无额外自由度广义有限元的近不可压弹-塑性分析

马今伟; 段庆林

doi:10.21656/1000-0887.440067

无额外自由度广义有限元的近不可压弹-塑性分析

doi: 10.21656/1000-0887.440067

马今伟^{1, 2,},
段庆林^{1, 2, ,}

1.
大连理工大学工业装备结构分析优化与CAE软件全国重点实验室，辽宁大连 116024
2.
大连理工大学大连理工大学白俄罗斯国立大学联合学院，辽宁大连 116024

基金项目:

中央高校基本科研业务费 DUT21GF304

科学挑战专题 TZ2018002

详细信息

作者简介:
马今伟(1992—)，男，博士生(E-mail: majinwei_1234@163.com)

通讯作者:
段庆林(1979—), 男，副教授，博士，博士生导师(通讯作者. E-mail: qinglingduan@dlut.edu.cn)

中图分类号: O302
计量
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- 被引次数: 0
出版历程
- 收稿日期: 2023-03-15
- 修回日期: 2023-06-13
- 刊出日期: 2024-02-01

Nearly Incompressible Elasto-Plastic Analysis of Extra-DOF-Free Generalized Finite Elements

MA Jinwei^{1, 2
,},
DUAN Qinglin^{1, 2
, ,}

1.
State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, Dalian University of Technology, Dalian, Liaoning 116024, P.R.China
2.
DUT-BSU Joint Institute, Dalian University of Technology, Dalian, Liaoning 116024, P.R.China

摘要

摘要: 研究了常规有限元方法在近不可压弹-塑性分析中的体积自锁问题，并在广义有限元框架下引入无额外自由度的强化函数对此问题进行了改进.一方面，插值函数在引入强化函数后获得了更加丰富的近似空间，提高了在体积近似不变约束下正确反映结构变形的能力；另一方面，强化函数的建立不依赖额外自由度，从而消除了传统广义有限元方法中的线性相关性问题.分析并验证了常规有限元在线弹性、超弹性和塑性分析中的体积自锁问题具有不同的触发条件和表现形式.3个典型的数值算例表明，无额外自由广义有限元能有效地缓解体积自锁并得到准确合理的计算结果.
- 广义有限元 /
- 额外自由度 /
- 近不可压 /
- 体积自锁 /
- 弹塑性 /
- 超弹性
Abstract: The volumetric locking of the conventional finite element method (FEM) was investigated through the nearly incompressible elasto-plastic analysis, and the extra-DOF-free reinforcement function was introduced in the framework of the generalized finite element method (GFEM) for improvement of this problem. Through the introduction of the reinforcement function, a richer approximation space was obtained for the interpolation function, of which the ability of correctly reflecting structural deformations was promoted under the approximate volumetric invariance constraint. Furthermore, the construction of the reinforcement function is independent of the extra degrees of freedom, which eliminates the linear dependency in the traditional GFEM. Different triggering conditions and manifestations of volumetric locking of the conventional FEM were found in linear elastic, hyperelastic, and plastic analyses. Three classical numerical examples show that, the extra-DOF-free GFEM can effectively alleviate the volumetric locking in all analyses, and obtain accurate and reasonable results.
- generalized finite element /
- extra degree of freedom /
- nearly incompressible /
- volumetric locking /
- elastoplasticity /
- hyperelasticity

HTML全文

图 1 节点patch示意图

Figure 1. Schematic diagram of patches

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图 2 Cook膜算例中GFEM采用3种基向量计算的位移场和变形

Figure 2. Displacement fields and deformations obtained with the GFEM for constant, quadratic and cubic bases in the Cook membrane exmaple

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图 3 Cook膜算例中GFEM采用3种基向量在不同密度网格下收敛性测试结果

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. Convergence results of the GFEM for constant, quadratic and cubic bases in the Cook membrane example

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图 4 FEM在纯弯曲橡胶梁中的变形

Figure 4. Deformations obtained with the FEM in the pure bending rubber beam example

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图 5 GFEM在纯弯曲橡胶梁中的变形

Figure 5. Deformations obtained with the GFEM in the pure bending rubber beam example

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图 6 圆杆颈缩算例

Figure 6. Displacement fields and deformations obtained in the necking bar example

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图 7 圆杆颈缩算例中FEM和GFEM数值结果与试验结果^[15]的对比

Figure 7. Comparison between experimental data^[15] and numerical results in the necking circular bar example

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参考文献(17)

[1]	李嘉钰, 陈梦成, 王开心. 基于剪切效应纤维梁单元的结构非线性有限元数值模拟[J]. 应用数学和力学, 2022, 43(1): 34-48. doi: 10.21656/1000-0887.420032 LI Jiayu, CHEN Mengcheng, WANG Kaixin. Nonlinear numerical simulation of finite elements based on fiber beam elements with shear effects for structures[J]. Applied Mathematics and Mechanics, 2022, 43(1): 34-48. (in Chinese) doi: 10.21656/1000-0887.420032
[2]	SUSSMAN T, BATHE K J. A finite element formulation for nonlinear incompressible elastic and inelastic analysis[J]. Computers & Structures, 1987, 26(1/2): 357-409.
[3]	HUGHES T J R. Generalization of selective integration procedures to anisotropic and nonlinear media[J]. International Journal for Numerical Methods in Engineering, 1980, 15(9): 1413-1418. doi: 10.1002/nme.1620150914
[4]	NAGTEGAAL J C, PARKS D M, RICE J R. On numerically accurate finite element solutions in the fully plastic range[J]. Computer Methods in Applied Mechanics and Engineering, 1974, 4(2): 153-177. doi: 10.1016/0045-7825(74)90032-2
[5]	SHAUER N, DUARTE C A. Improved algorithms for generalized finite element simulations of three-dimensional hydraulic fracture propagation[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 2019, 43(18): 2707-2742. doi: 10.1002/nag.2977
[6]	毛晓敏, 张慧化, 纪晓磊, 等. 基于XFEM和GA-BP神经网络的裂纹智能识别研究[J]. 应用数学和力学, 2022, 43(11): 1268-1280. doi: 10.21656/1000-0887.420250 MAO Xiaomin, ZHANG Huihua, JI Xiaolei, et al. Intelligent crack recognition based on XFEM and GA-BP neural networks[J]. Applied Mathematics and Mechanics, 2022, 43(11): 1268-1280. (in Chinese) doi: 10.21656/1000-0887.420250
[7]	LI H, DUARTE C A. A two-scale generalized finite element method for parallel simulations of spot welds in large structures[J]. Computer Methods in Applied Mechanics and Engineering, 2018, 337: 28-65. doi: 10.1016/j.cma.2018.03.030
[8]	BABUŠKA I, BANERJEE U. Stable generalized finite element method (SGFEM)[J]. Computer Methods in Applied Mechanics and Engineering, 2012, 201/204: 91-111.
[9]	SILLEM A, SIMONE A, SLUYS L J. The orthonormalized generalized finite element method-OGFEM: efficient and stable reduction of approximation errors through multiple orthonormalized enriched basis functions[J]. Computer Methods in Applied Mechanics and Engineering, 2015, 287: 112-149. doi: 10.1016/j.cma.2014.11.043
[10]	TIAN R. Extra-DOF-free and linearly independent enrichments in GFEM[J]. Computer Methods in Applied Mechanics and Engineering, 2013, 266: 1-22. doi: 10.1016/j.cma.2013.07.005
[11]	XIAO G Z, WEN L F, TIAN R, et al. Improved XFEM (IXFEM): 3D dynamic crack propagation under impact loading[J]. Computer Methods in Applied Mechanics and Engineering, 2023, 405: 115844. doi: 10.1016/j.cma.2022.115844
[12]	MA J W, DUAN Q L, TIAN R. A generalized finite element method without extra degrees of freedom for large deformation analysis of three-dimensional elastic and elastoplastic solids[J]. Computer Methods in Applied Mechanics and Engineering, 2022, 392: 114639. doi: 10.1016/j.cma.2022.114639
[13]	马今伟, 段庆林, 陈嵩涛. 无额外自由度广义有限元非线性分析[J]. 计算力学学报, 2021, 38(1): 60-65. https://www.cnki.com.cn/Article/CJFDTOTAL-JSJG202101009.htm MA Jinwei, DUAN Qinglin, CHEN Songtao. Extra-DOF-free generalized finite element method for non-linear analysis[J]. Chinese Journal of Computational Mechanics, 2021, 38(1): 60-65. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-JSJG202101009.htm
[14]	ELGUEDJ T, BAZILEVS Y, CALO V M, et al. B-bar and F-bar projection methods for nearly incompressible linear and non-linear elasticity and plasticity using higher-order NURBS elements[J]. Computer Methods in Applied Mechanics and Engineering, 2008, 197(33/40): 2732-2762.
[15]	CHEN J S, PAN C, WU C T, et al. Reproducing kernel particle methods for large deformation analysis of non-linear structures[J]. Computer Methods in Applied Mechanics and Engineering, 1996, 139: 195-227. doi: 10.1016/S0045-7825(96)01083-3
[16]	BELYTSCHKO T, LIU W K, MORAN B, et al. Nonlinear Finite Elements for Continua and Structures[M]. 2nd ed. UK: John Wiley & Sons, 2014.
[17]	SIMO J C, HUGHES T J R. Computational Inelasticity[M]. New York: Springer-Verlag, 1998.