一种结构不确定分析的改进多维平行六面体模型

doi:10.12382/bgxb.2023.0085

摘要/Abstract

摘要：

多维平行六面体模型是一种能够同时考虑相关变量和独立变量的非概率凸集模型,更适用于工程结构中常见的多源不确定性问题。为更加有效合理地度量结构不确定性,提出一种改进多维平行六面体模型。通过定义区间变量的相关角和边缘区间,给出模型不确定域的显式表达式,进而给出依据实验样本点构建多维平行六面体模型的方法。3个算例分析结果表明,改进多维平行六面体模型能够较好地反映区间变量之间的相关性,是一种比传统多维平行六面体模型更为紧凑合理的模型。

关键词: 多维平行六面体模型, 非概率凸集模型, 变量相关性, 不确定域显式表达式

Abstract:

The multidimensional parallelepiped model is a non-probabilistic convex set model that can take into account dependent and independent variables simultaneously. Therefore, it is more suitable for the “multi-source uncertainty” problems in engineering structures. An improved multidimensional parallelepiped model is presented to more reasonably and efficiently quantify the structural uncertainties. An explicit expression for the uncertainty domain of the parallelepiped model is given by defining the correlation angle and marginal intervals of interval variables. A method is further formulated to construct a multidimensional parallelepiped model based on the experimental sample points. The analyzed results of three numerical examples show that the proposed model can better reflect the correlation of interval variables, and has a more compactness and rationality than the traditional multidimensional parallelepiped model.

Key words: multidimensional parallelepiped model, non-probabilistic convex set model, correlation of variables, explicit expression of uncertainty domain

中图分类号:

TB114.3

乔心州, 张锦瑞, 方秀荣, 刘鹏. 一种结构不确定分析的改进多维平行六面体模型[J]. 兵工学报, 2024, 45(6): 1877-1888.

QIAO Xinzhou, ZHANG Jinrui, FANG Xiurong, LIU Peng. An Improved Multidimensional Parallelepiped Model for Structural Uncertainty Analysis[J]. Acta Armamentarii, 2024, 45(6): 1877-1888.

图/表 17

图1 两种多维平行六面体模型(二维情况)

Fig.1 Two types of multidimensional parallelepiped model (two-dimensional case)

图2 多维平行六面体模型(二维情况)

Fig.2 Multidimensional parallelepiped model (two-dimensional case)

图3 不同参数空间的平行四边形凸模型

Fig.3 A parallelogram convex model in different parameter spaces

图4 不同相关系数下的不确定域

Fig.4 Uncertainty domain under different correlation coefficients

图5 从单位正方形到平行四边形的线性变换

Fig.5 Linear transformation from a unit square to a parallelogram

图6 根据实验样本点构建改进多维平行六面体模型的流程图

Fig.6 Flow chart of creating the improved multidimensional parallelepiped model from the experimental sample points

图7 平行六面体不确定域的投影(情况1)

Fig.7 Projections of uncertainty domain of a three-dimensional parallelepiped model (Case 1)

图8 平行六面体不确定域的投影(情况2)

Fig.8 Projections of uncertainty domain of a three-dimensional parallelepiped model (Case 2)

图9 悬臂梁

Fig.9 Cantilever beam

表1 算例2的50个样本点

Table 1 50 sample points for Example 2

编号	F₁/kN	F₂/kN	F₃/kN
1	5.1755	1.9634	0.8777
2	5.2654	1.9122	0.8561
3	4.8108	2.1068	1.1634
4	5.2290	1.9991	1.1371
5	5.2098	2.0145	1.0898
6	4.7566	2.0932	1.1487
7	5.0843	2.0677	1.0603
8	4.6462	2.1643	1.1716
9	5.0339	2.0647	1.0866
10	5.2625	1.9796	1.1410
11	5.1978	1.8782	0.9033
12	4.7551	2.0395	1.1092
13	5.0658	2.0396	1.1424
14	4.7695	2.0084	0.9699
15	5.1534	1.9629	0.8958
16	4.9807	2.0196	1.0484
17	5.1152	1.9779	0.9264
18	5.1263	1.9462	0.8852
19	4.7291	2.1136	1.1668
20	4.9409	1.9753	0.9128
21	5.1649	1.9466	0.9874
22	5.2830	1.9385	0.8325
23	5.1162	1.8831	0.8674
24	4.8616	2.0229	0.9337
25	5.1460	2.1180	1.1740
26	4.9879	1.9256	0.9092
27	5.2313	2.0132	1.1543
28	5.1637	1.9937	1.0110
29	5.2173	1.9824	1.0697
30	5.1431	1.9749	1.0125
31	4.8166	1.9785	1.0072
32	5.1524	1.9708	0.8805
33	5.1082	1.9025	0.8307
34	4.9102	2.0194	1.0937
35	4.9396	1.9594	0.8919
36	5.0622	1.9865	1.0421
37	5.0810	2.0100	0.9862
38	4.7478	2.0708	1.1432
39	4.7670	2.0805	1.0329
40	5.3279	1.8907	0.9536
41	4.7995	2.0072	0.8565
42	5.0622	2.0877	1.1288
43	4.8180	1.9770	0.9258
44	5.1153	2.0397	1.1299
45	4.9028	1.9297	0.8799
46	4.7473	2.0953	1.0488
47	5.0202	2.0102	1.1075
48	5.0822	1.9717	0.9964
49	5.0318	1.9026	0.9001
50	4.8672	1.9404	0.9110

图10 样本点在不同二维平面上的投影

Fig.10 Projections of sample points on different two-dimensional planes

图11 改进多维平行六面体模型和传统多维平行六面体模型

Fig.11 The improved and conventional multidimensional parallelepiped models

图12 悬臂梁

Fig.12 Cantilever beam

表2 算例3的实验样本点

Table 2 100 experimental sample points for Example 3

编号	L/mm	b/mm	h/mm	E/GPa	ν
1	1025.53	103.44	214.91	198.524	0.2758
2	952.90	107.51	206.42	203.360	0.2829
3	1013.28	103.72	207.80	209.866	0.2783
4	1066.47	96.76	200.90	195.081	0.2700
5	982.61	99.88	200.40	210.729	0.2808
6	1065.85	109.91	222.70	197.042	0.2715
7	1026.92	95.88	205.41	204.181	0.2884
8	1002.17	100.36	208.01	210.394	0.2781
9	973.49	96.22	202.42	199.066	0.2686
10	1037.02	97.87	205.42	209.370	0.2742
11	1042.90	100.93	211.73	209.605	0.2763
12	920.06	97.61	195.60	198.936	0.2725
13	1062.40	102.36	217.78	200.788	0.2869
14	997.81	106.55	211.73	201.663	0.2724
15	1052.91	107.82	208.48	208.766	0.2764
16	1052.46	104.69	204.13	212.470	0.2898
17	1013.91	107.28	210.12	195.943	0.2733
18	1000.11	105.96	213.03	212.103	0.2725
19	990.40	93.98	193.80	194.399	0.2861
20	995.63	96.77	202.67	196.081	0.2696
21	1070.52	104.48	220.08	209.102	0.2801
22	1072.24	109.27	223.27	190.472	0.2918
23	1030.12	108.80	210.87	203.000	0.2872
24	968.59	102.22	210.45	205.363	0.2877
25	1035.83	102.12	203.29	209.432	0.2671
26	1031.26	104.76	213.22	213.230	0.2894
27	1057.76	104.62	219.09	207.348	0.2659
28	947.68	105.79	202.23	213.765	0.2693
29	1001.32	101.60	196.20	207.474	0.2748
30	921.99	99.14	191.92	195.111	0.2737
31	928.24	94.40	187.66	208.183	0.2725
32	1012.06	104.24	207.06	205.249	0.2774
33	950.55	101.72	207.89	214.381	0.2707
34	960.16	102.20	204.78	202.711	0.2719
35	1056.59	94.68	199.28	195.688	0.2725
36	1011.81	101.00	197.18	214.841	0.2824
37	1041.45	96.75	213.33	214.354	0.2892
38	1065.55	106.25	212.42	196.091	0.2784
39	945.81	95.35	206.61	199.519	0.2749
40	978.55	101.05	205.94	191.163	0.2911
41	1014.49	106.05	205.08	203.798	0.2677
42	949.16	102.43	198.07	215.355	0.2665
43	1086.67	103.64	216.78	217.589	0.2704
44	994.52	101.60	209.66	213.319	0.2666
45	936.83	97.00	189.66	205.133	0.2746
46	954.96	97.91	207.71	208.806	0.2772
47	1067.32	97.27	213.42	210.172	0.2793
48	948.31	101.37	196.98	195.381	0.2738
49	1051.31	94.92	201.57	192.792	0.2803
50	923.81	105.03	194.50	194.116	0.2766
51	1000.28	97.24	212.81	200.311	0.2748
52	916.78	95.84	187.39	203.759	0.2860
53	1005.20	94.06	194.43	210.467	0.2881
54	1008.65	96.88	210.09	200.474	0.2747
55	1021.43	93.01	194.70	208.312	0.2808
56	1015.92	105.48	211.85	211.949	0.2718
57	1031.65	102.15	205.83	192.241	0.2892
58	984.24	103.45	209.14	194.812	0.2718
59	1019.77	102.64	214.52	211.409	0.2834
60	911.95	92.68	186.55	210.929	0.2889
61	1030.57	97.14	209.98	194.978	0.2714
62	1025.57	99.67	209.27	210.457	0.2797
63	1041.82	98.79	211.78	215.552	0.2858
64	930.04	98.06	195.52	208.767	0.2805
65	1038.92	98.89	201.46	192.908	0.2888
66	954.22	99.29	201.45	196.876	0.2845
67	1000.42	97.24	208.74	194.333	0.2762
68	950.39	99.75	192.51	202.633	0.2693
69	940.01	103.93	197.18	205.844	0.2907
70	1001.51	104.97	211.46	202.113	0.2836
71	939.76	105.72	195.88	197.492	0.2884
72	1002.64	104.84	204.57	199.361	0.2683
73	1027.59	100.46	205.89	208.023	0.2668
74	982.66	105.65	209.04	195.337	0.2717
75	925.74	93.04	194.20	207.727	0.2840
76	1003.35	96.16	205.67	195.358	0.2718
77	922.82	94.08	189.50	196.754	0.2762
78	961.30	95.92	194.91	213.908	0.2820
79	998.15	95.77	197.06	192.351	0.2926
80	1031.86	101.89	204.15	195.297	0.2844
81	1068.47	96.60	203.91	202.481	0.2685
82	1042.17	102.04	217.85	201.403	0.2829
83	936.04	97.65	191.86	210.048	0.2873
84	976.56	102.97	199.78	204.253	0.2877
85	1009.84	98.38	200.71	203.147	0.2774
86	986.89	102.78	209.11	202.402	0.2777
87	927.93	92.86	191.76	199.140	0.2843
88	1082.68	108.44	213.61	196.903	0.2877
89	1054.51	106.24	215.94	193.607	0.2863
90	980.95	95.58	193.43	213.291	0.2685
91	961.21	104.51	210.65	204.178	0.2843
92	956.04	107.39	204.62	209.881	0.2658
93	1047.00	104.17	202.58	205.237	0.2744
94	1005.73	99.63	203.10	196.491	0.2746
95	936.95	107.10	205.13	216.136	0.2675
96	1077.32	106.75	214.79	203.190	0.2731
97	1026.77	96.27	196.73	210.732	0.2819
98	1037.44	103.53	207.27	200.672	0.2881
99	986.03	106.60	211.46	213.297	0.2771
100	1104.41	107.96	209.01	192.008	0.2915

表2 算例3的实验样本点

Table 2 100 experimental sample points for Example 3

编号	L/mm	b/mm	h/mm	E/GPa	ν
1	1025.53	103.44	214.91	198.524	0.2758
2	952.90	107.51	206.42	203.360	0.2829
3	1013.28	103.72	207.80	209.866	0.2783
4	1066.47	96.76	200.90	195.081	0.2700
5	982.61	99.88	200.40	210.729	0.2808
6	1065.85	109.91	222.70	197.042	0.2715
7	1026.92	95.88	205.41	204.181	0.2884
8	1002.17	100.36	208.01	210.394	0.2781
9	973.49	96.22	202.42	199.066	0.2686
10	1037.02	97.87	205.42	209.370	0.2742
11	1042.90	100.93	211.73	209.605	0.2763
12	920.06	97.61	195.60	198.936	0.2725
13	1062.40	102.36	217.78	200.788	0.2869
14	997.81	106.55	211.73	201.663	0.2724
15	1052.91	107.82	208.48	208.766	0.2764
16	1052.46	104.69	204.13	212.470	0.2898
17	1013.91	107.28	210.12	195.943	0.2733
18	1000.11	105.96	213.03	212.103	0.2725
19	990.40	93.98	193.80	194.399	0.2861
20	995.63	96.77	202.67	196.081	0.2696
21	1070.52	104.48	220.08	209.102	0.2801
22	1072.24	109.27	223.27	190.472	0.2918
23	1030.12	108.80	210.87	203.000	0.2872
24	968.59	102.22	210.45	205.363	0.2877
25	1035.83	102.12	203.29	209.432	0.2671
26	1031.26	104.76	213.22	213.230	0.2894
27	1057.76	104.62	219.09	207.348	0.2659
28	947.68	105.79	202.23	213.765	0.2693
29	1001.32	101.60	196.20	207.474	0.2748
30	921.99	99.14	191.92	195.111	0.2737
31	928.24	94.40	187.66	208.183	0.2725
32	1012.06	104.24	207.06	205.249	0.2774
33	950.55	101.72	207.89	214.381	0.2707
34	960.16	102.20	204.78	202.711	0.2719
35	1056.59	94.68	199.28	195.688	0.2725
36	1011.81	101.00	197.18	214.841	0.2824
37	1041.45	96.75	213.33	214.354	0.2892
38	1065.55	106.25	212.42	196.091	0.2784
39	945.81	95.35	206.61	199.519	0.2749
40	978.55	101.05	205.94	191.163	0.2911
41	1014.49	106.05	205.08	203.798	0.2677
42	949.16	102.43	198.07	215.355	0.2665
43	1086.67	103.64	216.78	217.589	0.2704
44	994.52	101.60	209.66	213.319	0.2666
45	936.83	97.00	189.66	205.133	0.2746
46	954.96	97.91	207.71	208.806	0.2772
47	1067.32	97.27	213.42	210.172	0.2793
48	948.31	101.37	196.98	195.381	0.2738
49	1051.31	94.92	201.57	192.792	0.2803
50	923.81	105.03	194.50	194.116	0.2766
51	1000.28	97.24	212.81	200.311	0.2748
52	916.78	95.84	187.39	203.759	0.2860
53	1005.20	94.06	194.43	210.467	0.2881
54	1008.65	96.88	210.09	200.474	0.2747
55	1021.43	93.01	194.70	208.312	0.2808
56	1015.92	105.48	211.85	211.949	0.2718
57	1031.65	102.15	205.83	192.241	0.2892
58	984.24	103.45	209.14	194.812	0.2718
59	1019.77	102.64	214.52	211.409	0.2834
60	911.95	92.68	186.55	210.929	0.2889
61	1030.57	97.14	209.98	194.978	0.2714
62	1025.57	99.67	209.27	210.457	0.2797
63	1041.82	98.79	211.78	215.552	0.2858
64	930.04	98.06	195.52	208.767	0.2805
65	1038.92	98.89	201.46	192.908	0.2888
66	954.22	99.29	201.45	196.876	0.2845
67	1000.42	97.24	208.74	194.333	0.2762
68	950.39	99.75	192.51	202.633	0.2693
69	940.01	103.93	197.18	205.844	0.2907
70	1001.51	104.97	211.46	202.113	0.2836
71	939.76	105.72	195.88	197.492	0.2884
72	1002.64	104.84	204.57	199.361	0.2683
73	1027.59	100.46	205.89	208.023	0.2668
74	982.66	105.65	209.04	195.337	0.2717
75	925.74	93.04	194.20	207.727	0.2840
76	1003.35	96.16	205.67	195.358	0.2718
77	922.82	94.08	189.50	196.754	0.2762
78	961.30	95.92	194.91	213.908	0.2820
79	998.15	95.77	197.06	192.351	0.2926
80	1031.86	101.89	204.15	195.297	0.2844
81	1068.47	96.60	203.91	202.481	0.2685
82	1042.17	102.04	217.85	201.403	0.2829
83	936.04	97.65	191.86	210.048	0.2873
84	976.56	102.97	199.78	204.253	0.2877
85	1009.84	98.38	200.71	203.147	0.2774
86	986.89	102.78	209.11	202.402	0.2777
87	927.93	92.86	191.76	199.140	0.2843
88	1082.68	108.44	213.61	196.903	0.2877
89	1054.51	106.24	215.94	193.607	0.2863
90	980.95	95.58	193.43	213.291	0.2685
91	961.21	104.51	210.65	204.178	0.2843
92	956.04	107.39	204.62	209.881	0.2658
93	1047.00	104.17	202.58	205.237	0.2744
94	1005.73	99.63	203.10	196.491	0.2746
95	936.95	107.10	205.13	216.136	0.2675
96	1077.32	106.75	214.79	203.190	0.2731
97	1026.77	96.27	196.73	210.732	0.2819
98	1037.44	103.53	207.27	200.672	0.2881
99	986.03	106.60	211.46	213.297	0.2771
100	1104.41	107.96	209.01	192.008	0.2915

图13 测试弹性模量E和泊松比ν的实验装置

Fig.13 Experimental device for testing elastic modulus E and Poisson’s ratio ν

图14 样本点在不同二维平面上的投影

Fig.14 Projections of sample points on different two-dimensional planes

表3 两种多维平行六面体模型的评价指标

Table 3 Evaluation indexes of two multidimensional parallelepiped models

模型	未被包络的样本点编号	适应度 κ/%	体积比 $V ¯$ /%
传统多维平行六面体模型	4,31,55, 81,96,100	94	27.20
改进多维平行六面体模型	31,81,100	97	18.46

表3 两种多维平行六面体模型的评价指标

Table 3 Evaluation indexes of two multidimensional parallelepiped models

模型	未被包络的样本点编号	适应度 κ/%	体积比 $V ¯$ /%
传统多维平行六面体模型	4,31,55, 81,96,100	94	27.20
改进多维平行六面体模型	31,81,100	97	18.46

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