高阶等参元在薄膜结构自由振动中的应用

乔海青; 鲍四元; 邓子辰; 王博

doi:10.21656/1000-0887.450025

高阶等参元在薄膜结构自由振动中的应用

doi: 10.21656/1000-0887.450025

1.
苏州科技大学土木工程学院，江苏苏州 215011
2.
西北工业大学力学与土木建筑学院，西安 710072

（我刊编委邓子辰来稿）

基金项目:

国家自然科学基金 12172282

详细信息

作者简介:
乔海青(1999—)，男，硕士生(E-mail: 2640403550@qq.com)

通讯作者:
鲍四元(1980—)，男，副教授，博士，硕士生导师(通讯作者. E-mail: bsiyuan@126.com)

中图分类号: O327
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- 文章访问数: 70
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- 被引次数: 0
出版历程
- 收稿日期: 2024-01-31
- 修回日期: 2024-04-29
- 刊出日期: 2025-02-01

Application of High-Order Isoparametric Elements in Free Vibration of Membrane Structures

1.
School of Civil Engineering, Suzhou University of Science and Technology, Suzhou, Jiangsu 215011, P.R.China
2.
School of Mechanics and Civil Engineering, Northwestern Polytechnic University, Xi'an 710072, P.R.China

(Contributed by DENG Zichen, M.AMM Editorial Board)

摘要

摘要: 薄膜结构是工程中广泛应用的结构之一，其自振特性的理论解与三角函数族有关，采用常规低阶单元分析时，有限元解精度不高. 虽然h型有限元法将结构的网格细化后可提高有限元解的精度，但是其前处理相对困难，如果细化网格时出现畸变网格，那么有限元解的精度可能降低. 基于p型有限元法构造两种用于研究薄膜结构自由振动特性的四边形高阶等参元，即节点数为16的Q16等参元和节点数为13的Q13等参元. 不同形状和不同边界条件的薄膜结构算例表明，所提单元较常规低阶等参元有较快的收敛速率和较高的精度及计算效率.
- 薄膜自由振动 /
- p型有限元 /
- 自振特性
Abstract: The membrane is one of the most widely used structures in engineering. Because the theoretical solution of the structure's natural vibration characteristics is related to the trigonometric function family, the accuracy of the finite element solution is not very high by the conventional low-order element analysis. Although the h-type finite element method can improve the accuracy of the finite element solution with refined meshing of the structure, the corresponding pre-processing is relatively difficult, and the accuracy of the finite element solution may be reduced if the refined mesh is distorted. Based on the p-type finite element method, 2 quadrilateral high order isoparametric elements, i.e., isoparametric element Q16 with 16 nodes and isoparametric element Q13 with 13 nodes, were constructed to study the free vibration characteristics of membranes. Examples of membranes with different shapes and different boundary conditions show that, the proposed elements have faster convergence rates, higher computational accuracies and efficiencies than conventional low-order isoparametric elements.
- membrane free vibration /
- p-type finite element /
- vibration characteristic
edited-by

edited-by
注释:

1) （我刊编委邓子辰来稿）

HTML全文

图 1 单元的映射关系

Figure 1. Mapping relationships of the elements

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图 2 三边固定一边自由的方膜

Figure 2. A square membrane with 3 fixed sides and 1 free

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图 3 三边固定一边自由方膜前6阶振型

注为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.

Figure 3. The 1st 6 modes of a membrane with 3 sides fixed and one side free

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图 4 裂缝薄膜结构

Figure 4. A square membrane with an inside crack

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图 5 裂缝薄膜前3阶振型

Figure 5. The 1st 3 modes of a square membrane with an inside crack

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图 6 环扇形薄膜

Figure 6. An annular membrane

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图 7 环扇形薄膜前6阶振型

Figure 7. The 1st 6 modes of the annular membrane

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表 1 三边固定一边自由方膜的前10阶自振频率(基于Q13、Q9、Q8和Q4)(单位: Hz)

Table 1. The 1st 10 natural frequencies of the 3-sides-fixed and one-side-free membrane based on Q13, Q9, Q8 and Q4 (unit: Hz)

N-F(m, n)	element type
	Q13				Q9 N_e=100 (u_dof=380)	Q8 N_e=100 (u_dof=280)	Q4 N_e=900 (u_dof=870)	exact^[21]
	N_e=2 (u_dof=6)	N_e=4 (u_dof=18)	N_e=16 (u_dof=84)	N_e=100 (u_dof=570)	Q9 N_e=100 (u_dof=380)	Q8 N_e=100 (u_dof=280)	Q4 N_e=900 (u_dof=870)	exact^[21]
1(m=1, n=1)	0.342	0.340	0.340	0.340	0.340	0.340	0.340	0.340
2(m=2, n=1)	0.556	0.549	0.549	0.549	0.549	0.549	0.549	0.549
3(m=1, n=2)	0.647	0.631	0.627	0.627	0.627	0.627	0.628	0.627
4(m=2, n=2)	0.831	0.768	0.761	0.761	0.761	0.761	0.762	0.761
5(m=3, n=1)	0.904	0.829	0.820	0.819	0.820	0.820	0.821	0.819
6(m=1, n=3)	1.399	0.949	0.926	0.926	0.926	0.926	0.929	0.926
7(m=3, n=2)	-	1.048	0.975	0.974	0.974	0.974	0.977	0.974
8(m=2, n=3)	-	1.064	1.021	1.021	1.021	1.021	1.024	1.021
9(m=4, n=1)	-	1.164	1.109	1.108	1.109	1.109	1.113	1.108
10(m=3, n=3)	-	1.265	1.190	1.188	1.189	1.189	1.193	1.188
time cost /s	3.25	5.85	24.66	122.75	46.69	42.19	182.17	-

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表 2 三边固定一边自由方膜的前10阶自振频率(基于Q16、Q9、Q8和Q4) (单位: Hz)

Table 2. The 1st 10 natural frequencies of the 3-sides-fixed and one-side-free membrane based on Q16, Q9, Q8 and Q4 (unit: Hz)

N-F(m, n)	element type
	Q16				Q9 N_e=100 (u_dof=380)	Q8 N_e=100 (u_dof=280)	Q4 N_e=900 (u_dof=870)	exact^[21]
	N_e=2 (u_dof=12)	N_e=4 (u_dof=30)	N_e=16 (u_dof=132)	N_e=100 (u_dof=870)	Q9 N_e=100 (u_dof=380)	Q8 N_e=100 (u_dof=280)	Q4 N_e=900 (u_dof=870)	exact^[21]
1(m=1, n=1)	0.342	0.340	0.340	0.340	0.340	0.340	0.340	0.340
2(m=2, n=1)	0.550	0.549	0.549	0.549	0.549	0.549	0.549	0.549
3(m=1, n=2)	0.646	0.627	0.627	0.627	0.627	0.627	0.628	0.627
4(m=2, n=2)	0.776	0.761	0.761	0.761	0.761	0.761	0.762	0.761
5(m=3, n=1)	0.828	0.820	0.820	0.819	0.820	0.820	0.821	0.819
6(m=1, n=3)	0.993	0.926	0.926	0.925	0.926	0.926	0.929	0.926
7(m=3, n=2)	1.160	0.974	0.974	0.974	0.974	0.974	0.977	0.974
8(m=2, n=3)	1.282	1.021	1.021	1.021	1.021	1.021	1.024	1.021
9(m=4, n=1)	1.719	1.109	1.109	1.108	1.109	1.109	1.113	1.108
10(m=3, n=3)	1.804	1.189	1.189	1.188	1.189	1.189	1.193	1.188
time cost /s	9.55	19.37	77.45	428.38	46.69	42.19	182.17	-

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表 3 裂缝薄膜的前5阶特征值(基于Q13、Q9、Q8和Q4单元)

Table 3. The 1st 5 eigenvalues of the cracked membrane based on Q13, Q9, Q8 and Q4

N-F	element type
	Q13				Q9 N_e=225 (u_dof=855)	Q8 N_e=225 (u_dof=630)	Q4 N_e=900 (u_dof=855)	exact
	N_e=2 (u_dof=6)	N_e=4 (u_dof=15)	N_e=16 (u_dof=78)	N_e=64 (u_dof=348)	Q9 N_e=225 (u_dof=855)	Q8 N_e=225 (u_dof=630)	Q4 N_e=900 (u_dof=855)	exact
1	8.842 2	8.726 3	8.539 2	8.454 6	8.563 7	8.580 8	8.464 7	8.371 3^[22]
2	16.999 2	16.815 2	16.648 5	16.645 6	16.644 3	16.644 4	16.685 5	16.645 3^[22]
3	31.107 5	28.807 3	27.996 6	27.977 9	27.979 7	27.979 7	28.098 2	27.977 7^[5]
4	33.498 5	32.188 4	31.024 0	30.759 0	31.049 5	31.093 7	30.916 2	30.536 4^[5]
5	63.512 3	44.514 7	42.798 4	42.551 3	42.696 7	42.718 2	42.845 3	42.447 9^[5]
time cost /s	3.42	5.53	19.91	77.70	244.89	125.58	178.03

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表 4 裂缝薄膜的前5阶特征值(基于Q16、Q9、Q8和Q4单元)

Table 4. The 1st 5 eigenvalues of the cracked membrane based on Q16, Q9, Q8 and Q4

N-F	element type
	Q16				Q9 N_e=225 (u_dof=855)	Q8 N_e=225 (u_dof=630)	Q4 N_e=900 (u_dof=855)	exact
	N_e=2 (u_dof=12)	N_e=4 (u_dof=27)	N_e=16 (u_dof=126)	N_e=64 (u_dof=540)	Q9 N_e=225 (u_dof=855)	Q8 N_e=225 (u_dof=630)	Q4 N_e=900 (u_dof=855)	exact
1	8.652 0	8.597 0	8.476 5	8.423 8	8.563 7	8.580 8	8.464 7	8.371 3^[22]
2	16.940 1	16.792 6	16.647 4	16.645 4	16.644 3	16.644 4	16.685 5	16.645 3^[22]
3	29.159 8	28.500 7	27.994 8	27.977 8	27.979 7	27.979 7	28.098 2	27.977 7^[5]
4	32.424 9	31.835 6	30.853 0	30.676 6	31.049 5	31.093 7	30.916 2	30.536 4^[5]
5	45.050 7	43.884 1	42.725 3	42.513 4	42.696 7	42.718 2	42.845 3	42.447 9^[5]
time cost /s	9.20	17.15	66.36	269.39	244.89	125.58	178.03

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表 5 环扇形薄膜的前10阶自振频率参数(基于Q13、Q9、Q8和Q4单元)

Table 5. The 1st 10 natural vibration frequency parameters of an annular membrane (based on Q13, Q9, Q8 and Q4)

N-F	element type
	Q13						Q9 N_e=225 (u_dof=841)	Q8 N_e=225 (u_dof=616)	Q4 N_e=400 (u_dof=361)	exact^[7]
	N_e=4 (u_dof=10)	N_e=8 (u_dof=22)	N_e=12 (u_dof=49)	N_e=25 (u_dof=121)	N_e=36 (u_dof=181)	N_e=64 (u_dof=337)	Q9 N_e=225 (u_dof=841)	Q8 N_e=225 (u_dof=616)	Q4 N_e=400 (u_dof=361)	exact^[7]
1	6.842 6	6.842 3	6.814 1	6.813 8	6.813 8	6.813 8	6.813 9	6.813 9	6.826 0	6.814 0
2	8.313 3	8.301 0	8.267 3	8.266 8	8.266 7	8.266 7	8.266 9	8.266 8	8.291 6	8.267 0
3	10.338 0	10.237 7	10.191 6	10.190 2	10.189 4	10.189 0	10.189 8	10.189 6	10.257 0	10.189 0
4	12.537 2	12.407 7	12.323 0	12.320 4	12.314 5	12.311 9	12.314 5	12.314 1	12.467 2	12.311 0
5	13.281 4	13.270 6	12.929 4	12.855 7	12.855 6	12.855 5	12.856 2	12.855 8	12.916 8	12.856 0
6	14.330 5	14.168 6	13.816 7	13.742 7	13.742 4	13.742 3	13.743 1	13.742 6	13.809 4	13.742 0
7	15.739 4	14.739 1	14.546 8	14.576 8	14.516 6	14.505 1	14.511 6	14.511 0	14.802 0	14.502 0
8	16.943 0	15.710 6	15.198 1	15.112 3	15.110 6	15.110 1	15.111 4	15.110 7	15.207 8	15.110 0
9	20.438 0	17.245 1	16.806 1	16.829 1	16.794 4	16.715 3	16.727 9	16.727 0	17.017 1	16.706 0
10	24.941 9	17.922 7	16.987 4	16.854 1	16.845 1	16.841 8	16.844 7	16.843 8	17.216 0	16.841 0
time cost /s	28.13	55.09	83.25	174.13	257.67	462.43	447.65	333.32	104.52	-

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表 6 环扇形薄膜的前10阶自振频率参数(基于Q16、Q9、Q8和Q4单元)

Table 6. The 1st 10 natural vibration frequency parameters of an annular membrane(based on Q16, Q9, Q8 and Q4)

N-F	element type
	Q16						Q9 N_e=225 (u_dof=841)	Q8 N_e=225 (u_dof=616)	Q4 N_e=400 (u_dof=361)	exact^[7]
	N_e=4 (u_dof=22)	N_e=8 (u_dof=46)	N_e=12 (u_dof=85)	N_e=25 (u_dof=196)	N_e=36 (u_dof=289)	N_e=64 (u_dof=529)	Q9 N_e=225 (u_dof=841)	Q8 N_e=225 (u_dof=616)	Q4 N_e=400 (u_dof=361)	exact^[7]
1	6.842 2	6.842 2	6.814 1	6.813 8	6.813 8	6.813 8	6.813 9	6.813 9	6.826 0	6.814 0
2	8.300 3	8.300 0	8.267 1	8.266 8	8.266 7	8.266 7	8.266 9	8.266 8	8.291 6	8.267 0
3	10.233 1	10.228 9	10.190 6	10.190 1	10.189 3	10.189 0	10.189 8	10.189 6	10.257 0	10.189 0
4	12.412 6	12.355 9	12.317 8	12.320 0	12.314 4	12.311 9	12.314 5	12.314 1	12.467 2	12.311 0
5	13.267 0	13.267 1	12.929 4	12.855 7	12.855 6	12.855 5	12.856 2	12.855 8	12.916 8	12.856 0
6	14.114 2	14.114 1	13.815 0	13.742 6	13.742 4	13.742 3	13.743 1	13.742 6	13.809 4	13.742 0
7	14.667 0	14.549 7	14.522 7	14.574 9	14.516 1	14.505 1	14.511 6	14.511 0	14.802 0	14.502 0
8	15.460 1	15.457 0	15.182 3	15.111 1	15.110 3	15.110 0	15.111 4	15.110 7	15.207 8	15.110 0
9	17.101 6	16.762 2	16.803 9	16.823 3	16.792 9	16.715 2	16.727 9	16.727 0	17.017 1	16.706 0
10	17.270 9	17.220 1	16.917 5	16.848 8	16.843 9	16.841 7	16.844 7	16.843 8	17.216 0	16.841 0
time cost /s	72.13	144.05	214.37	444.51	661.42	1176.72	447.65	333.32	104.52	-

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