以频率误差控制为目标的自由振动问题自适应有限元分析

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振动与冲击 ›› 2023, Vol. 42 ›› Issue (4) : 106-115.

论文

孙浩涵1,2，袁驷3

作者信息 +

Adaptive finite element analysis of free vibration problems with frequency error control alone as the objective

SUN Haohan1,2，YUAN Si3

Author information +

文章历史 +

摘要

对于自由振动问题，基于单元能量投影（element energy projection ,EEP）技术，对频率和模态同时进行误差控制的自适应有限元分析已建立，并被证明可靠且高效。在实际应用中，也存在另一类需求，即只需保证频率的精度，而并不关心模态误差大小。本文提出了频率超收敛计算方案，继而建立了整体频率误差和局部模态误差的转换关系，从而在整体上以频率误差估计控制算法停机，在局部上以模态误差估计驱动网格更新，最终建立了以频率误差控制为目标的自由振动问题自适应有限元分析策略。本方法的有效性在二阶Sturm-Liouville问题及弹性薄膜自由振动问题上得到了应用验证。

Abstract

Adaptive finite element analysis of free vibration problems based on element energy projection (EEP) method has been proved to be reliable and efficient, where the errors of frequencies and modes are simultaneously controlled. In practical applications, the error evaluation of frequencies can sometimes be more required than the error distribution of modes. The paper proposes an approach to calculate the super-convergent frequencies, and then sets up the transformation relation between the global frequency error and the local mode error, so that the former is used to control the termination of the adaptive algorithm on the whole and the latter is used to drive mesh refinement locally. As a result, an adaptive finite element analysis of free vibration problem with frequency error control alone as the objective is established. The method is verified by its application to second order Sturm-Liouville problem and free vibration problem of elastic membrane.

导出引用

孙浩涵1,2，袁驷3. 以频率误差控制为目标的自由振动问题自适应有限元分析[J]. 振动与冲击, 2023, 42(4): 106-115

SUN Haohan1,2，YUAN Si3. Adaptive finite element analysis of free vibration problems with frequency error control alone as the objective[J]. Journal of Vibration and Shock, 2023, 42(4): 106-115

参考文献

[1] 周文鑫, 周叮, 张建东, 等. 多跨高墩变截面梁桥的动力学特性研究[J]. 振动与冲击, 2021, 40(16): 111-117, 118.
ZHOU Wenxin，ZHOU Ding，ZHANG Jiandong，et al. Dynamic characteristics of a multi-span high-pier bridge with variable cross-sections[J]. Journal of Vibration and Shock, 2021, 40(16): 111-117.
[2] 杨翼, 王旭荣, 王明坤, 等. 船舶推进轴系的扭转-纵向冲击响应[J]. 振动与冲击, 2017, 36(13): 96-102.
YANG Yi, WANG Xurong, WANG Mingkun, et al. Torsional-Longitudinal Shock Response of Ship Propulsion Shaft[J]. Journal of Vibration and Shock, 2017, 36(13): 96-102.
[3] Babuška I, Rheinboldt W C. A posteriori error estimates for the finite element method[J]. International Journal for Numerical Methods in Engineering, 1978, 12(10): 1597-1615.
[4] Babuška I, Rheinboldt W C. Adaptive approaches and reliability estimations in finite element analysis [J]. Computer Methods in Applied Mechanics and Engineering, 1979, 17: 519-540.
[5] Strang G, Fix G J. An analysis of the finite element method[M]. Englewood Cliffs, NJ: Prentice-hall, 1973.
[6] Babuška I, Osborn J. Eigenvalue problems[M], Handbook of Numerical Analysis, 1991: 641-787.
[7] Knyazev A V, Osborn J E. New a priori FEM error estimates for eigenvalues[J]. SIAM Journal on Numerical Analysis, 2006, 43(6): 2647-2667.
[8] Heuveline V, Rannacher R. A posteriori error control for finite element approximations of elliptic eigenvalue problems[J]. Advances in Computational Mathematics, 2001, 15(1-4): 107-138.
[9] Ladevèze P, Pelle J P. Estimation of discretization errors in dynamics[J]. Computers & structures, 2003, 81(12): 1133-1148.
[10] Friberg O, Moller P, Makovička D, et al. An adaptive procedure for eigenvalue problems using the hierarchical finite element method[J]. International journal for numerical methods in engineering, 1987, 24(2): 319-335.
[11] Wiberg N E, Bausys R, Hager P. Improved eigenfrequencies and eigenmodes in free vibration analysis[J]. Computers & structures, 1999, 73(1-5): 79-89.
[12] Wiberg N E, Bausys R, Hager P. Adaptive h-version eigenfrequency analysis[J]. Computers & structures, 1999, 71(5): 565-584.
[13] 王永亮. 中厚圆柱壳振型的有限元超收敛拼片恢复解和网格自适应分析[J]. 振动与冲击, 2021, 40(18): 112-118, 147.
WANG Yongliang. Superconvergent patch recovery solutions and adaptive mesh refinement analysis of finite element method for vibration modes of moderately thick circular cylindrical shells[J]. Journal of Vibration and Shock, 2021, 40(18): 112-118.
[14] Zienkiewicz O C, Zhu J Z. The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique[J]. International Journal for Numerical Methods in Engineering, 1992, 33(7): 1331-1364.
[15] 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.
[16] 袁驷, 王枚. 一维有限元后处理超收敛解答计算的EEP法[J]. 工程力学, 2004, 21(2): 1―9.
YUAN Si,WANG Mei. An element-energy-projection method for post-computation of super-convergent solutions in one-dimensional fem[J]. Engineering Mechanics, 2004, 21(2): 1―9.
[17] Sun H, Yuan S. Adaptive finite element analysis of free vibration of elastic membranes via element energy projection technique[J], Engineering Computations, 2021, ahead-of-print(ahead-of-print).
[18] Yuan S, Ye K, Wang Y, Kennedy D, Williams W F. Adaptive finite element method for eigensolutions of regular second and fourth order Sturm-Liouville problems via the element energy projection technique, Engineering Computations, 2017, 34(8): 2862-2876.
[19] Yuan S, Sun H. A general adaptive finite element eigen-algorithm stemming from Wittrick-Williams algorithm[J]. Thin-Walled Structures, 2021, 161(107448).
[20] Yuan, S., Wu, Y., Xing, Q. Recursive super-convergence computation for multi-dimensional problems via one-dimensional element energy projection technique[J]. Applied Mathematics and Mechanics. 2018, 39(7): 1031–1044.
[21] Jiang, K., Yuan, S., Xing, Q. An adaptive nonlinear finite element analysis of minimal surface problem based on element energy projection technique[J]. Engineering Computations. 2020, ahead-of-print.
[22] Dong Y, Yuan S, Xing Q. 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.
[23] Greaves D, Borthwick A. Hierarchical tree-based finite element mesh generation[J]. International Journal for Numerical Methods in Engineering. 1999, 45(4): 447–471.