平面曲梁面外自由振动有限元分析的<i>p</i>型超收敛算法

叶康生; 梁童

doi:10.6052/j.issn.1000-4750.2019.11.0694

平面曲梁面外自由振动有限元分析的p型超收敛算法

叶康生^,,
梁童

清华大学土木工程系，土木工程安全与耐久教育部重点实验室，北京 100084

基金项目: 国家自然科学基金项目(51078198)；清华大学自主科研计划项目(2011THZ03)

详细信息

作者简介:
梁　童(1994−)，男，山东人，硕士生，主要从事结构工程研究(E-mail: liangt17@mails.tsinghua.edu.cn)

通讯作者:
叶康生(1972−)，男，江苏人，副教授，博士，主要从事结构工程研究(E-mail: yeks@tsinghua.edu.cn)

中图分类号: TU311.3
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出版历程
- 收稿日期: 2019-11-24
- 修回日期: 2020-03-04
- 网络出版日期: 2020-05-24
- 刊出日期: 2020-10-09

A p-TYPE SUPERCONVERGENT RECOVERY METHOD FOR FINITE ELEMENT ANALYSIS OF OUT-OF-PLANE FREE VIBRATIONS OF PLANAR CURVED BEAMS

YE Kang-sheng^,,
LIANG Tong

Department of Civil Engineering, Tsinghua University,Key Laboratory of Civil Engineering Safety and Durability of China Education Ministry, Beijing 100084, China

摘要

摘要: 该文用p型超收敛算法对平面曲梁面外自由振动问题进行求解。该法基于频率和振型结点位移在有限元解答中的超收敛特性，在单元上建立振型近似满足的线性常微分方程边值问题，用更高次元对该线性边值问题进行有限元求解获得各单元上振型的超收敛解，将振型的超收敛解代入Rayleigh商，得到频率的超收敛解。该法作为后处理法，修复计算分别在各个单元上单独进行，故通过少量计算即能显著提高频率和振型的精度和收敛阶。数值算例显示该法稳定、高效，值得进一步研究和推广。
- 平面曲梁 /
- 面外自由振动 /
- 有限元 /
- p型超收敛 /
- Rayleigh商
Abstract: It extends the p-type superconvergence recovery method to the finite element analysis of the out-of-plane free vibrations of planar curved beams. Based on the superconvergence properties on frequencies and nodal displacements in modes, a linear ordinary differential boundary value problem (BVP) which approximately governs the mode on each element is set up. This linear BVP is solved by using a higher order element from which the mode on each element is recovered. By substituting the recovered mode into the Rayleigh quotient, the frequency is recovered. This method is a post-processing approach. Its recovery computation for each element is handled only on its own domain. It can enhance the accuracy and convergence rate of the frequencies and modes significantly with a small computation cost. Numerical examples demonstrate that the method is stable, efficient and worth further exploring.
- planar curved beam /
- out-of-plane free vibration /
- finite element method /
- p-type superconvergence /
- Rayleigh quotient

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图 1 平面曲梁

Figure 1. Planar curved beam

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图 2 有限元网格划分

Figure 2. Finite element mesh

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图 3 例1常截面圆弧曲梁

Figure 3. Uniform arch beam in Example 1

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图 4 例1第三阶频率、振型和结点位移的收敛阶

Figure 4. Rate of convergence of 3rd frequence, mode and its nodal displacement in Example 1

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图 5 例1第三阶振型的有限元解与超收敛解比较

Figure 5. Comparison of FE solution with recovered solution on 3rd mode in Example 1

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图 6 例1第三阶特征对有限元解与超收敛解收敛阶

Figure 6. Rate of convergence of FE solution and recovered solutions on 3rd eigenpair in Example 1

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图 7 例2常截面圆弧曲梁

Figure 7. Uniform arch beam in Example 2

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图 8 例2第26阶特征对有限元解与超收敛解收敛阶

Figure 8. Rate of convergence of FE solution and recovered solutions on 26th eigenpair in Example 2

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图 9 三类轴线曲梁

Figure 9. Three types of curved beams

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表 1 例1第三阶频率、振型和结点位移的收敛阶

Table 1 Rate of convergence of 3rd frequency, mode and its nodal displacement in Example 1

ne	$\|\varOmega {\rm{ - }}{\varOmega ^h}\|$	r	$\|\|{{{d}}_c}{\rm{ - }}{{d}}_c^h\|\|$	r	$\|\|{{d}}{\rm{ - }}{{{d}}^h}\|\|$	r
16	2.8821×10⁻³	−	7.5917×10⁻⁶	−	4.9237×10⁻⁴	−
32	1.8948×10⁻⁴	3.9	5.7596×10⁻⁷	3.7	6.2308×10⁻⁵	3.0
64	1.1998×10⁻⁵	4.0	3.7692×10⁻⁸	3.9	7.8069×10⁻⁶	3.0
128	7.5231×10⁻⁷	4.0	2.3827×10⁻⁹	4.0	9.7640×10⁻⁷	3.0
256	4.7058×10⁻⁸	4.0	1.4934×10⁻¹⁰	4.0	1.2207×10⁻⁷	3.0
512	2.9417×10⁻⁹	4.0	9.3404×10⁻¹²	4.0	1.5259×10⁻⁸	3.0
m=2		4		4		3

ne	$\|\varOmega {\rm{ - }}{\varOmega ^h}\|$	r	$\|\|{{{d}}_c}{\rm{ - }}{{d}}_c^h\|\|$	r	$\|\|{{d}}{\rm{ - }}{{{d}}^h}\|\|$	r
16	3.2911×10⁻⁶	−	2.0671×10⁻⁷	−	1.3848×10⁻⁵	−
32	5.2669×10⁻⁸	6.0	3.3117×10⁻⁹	6.0	8.1494×10⁻⁷	4.1
64	8.2793×10⁻¹⁰	6.0	5.2071×10⁻¹¹	6.0	5.0101×10⁻⁸	4.0
128	1.2956×10⁻¹¹	6.0	8.1489×10⁻¹³	6.0	3.1181×10⁻⁹	4.0
256	2.0251×10⁻¹³	6.0	1.2738×10⁻¹⁴	6.0	1.9468×10⁻¹⁰	4.0
512	3.1645×10⁻¹⁵	6.0	1.9905×10⁻¹⁶	6.0	1.2164×10⁻¹¹	4.0
m=3		6		6		4

ne	$\|\varOmega {\rm{ - }}{\varOmega ^h}\|$	r	$\|\|{{{d}}_c}{\rm{ - }}{{d}}_c^h\|\|$	r	$\|\|{{d}}{\rm{ - }}{{{d}}^h}\|\|$	r
16	2.0402×10⁻⁹	−	1.2643×10⁻¹⁰	−	2.3812×10⁻⁷	−
32	8.0841×10⁻¹²	8.0	4.9957×10⁻¹³	8.0	7.3852×10⁻⁹	5.0
64	3.1692×10⁻¹⁴	8.0	1.9571×10⁻¹⁵	8.0	2.3031×10⁻¹⁰	5.0
128	1.2390×10⁻¹⁶	8.0	7.6504×10⁻¹⁸	8.0	7.1933×10⁻¹²	5.0
256	4.8646×10⁻¹⁹	8.0	2.9890×10⁻²⁰	8.0	2.2476×10⁻¹³	5.0
512	1.8912×10⁻²¹	8.0	1.1676×10⁻²²	8.0	7.0236×10⁻¹⁵	5.0
m=4		8		8		5

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表 2 例1第三阶特征对有限元解与超收敛解收敛阶

Table 2 Rate of convergence of FE solution and recovered solutions on 3rd eigenpair in Example 1

$m = 2$
$ne$	$\|\varOmega {\rm{ - }}{\varOmega ^h}\|$	$r$	$\|\|{{d}} - {{{d}}^h}\|\|$	$r$
16	2.8821×10⁻³	−	4.9237×10⁻⁴	−
32	1.8948×10⁻⁴	3.9	6.2308×10⁻⁵	3.0
64	1.1998×10⁻⁵	4.0	7.8069×10⁻⁶	3.0
128	7.5231×10⁻⁷	4.0	9.7640×10⁻⁷	3.0
256	4.7058×10⁻⁸	4.0	1.2207×10⁻⁷	3.0
512	2.9417×10⁻⁹	4.0	1.5259×10⁻⁸	3.0
理论值		4		3

$\bar m = 3$
$ne$	$\|\varOmega {\rm{ - }}{\varOmega ^{\rm{*}}}\|$	$r$	$\|\|{{d}} - {{{d}}^{\rm{*}}}\|\|$	$r$
16	4.1065×10⁻⁵	−	1.4864×10⁻⁴	−
32	2.5175×10⁻⁷	7.3	9.6899×10⁻⁶	3.9
64	1.6723×10⁻⁹	7.2	6.1437×10⁻⁷	4.0
128	1.6325×10⁻¹¹	6.7	3.8746×10⁻⁸	4.0
256	2.1574×10⁻¹³	6.2	2.4284×10⁻⁹	4.0
512	3.2163×10⁻¹⁵	6.1	1.5188×10⁻¹⁰	4.0
理论值		6		4

$\bar m = 4$
$ne$	$\|\varOmega {\rm{ - }}{\varOmega ^{\rm{*}}}\|$	$r$	$\|\|{{d}} - {{{d}}^{\rm{*}}}\|\|$	$r$
16	3.7774×10⁻⁵	−	1.4864×10⁻⁴	−
32	1.9909×10⁻⁷	7.6	9.6899×10⁻⁶	3.9
64	8.4441×10⁻¹⁰	7.9	6.1233×10⁻⁷	4.0
128	3.3689×10⁻¹²	8.0	3.8378×10⁻⁸	4.0
256	1.3230×10⁻¹⁴	8.0	2.4003×10⁻⁹	4.0
512	5.1747×10⁻¹⁷	8.0	1.5004×10⁻¹⁰	4.0
理论值		8		4

$\bar m = 5$
$ne$	$\|\varOmega {\rm{ - }}{\varOmega ^{\rm{*}}}\|$	$r$	$\|\|{{d}} - {{{d}}^{\rm{*}}}\|\|$	$r$
16	3.7772×10⁻⁵	−	1.4864×10⁻⁴	−
32	1.9908×10⁻⁷	7.6	9.6899×10⁻⁶	3.9
64	8.4438×10⁻¹⁰	7.9	6.1233×10⁻⁷	4.0
128	3.3687×10⁻¹²	8.0	3.8378×10⁻⁸	4.0
256	1.3229×10⁻¹⁴	8.0	2.4003×10⁻⁹	4.0
512	5.1745×10⁻¹⁷	8.0	1.5004×10⁻¹⁰	4.0
理论值		8		4

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表 3 例1第三阶特征对SPRD法收敛阶

Table 3 Rate of convergence of recovered solutions on 3rd eigenpair in Example 1 with SPRD method

$ne$	$\|\varOmega {\rm{ - }}{\varOmega ^{\rm{*}}}\|$	$r$	$\|\|{{d}} - {{{d}}^{\rm{*}}}\|\|$	$r$
32	1.9168×10⁻⁵	−	6.1793×10⁻⁵	−
64	2.6890×10⁻⁷	6.2	3.9614×10⁻⁶	4.0
128	3.9339×10⁻⁹	6.1	2.4915×10⁻⁷	4.0
256	5.9333×10⁻¹¹	6.1	1.5596×10⁻⁸	4.0
512	9.1038×10⁻¹³	6.0	9.7516×10⁻¹⁰	4.0
1024	1.4094×10⁻¹⁴	6.0	6.0954×10⁻¹¹	4.0

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表 4 例2前12阶特征对有限元解与超收敛解误差

Table 4 Errors of FE solution and recovered solutions on first 12 eigenpairs in Example 2

阶数	$\varOmega$	${\varOmega ^h}$	$\|\varOmega - {\varOmega ^h}\|$	$\|\|{{d}} - {{{d}}^h}\|\|$	${\varOmega ^{\rm{*}}}$	$\|\varOmega {\rm{ - }}{\varOmega ^{\rm{*}}}\|$	$\dfrac{{\|\varOmega - {\varOmega ^{\rm{*}}}\|}}{{\|\varOmega - {\varOmega ^h}\|}}/( \text{%})$	$\|\|{{d}} - {{{d}}^{\rm{*}}}\|\|$	$\dfrac{{\|\|{{d}} - {{{d}}^{\rm{*}}}\|\|}}{{\|\|{{d}} - {{{d}}^h}\|\|}}/( \text{%})$	n
1	0.000000000	0.278903246	2.79×10⁻¹	1.21×10⁻³	0.003516615	3.52×10⁻³	1.26	8.75×10⁻⁶	0.72	1
2	2.096810333	2.379666689	2.83×10⁻¹	4.89×10⁻³	2.097006671	1.96×10⁻⁴	0.07	2.11×10⁻⁴	4.32	2
3	6.316663854	6.817878189	5.01×10⁻¹	1.11×10⁻²	6.317460674	7.97×10⁻⁴	0.16	9.77×10⁻⁴	8.76	3
4	12.156595065	13.009811658	8.53×10⁻¹	1.95×10⁻²	12.159147906	2.55×10⁻³	0.30	2.62×10⁻³	13.41	4
5	17.335703469	17.337172055	1.47×10⁻³	1.20×10⁻³	17.335703469	0.00×10⁰	0.00	7.65×10⁻⁶	0.64	1
6	19.209142151	20.568557976	1.36×10⁰	3.09×10⁻²	19.216004983	6.86×10⁻³	0.50	5.86×10⁻³	18.93	5
7	22.125304755	22.143921945	1.86×10⁻²	4.81×10⁻³	22.125316688	1.19×10⁻⁵	0.06	8.89×10⁻⁵	1.85	2
8	28.486387871	28.560421560	7.40×10⁻²	1.09×10⁻²	28.486493560	1.06×10⁻⁴	0.14	4.06×10⁻⁴	3.74	3
9	27.163609617	29.207708099	2.04×10⁰	4.41×10⁻²	27.179901316	1.63×10⁻²	0.80	1.15×10⁻²	26.16	6
10	35.578254236	35.767316820	1.89×10⁻¹	1.90×10⁻²	35.578729685	4.75×10⁻⁴	0.25	1.09×10⁻³	5.76	4
11	35.780085885	38.709492937	2.93×10⁰	5.95×10⁻²	35.815415938	3.53×10⁻²	1.21	2.16×10⁻²	36.35	7
12	43.038408009	43.423183936	3.85×10⁻¹	3.01×10⁻²	43.039902934	1.49×10⁻³	0.39	2.41×10⁻³	8.00	5

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表 5 例2第26阶特征对有限元解与超收敛解收敛阶

Table 5 Rate of convergence of FE solution and recovered solutions on 26th eigenpair in Example 2

$m = 3$					$\bar m = {\rm{4}}$
$ne$	$\|\varOmega {\rm{ - }}{\varOmega ^h}\|$	$r$	$\|\|{{d}} - {{{d}}^h}\|\|$	$r$	$ne$	$\|\varOmega {\rm{ - }}{\varOmega ^{\rm{*}}}\|$	$r$	$\|\|{{d}} - {{{d}}^{\rm{*}}}\|\|$	$r$
16	3.0092×10⁻¹	−	8.6231×10⁻¹	−	16	1.1674×10⁻²	−	8.4742×10⁻¹	−
32	5.8019×10⁻³	5.7	2.5207×10⁻³	8.4	32	4.4443×10⁻⁵	8.0	2.0138×10⁻⁴	12.0
64	9.5518×10⁻⁵	5.9	1.4991×10⁻⁴	4.1	64	1.7980×10⁻⁷	7.9	5.2408×10⁻⁶	5.3
128	1.5120×10⁻⁶	6.0	9.2470×10⁻⁶	4.0	128	7.0914×10⁻¹⁰	8.0	1.5277×10⁻⁷	5.1
256	2.3701×10⁻⁸	6.0	5.7601×10⁻⁷	4.0	256	2.7769×10⁻¹²	8.0	4.6824×10⁻⁹	5.0
512	3.7063×10⁻¹⁰	6.0	3.5970×10⁻⁸	4.0	512	1.0854×10⁻¹⁴	8.0	1.4560×10⁻¹⁰	5.0
理论值		6		4	理论值		8		5

$\bar m = {\rm{5}}$					$\bar m = {\rm{6}}$
$ne$	$\|\varOmega {\rm{ - }}{\varOmega ^{\rm{*}}}\|$	$r$	$\|\|{{d}} - {{{d}}^{\rm{*}}}\|\|$	$r$	$ne$	$\|\varOmega {\rm{ - }}{\varOmega ^{\rm{*}}}\|$	$r$	$\|\|{{d}} - {{{d}}^{\rm{*}}}\|\|$	$r$
16	2.2943×10⁻³	−	8.4742×10⁻¹	−	16	2.1079×10⁻³	−	8.4742×10⁻¹	−
32	4.2573×10⁻⁷	12.4	1.4481×10⁻⁴	12.5	32	2.1331×10⁻⁷	13.3	1.4365×10⁻⁴	12.5
64	2.5609×10⁻¹⁰	10.7	2.4027×10⁻⁶	5.9	64	4.1349×10⁻¹¹	12.3	2.3581×10⁻⁶	5.9
128	2.2096×10⁻¹³	10.2	3.8469×10⁻⁸	6.0	128	9.4404×10⁻¹⁵	12.1	3.7307×10⁻⁸	6.0
256	2.0927×10⁻¹⁶	10.0	6.0524×10⁻¹⁰	6.0	256	2.2647×10⁻¹⁸	12.0	5.8473×10⁻¹⁰	6.0
512	2.0281×10⁻¹⁹	10.0	9.4732×10⁻¹²	6.0	512	5.5045×10⁻²²	12.0	9.1435×10⁻¹²	6.0
理论值		10		6	理论值		12		6

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表 6 例3三类轴线曲梁的无量纲频率

Table 6 Dimensionless frequencies for three curved beams of Example 3

轴线与支座	阶数	文献[27]	文献[28]	$m = {\rm{1}}$ ， $\bar m = {\rm{2}}$				$m = {\rm{2}}$ ， $\bar m = {\rm{3}}$
轴线与支座	阶数	文献[27]	文献[28]	有限元解	误差/(%)	超收敛解	误差/(%)	有限元解	误差/(%)	超收敛解	误差/(%)
抛物线铰支-铰支	1	6.0825	6.090	7.813	28.45	6.095	0.21	6.083	0.00	6.082	0.00
	2	30.4026	30.40	35.739	17.55	30.426	0.08	30.405	0.01	30.402	0.00
	3	70.0449	70.03	81.753	16.71	70.122	0.11	70.048	0.00	70.032	−0.02
三角正弦线固支-固支	1	17.0609	17.14	20.088	17.74	17.068	0.04	17.064	0.02	17.061	0.00
	2	48.5761	48.95	56.542	16.40	48.609	0.07	48.593	0.03	48.575	0.00
	3	95.0274	96.05	107.806	13.45	101.690	7.01	95.075	0.05	95.020	−0.01
椭圆线铰支-固支	1	11.0158	11.04	13.299	20.73	11.034	0.16	11.017	0.01	11.016	0.00
	2	38.6820	38.82	45.353	17.24	38.722	0.10	38.691	0.02	38.682	0.00
	3	81.9146	82.33	95.626	16.74	82.051	0.17	81.947	0.04	81.911	0.00

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表 7 例3前20阶振型计算总耗时

Table 7 Total computation time for first 20 modes of Example 3 /s

轴线	有限元解1 (m=1)	超收敛解1 ( $\bar m = 2$ )	有限元解2 (m=2)	超收敛解2 ( $\bar m = 3$ )
a)	12.204	0.063	30.223	0.105
b)	13.180	0.067	37.097	0.162
c)	14.292	0.084	38.639	0.146

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表 8 超收敛解的收敛阶

Table 8 Convergence order of recovered solutions

收敛阶	${\omega ^*}$	${{{d}}^{\rm{*}}}$
$r$	$\min (2\bar m,4m)$	$\min (\bar m + 1,2m)$

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