一维有限元的EEP单元及其自适应分析

杨帅; 袁驷

doi:10.21656/1000-0887.450036

一维有限元的EEP单元及其自适应分析

doi: 10.21656/1000-0887.450036

杨帅^,,
袁驷

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 51878383

国家自然科学基金 51378293

详细信息

作者简介:
袁驷(1953—)，男，教授，博士

通讯作者:
杨帅(1997—)，男，博士生(通讯作者. E-mail: s-yang20@mails.tsinghua.edu.cn)

中图分类号: O342
计量
- 文章访问数: 27
- HTML全文浏览量: 15
- PDF下载量: 6
- 被引次数: 0
出版历程
- 收稿日期: 2024-02-18
- 修回日期: 2024-05-08
- 刊出日期: 2025-01-01

EEP Elements for the 1 D Finite Element Method and the Adaptivity Analysis

YANG Shuai^,,
YUAN Si

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

摘要

摘要: 对m(>1)次单元，基于单元能量投影(element energy projection，简称EEP)法提出的简约格式位移解u^*具有比常规有限元解u^h至少高一阶的精度，据此提出了EEP单元概念，并给出以EEP单元作为最终解的自适应有限元求解策略. 通过编制相应的计算程序分析了一维非自伴随问题，计算结果与理论预期吻合较好，验证了自适应求解策略的有效性和可靠性. 研究结果表明：该法可以给出按最大模度量、逐点满足误差限的解答，相较于常规单元，最终的求解单元数更少.
- 一维有限元 /
- 单元能量投影(EEP) /
- EEP单元 /
- 自适应有限元法
Abstract: For the elements of degree m(>1), simplified form solution u^* based on the element energy projection (EEP) method has at least 1-order higher accuracy than conventional finite element solution u^h. As a result, the EEP element, with simplified form EEP solution u^* in as the final solution, was proposed, and a corresponding adaptive finite element analysis strategy for EEP elements was developed. By means of the developed algorithm, the 1D 2-point boundary value problem was analyzed, and the computation results are in good agreement with theoretical solutions, verifying the effectiveness and reliability of the proposed adaptivity strategy. The theoretical study and numerical experiments show that, the proposed method provides an EEP element solution satisfying the preset error tolerances in the maximum norm with fewer elements and less adaptive steps compared to conventional finite elements.
- 1D problem /
- element energy projection (EEP) /
- EEP element /
- adaptive finite element method

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图 1 误差分布

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 1. The error distribution

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图 2 问题的解析解

Figure 2. The analytical solution to the problem

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图 3 单元分布示意图

Figure 3. The element distribution diagram

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图 4 三次单元自适应误差图

Figure 4. Error distributions of cubic elements

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图 5 3.1.2小节问题的解析解

Figure 5. Analytical solutions to the problem of section 3.1.2

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图 6 案例1的单元分布示意图

Figure 6. Element distributions of case 1

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图 7 案例1的三次单元自适应误差图

Figure 7. Error distributions of cubic elements of case 1

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图 8 案例2的三次EEP单元分布

Figure 8. The cubic EEP element distribution of case 2

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图 9 案例2的三次EEP单元误差图

Figure 9. The error distribution of cubic EEP elements of case 2

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图 10 3.1.3小节问题的解析解

Figure 10. The analytical solution to the problem of section 3.1.3

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图 11 3.1.3小节单元分布示意图

Figure 11. The element distribution diagram of section 3.1.3

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图 12 3.1.3小节三次单元自适应误差图

Figure 12. Error distributions of cubic elements of section 3.1.3

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图 13 自适应收敛率

Figure 13. Adaptive convergence rates

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表 1 常规单元自适应求解策略结果(T_l=1×10^－8)

Table 1. Results of the conventional element adaptive strategy (T_l=1×10^－8)

m	N_e	N_dof	N_adp	h_max	h_min	e_max^h
3	26	79	6	0.125 0	0.015 6	0.804 0
4	11	45	5	0.250 0	0.031 3	0.862 9
5	7	36	4	0.250 0	0.062 5	0.562 6

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表 2 EEP单元自适应求解策略结果(T_l=1×10^－8)

Table 2. Results of the EEP element adaptive strategy (T_l=1×10^－8)

m	N_e	N_dof	N_adp	h_max	h_min	e_max^h
3	15	46	5	0.125 0	0.031 3	0.796 7
4	9	37	4	0.250 0	0.062 5	0.458 0
5	6	31	3	0.250 0	0.125 0	0.228 7

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表 3 常规单元和EEP单元结果对比(T_l=1×10^－10)

Table 3. Comparison of results between conventional elements and EEP elements(T_l=1×10^－10)

element type	convergence order	N_e	N_dof	N_adp	e_max^h(e_max^*)
quartic conventional element^[11]	h⁵	30	121	6	0.870 0
cubic EEP element	h⁵	36	109	6	0.932 0

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表 4 常规单元自适应求解策略结果(案例1)

Table 4. Results of the conventional element adaptive strategy (case 1)

m	N_e	N_dof	N_adp	h_max	h_min	e_max^h
3	48	145	7	0.061 3	0.009 4	0.829 6
4	19	77	6	0.148 5	0.020 4	0.810 9
5	9	46	5	0.240 0	0.040 5	0.733 2

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表 5 EEP单元自适应求解策略结果(案例1)

Table 5. Results of the EEP element adaptive strategy (case 1)

m	N_e	N_dof	N_adp	h_max	h_min	e_max^h
3	27	82	7	0.091 0	0.018 8	0.971 3
4	15	61	5	0.113 8	0.047 5	0.430 5
5	9	46	4	0.250 0	0.067 5	0.232 0

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表 6 常规单元和EEP单元结果对比(T_l=1×10^－9)

Table 6. Comparison of results between conventional elements and EEP elements(T_l=1×10^－9)

element type	convergence order	N_e	N_dof	N_adp	e_max^h(e_max^*)
quartic conventional element^[11]	h⁵	45	181	7	0.838 8
cubic EEP element	h⁵	66	199	8	0.891 9

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表 7 常规单元自适应求解策略结果(案例2)

Table 7. Results of the conventional element adaptive strategy (case 2)

m	N_e	N_dof	N_adp	h_max	h_min	e_max^h
3	125	376	11	0.030 9	5.9×10^-4	1.037 8
4	50	201	10	0.061 9	1.3×10^-3	0.845 2
5	31	156	9	0.091 1	2.6×10^-3	0.758 3

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表 8 EEP单元自适应求解策略结果(案例2)

Table 8. Results of the EEP element adaptive strategy (case 2)

m	N_e	N_dof	N_adp	h_max	h_min	e_max^h
3	81	244	11	0.036 4	1.4×10^-3	1.019 3
4	42	169	10	0.064 4	2.2×10^-3	0.723 5
5	30	151	8	0.072 6	7.5×10^-3	0.673 5

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表 9 常规单元和EEP单元结果对比(T_l=1×10^－10)

Table 9. Comparison of results between conventional elements and EEP elements(T_l=1×10^－10)

element type	convergence order	N_e	N_dof	N_adp	e_max^h(e_max^*)
quartic conventional element^[10]	h⁵	121	485	11	0.948 2
cubic EEP element	h⁵	200	601	13	0.959 3

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表 10 常规单元自适应求解策略结果(T_l=1×10^－3)

Table 10. Results of the conventional element adaptive strategy (T_l=1×10^－3)

m	N_e	N_dof	N_adp	h_max	h_min	e_max^h
3	10	31	9	0.650 0	7.9×10^-5	0.740 0
4	9	37	8	0.650 0	2.3×10^-4	0.706 8
5	8	41	7	0.650 0	6.4×10^-4	0.894 2

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表 11 EEP单元自适应求解策略结果(T_l=1×10^－3)

Table 11. Results of the EEP element adaptive strategy (T_l=1×10^－3)

m	N_e	N_dof	N_adp	h_max	h_min	e_max^h
3	3	10	2	0.660 0	0.129 2	0.456 3
4	3	13	2	0.630 0	0.144 3	0.252 8
5	2	11	1	0.620 0	0.380 0	0.649 1

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表 12 常规单元和EEP单元结果对比(T_l=1×10^－4)

Table 12. Comparison of results between conventional elements and EEP elements(T_l=1×10^－4)

element type	convergence order	N_e	N_dof	N_adp	e_max^h(e_max^*)
quartic conventional element^[11]	h⁵	13	53	12	0.866 5
cubic EEP element	h⁵	5	16	4	0.459 2

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