降阶单元的新进展：内置了最大模误差估计器的自适应有限元法初探

袁驷; 杨帅; 袁全; 王亦平

doi:10.6052/j.issn.1000-4750.2023.07.ST04

降阶单元的新进展：内置了最大模误差估计器的自适应有限元法初探

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(51878383，51378293)

详细信息

作者简介:
杨　帅(1997−)，男，河北人，博士生，主要从事结构工程研究(E-mail: s-yang20@mails.tsinghua.edu.cn)

袁　全(1993−)，男，北京人，博士，主要从事结构工程研究(E-mail: quany@tsinghua.edu.cn)

王亦平(2000−)，男，湖南人，博士生，主要从事结构工程研究(E-mail: wangyipi22@mails.tsinghua.edu.cn)

通讯作者:
袁　驷(1953−)，男，北京人，教授，博士，主要从事结构工程研究(E-mail: yuans@tsinghua.edu.cn)

中图分类号: O342
计量
- 文章访问数: 319
- HTML全文浏览量: 54
- PDF下载量: 107
出版历程
- 收稿日期: 2023-07-13
- 修回日期: 2023-07-24
- 网络出版日期: 2023-12-26
- 刊出日期: 2024-03-24

NEW PROGRESS IN REDUCED ELEMENT: AN ADAPTIVE FINITE ELEMENT METHOD WITH BUILT-IN ERROR ESTIMATOR IN MAXIMUM NORM

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:
该文基于初值问题中降阶单元的成功实践，进一步对一般边值问题提出无需超收敛计算、无需结构化网格、无需结点位移修正的降阶单元；进而提出以降阶单元作为最终解，且内置了最大模误差估计器的自适应有限元算法。该文对这一研究进展做简要介绍，并给出一维和二维边值问题的初步算例，以展示本法的可行性、有效性和可靠性。
- 有限元法 /
- 边值问题 /
- 降阶单元 /
- 最大模 /
- 自适应有限元法
Abstract:
Based on the successful performance of the reduced element for initial value problems, the reduced element for general boundary value problems is proposed without the needs for super-convergence calculations, structural meshes and nodal accuracy improvement. An adaptive finite element algorithm with the solution of the reduced element and a built-in maximum norm error estimator as the final results is subsequently proposed. This paper gives a brief report to this research progress and provides preliminary numerical examples in one- and two-dimensional boundary value problems to exhibit the feasibility, effectiveness and reliability of the proposed algorithm.
- finite element method /
- boundary value problem /
- reduced element /
- maximum norm /
- adaptive finite element method

HTML全文

图 1 一维降阶单元示意图

Figure 1. Schematic diagram of one-dimensional reduced element

下载: 全尺寸图片幻灯片

图 2 二维降阶单元示意图

Figure 2. Schematic diagram of two-dimensional reduced element

下载: 全尺寸图片幻灯片

图 3 例1的位移精确解

Figure 3. Exact displacement solution for example 1

下载: 全尺寸图片幻灯片

降阶法误差比( ${N_e} = 16,{N_{{\text{adp}}}} = 0$ )

Error ratio of reduced order method ( ${N_e} = 16,{N_{{\text{adp}}}} = 0$ )

下载: 全尺寸图片幻灯片

双阶法误差比( ${N_e} = 16,{N_{{\text{adp}}}} = 0$ )

Error ratio of double order method ( ${N_e} = 16,{N_{{\text{adp}}}} = 0$ )

下载: 全尺寸图片幻灯片

降阶法误差比( ${N_e} = 32,{N_{{\text{adp}}}} = 1$ )

Error ratio of reduced order method ( ${N_e} = 32,{N_{{\text{adp}}}} = 1$ )

下载: 全尺寸图片幻灯片

双阶法误差比( ${N_e} = 32,{N_{{\text{adp}}}} = 1$ )

Error ratio of double order method ( ${N_e} = 32,{N_{{\text{adp}}}} = 1$ )

下载: 全尺寸图片幻灯片

降阶法误差比( ${N_e} = 123,{N_{{\text{adp}}}} = 5$ )

Error ratio of reduced order method ( ${N_e} = 123,{N_{{\text{adp}}}} = 5$ )

下载: 全尺寸图片幻灯片

双阶法误差比( ${N_e} = 349,{N_{{\text{adp}}}} = 8$ )

Error ratio of double order method ( ${N_e} = 349,{N_{{\text{adp}}}} = 8$ )

下载: 全尺寸图片幻灯片

图 10 四边形区域示意图及初始网格

Figure 10. Schematic diagram of quadrilateral area and initial mesh

下载: 全尺寸图片幻灯片

图 11 例2最终网格划分

Figure 11. Final mesh of example 2

下载: 全尺寸图片幻灯片

例2初始网格域内误差比 $\overline e_{\text{R}}^h$ 的分布

Error ratio $\overline e_{\text{R}}^h$ on the initial mesh of example 2

下载: 全尺寸图片幻灯片

例2最终网格域内误差比 $\overline e_{\text{R}}^h$ 的分布

Error ratio $\overline e_{\text{R}}^h$ on the final mesh of example 2

下载: 全尺寸图片幻灯片

图 14 例3最终网格划分

Figure 14. Final mesh of example 3

下载: 全尺寸图片幻灯片

例3域内误差比 $\overline e_{\text{R}}^h$ 的分布

Error ratio $\overline e_{\text{R}}^h$ of example 3

下载: 全尺寸图片幻灯片

图 16 Cook梁示意图及初始网格

Figure 16. Schematic diagram of Cook beam and initial mesh

下载: 全尺寸图片幻灯片

图 17 例4最终网格划分

Figure 17. Final mesh of example 4

下载: 全尺寸图片幻灯片

例4域内误差比 $\overline e_{\text{R}}^h(u)$ 的分布

Error ratio $\overline e_{\text{R}}^h(u)$ of example 4

下载: 全尺寸图片幻灯片

例4域内误差比 $\overline e_{\text{R}}^h(v)$ 的分布

Error ratio $\overline e_{\text{R}}^h(v)$ of example 4

下载: 全尺寸图片幻灯片

表 1 一维常规单元和降阶单元位移解的收敛阶

Table 1 Convergence orders of displacement solutions for one-dimensional conventional and reduced elements

单元类型内部收敛阶端点收敛阶

常规线性元 ${h^2}$ ${h^2}$
常规二次元 ${h^3}$ ${h^4}$
降阶线性元 ${h^2}$ ${h^4}$

下载: 导出CSV

参考文献(20)

[1]	BABUŠKA I, RHEINBOLDT W C. Adaptive approaches and reliability estimations in finite element analysis [J]. Computer Methods in Applied Mechanics and Engineering, 1979, 17/18: 519 − 540. doi: 10.1016/0045-7825(79)90042-2
[2]	ZIENKIEWICZ O C, ZHU J Z. A simple error estimator and adaptive procedure for practical engineering analysis [J]. International Journal for Numerical Methods in Engineering, 1987, 24(2): 337 − 357. doi: 10.1002/nme.1620240206
[3]	YUAN S, HE X F. Self-adaptive strategy for one-dimensional finite element method based on element energy projection method [J]. Applied Mathematics and Mechanics, 2006, 27(11): 1461 − 1474. doi: 10.1007/s10483-006-1103-1
[4]	YUAN S, DU Y, XING Q Y, et al. Self-adaptive one-dimensional nonlinear finite element method based on element energy projection method [J]. Applied Mathematics and Mechanics, 2014, 35(10): 1223 − 1232. doi: 10.1007/s10483-014-1869-9
[5]	DONG Y Y, YUAN S, XING Q Y. Adaptive finite element analysis with local mesh refinement based on a posteriori error estimate of element energy projection technique [J]. Engineering Computations, 2019, 36(6): 2010 − 2033. doi: 10.1108/EC-11-2018-0523
[6]	JIANG K F, YUAN S, XING Q Y. An adaptive nonlinear finite element analysis of minimal surface problem based on element energy projection technique [J]. Engineering Computations, 2020, 37(8): 2847 − 2869. doi: 10.1108/EC-08-2019-0369
[7]	YUAN S, SUN H H. A general adaptive finite element eigen-algorithm stemming from Wittrick-Williams algorithm [J]. Thin-Walled Structures, 2021, 161: 107448. doi: 10.1016/j.tws.2021.107448
[8]	SUN H H, YUAN S. Adaptive finite element analysis of free vibration of elastic membranes via element energy projection technique [J]. Engineering Computations, 2021, 38(8): 3492 − 3516.
[9]	HUANG Z M, YUAN S, XING Q Y. Nodal accuracy improvement technique for linear elements with application to adaptivity [J]. Applied Sciences, 2023, 13(5): 2844. doi: 10.3390/app13052844
[10]	袁驷, 刘泽洲, 邢沁妍. 二维变分不等式问题的自适应有限元分析[J]. 工程力学, 2016, 33(1): 11 − 17. doi: 10.6052/j.issn.1000-4750.2015.06.ST01 YUAN Si, LIU Zezhou, XING Qinyan. Self-adaptive FEM analysis for 2D variational inequality problems [J]. Engineering Mechanics, 2016, 33(1): 11 − 17. (in Chinese) doi: 10.6052/j.issn.1000-4750.2015.06.ST01
[11]	STRANG G, FIX G J. An analysis of the finite element method [M]. Englewood Clifs: Prentice-Hall, 1973: 39 − 50.
[12]	DOUGLAS J JR. Galerkin approximations for the two point boundary problem using continuous, piecewise polynomial spaces [J]. Numerische Mathematik, 1974, 22(2): 99 − 109. doi: 10.1007/BF01436724
[13]	HULME B L. One-step piecewise polynomial Galerkin methods for initial value problems [J]. Mathematics of Computation, 1972, 26(118): 415 − 426. doi: 10.1090/S0025-5718-1972-0321301-2
[14]	陈传淼, 黄云清. 有限元高精度理论 [M]. 长沙: 湖南科学技术出版社, 1995: 235 − 309. CHEN Chuanmiao, HUANG Yunqing. High accuracy theory of finite element methods [M]. Changsha: Hunan Science & Technology Press, 1995: 235 − 309. (in Chinese)
[15]	袁驷, 袁全. 自适应步长时程分析的新进展: 从凝聚单元到降阶单元[J]. 工程力学, 2023, 40(1): 32 − 39. doi: 10.6052/j.issn.1000-4750.2022.05.ST01 YUAN Si, YUAN Quan. New progress in adaptive time-stepping for time integration analysis: From condensed element to reduced element [J]. Engineering Mechanics, 2023, 40(1): 32 − 39. (in Chinese) doi: 10.6052/j.issn.1000-4750.2022.05.ST01
[16]	YUAN Si, YUAN Quan. Condensed Galerkin element of degree m for first-order initial-value problem with O( h^{2 m +2}) super-convergent nodal solutions [J]. Applied Mathematics and Mechanics, 2022, 43(4): 603 − 614. doi: 10.1007/s10483-022-2831-6
[17]	袁驷, 袁全. 运动方程自适应步长求解的高性能Galerkin时程单元初探[J]. 工程力学, 2022, 39(1): 21 − 26. YUAN Si, YUAN Quan. An initial study of a novel high-performance Galerkin finite element and applying for the adaptive time-step method in solving motion equation [J]. Engineering Mechanics, 2022, 39(1): 21 − 26. (in Chinese)
[18]	ZIENKIEWICZ O C, ZHU J Z. The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity [J]. International Journal for Numerical Methods in Engineering, 1992, 33(7): 1365 − 1382. doi: 10.1002/nme.1620330703
[19]	COOK R D. A plane hybrid element with rotational d. o. f. and adjustable stiffness [J]. International Journal for Numerical Methods in Engineering, 1987, 24(8): 1499 − 1508. doi: 10.1002/nme.1620240807
[20]	YUNUS S M, SAIGAL S, COOK R D. On improved hybrid finite elements with rotational degrees of freedom [J]. International Journal for Numerical Methods in Engineering, 1989, 28(4): 785 − 800. doi: 10.1002/nme.1620280405

施引文献(11)

期刊类型引用(5)

1.	袁驷，袁全. 时空降阶单元及其自适应分析初探. 工程力学. 2025(01): 10-19 . 本站查看
2.	杨帅，袁驷. 几种自适应有限元方法的对比分析. 工程力学. 2025(S1): 1-8 . 本站查看
3.	袁全，袁驷. Timoshenko梁厚薄通用降阶单元的构造及其自适应求解. 工程力学. 2025(S1): 23-29 . 本站查看
4.	袁驷，袁全，杨帅，王亦平，刘海阳. 内置了最大模误差估计器的自适应有限元法——研究进展与展望. 土木工程学报. 2024(06): 43-58 . 百度学术
5.	傅向荣，王钰，赵阳，陈璞，孙树立. 超协调元. 工程力学. 2024(S1): 7-14 . 本站查看

其他类型引用(6)

资源附件(0)

在线预览下载 (5.6MB)

图(19) / 表(1)

计量

文章访问数: 319
HTML全文浏览量: 54
PDF下载量: 107
被引次数: 11

WANG Yi-ping, wangyipi22@mails.tsinghua.edu.cn

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

单元类型	内部收敛阶	端点收敛阶
常规线性元	${h^2}$	${h^2}$
常规二次元	${h^3}$	${h^4}$
降阶线性元	${h^2}$	${h^4}$

降阶单元的新进展：内置了最大模误差估计器的自适应有限元法初探

通讯作者: 袁 驷(1953−)，男，北京人，教授，博士，主要从事结构工程研究(E-mail: yuans@tsinghua.edu.cn)

计量

出版历程

NEW PROGRESS IN REDUCED ELEMENT: AN ADAPTIVE FINITE ELEMENT METHOD WITH BUILT-IN ERROR ESTIMATOR IN MAXIMUM NORM

期刊类型引用(5)

其他类型引用(6)

计量

出版历程

目录

WANG Yi-ping, wangyipi22@mails.tsinghua.edu.cn

通讯作者:
袁　驷(1953−)，男，北京人，教授，博士，主要从事结构工程研究(E-mail: yuans@tsinghua.edu.cn)