	68.14	63	67.74	68.10	68.18	68.06	68.10
6	49.58	45	49.07	49.51	49.50	49.52	49.50
7	44.75	40	44.49	44.78	44.65	44.64	44.77
8	49.19	44	48.69	49.12	49.13	49.15	49.12
9	53.65	49	53.25	53.59	53.62	53.56	53.59
10	43.69	39	43.07	43.52	43.68	43.66	43.51
?	?	?	?	?	?	?	?
50	28.46	23	28.24	28.49	28.46	28.47	28.49
51	124.47	119	124.01	124.30	124.48	124.40	124.30
52	69.60	65	69.15	69.42	69.56	69.55	69.42
53	113.42	108	113.00	113.29	113.43	113.34	113.28
54	63.72	59	63.32	63.58	63.68	63.66	63.57
55	115.93	111	115.24	115.79	115.94	115.86	115.79
56	84.20	79	83.51	83.99	84.17	84.16	83.98
57	130.54	126	130.09	130.37	130.56	130.48	130.36
58	105.94	101	105.20	105.68	105.91	105.90	105.67
59	110.96	106	110.27	110.82	110.94	110.86	110.81
60	52.33	47	52.10	52.36	52.33	52.34	52.36

表1

图4

图5

图6

图7

图8

表2

首次迭代后不同设计参数组的位移误差向量均方值及误差"

参数	最优参数组	对照1组	对照2组	对照3组	对照4组
参数	$A 64, A 79, A 55 A 61, A 2, A 69, A 40$	$A 23 ~ A 29$	$A 34 ~ A 40$	$A 29, A 31, A 21 A 34, A 36, A 38, A 40$	$A 44 ~ A 50$
位移误差向量均方值 $φ r m s 2$	1 317.586	1 444.712	1 483.831	1 466.261	1 394.268
$ε l$ / $%$	0	9.648	12.617	11.283	5.819

表2

表3

每次迭代均基于投影优选法的多次迭代后的位移误差向量均方值"

迭代次数	1	100	200	300	400	500	621
位移误差向量均方值 $φ r m s 2$	1 317.586	437.361	159.224	63.776	44.071	28.926	5.838

表3

图9

表4

使用首次迭代中得到的优选参数组的多次迭代后的位移误差向量均方值"

迭代次数	1	5	7	10	20	50	100
位移误差向量均方值 $φ r m s 2$	1 317.586	340.721	223.508	267.741	1 412.205	11 806.770	14 707.850

表4

图10

表5

表6

参考文献 18

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序号	横截面积	序号	横截面积	序号	横截面积	序号	横截面积	序号	横截面积	序号	横截面积
1	62.952	15	172.531	29	63.797	43	161.846	57	63.961	71	115.542
2	153.797	16	70.945	30	99.991	44	105.291	58	137.521	72	79.695
3	10.000	17	71.079	31	199.330	45	113.434	59	98.761	73	67.712
4	88.784	18	89.382	32	193.597	46	10.330	60	93.915	74	106.772
5	113.478	19	103.615	33	95.602	47	10.000	61	113.185	75	86.581
6	120.717	20	10.000	34	86.251	48	18.099	62	56.444	76	61.457
7	90.509	21	91.390	35	84.332	49	198.212	63	103.598	77	113.593
8	97.674	22	100.071	36	31.613	50	166.611	64	97.915	78	113.169
9	120.828	23	115.448	37	104.569	51	268.330	65	98.745	79	106.079
10	120.367	24	124.762	38	58.331	52	125.733	66	124.250
11	125.845	25	104.920	39	86.912	53	119.008	67	134.266
12	161.714	26	101.842	40	10.000	54	64.402	68	89.820
13	151.285	27	118.143	41	86.408	55	57.780	69	62.970
14	11.480	28	121.187	42	73.131	56	127.696	70	119.673

序号	目标位移/mm	实际位移/mm	误差/%
1	115	115.279 05	0.24
2	64	64.104 563	0.16
3	93	93.137 407	0.15
4	48	48.153 887	0.32
5	63	63.117 785	0.19
6	45	44.800 59	0.44
7	40	40.153 148	0.38
8	44	44.226 436	0.51
9	49	49.081 9	0.17
10	39	38.928 128	0.18
?	?	?	?
50	23	22.987 67	0.05
51	119	119.206 21	0.17
52	65	65.012 716	0.02
53	108	108.241 13	0.22
54	59	59.069 813	0.12
55	111	111.284 53	0.26
56	79	79.022 354	0.03
57	126	126.213 33	0.17
58	101	101.017 09	0.02
59	106	106.198 05	0.19
60	47	47.069 806	0.15

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