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张腾, 刘铖, 张志娟, 刘绍奎. 基于SE(3)群局部标架的5/6 Dofs CB壳单元. 力学学报, 2022, 54(3): 746-745. DOI: 10.6052/0459-1879-21-584
引用本文: 张腾, 刘铖, 张志娟, 刘绍奎. 基于SE(3)群局部标架的5/6 Dofs CB壳单元. 力学学报, 2022, 54(3): 746-745. DOI: 10.6052/0459-1879-21-584
Zhang Teng, Liu Cheng, Zhang Zhijuan, Liu Shaokui. 5/6 dofs cb shell elements based on the local frame of se(3) group. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(3): 746-745. DOI: 10.6052/0459-1879-21-584
Citation: Zhang Teng, Liu Cheng, Zhang Zhijuan, Liu Shaokui. 5/6 dofs cb shell elements based on the local frame of se(3) group. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(3): 746-745. DOI: 10.6052/0459-1879-21-584

基于SE(3)群局部标架的5/6 Dofs CB壳单元

5/6 DOFS CB SHELL ELEMENTS BASED ON THE LOCAL FRAME OF SE(3) GROUP

  • 摘要: 基于李群局部标架(local frame of Lie group, LFLG)的多柔体系统动力学建模方法可自然消除刚体运动带来的几何非线性, 使系统的广义弹性力、广义惯性力及其雅可比矩阵满足刚体运动的不变性. 本文融合李群局部标架思想和基于连续体(continuum-based, CB)的壳理论, 提出基于SE(3)群局部标架的5自由度(degrees of freedom, Dofs) CB壳单元. 与SE(3)群局部标架下几何精确壳单元相比, 该单元大大简化了插值带来的复杂性, 离散应变张量自然满足客观性. 同时, 该单元在线性化过程中不存在有限元离散与变分操作先后顺序的问题, 进一步简化了广义弹性力及其雅可比矩阵的计算. 为方便处理组合体结构, 在5 Dofs CB壳单元基础上, 通过中面变形梯度张量的极分解建立壳单元中面运动与自转(drilling)自由度之间的关系, 提出基于SE(3)群局部标架的6 Dofs CB壳单元. 为提高单元收敛精度, 采用Hellinger-Reissner两场变分原理缓解单元面内闭锁, 采用假设自然应变(assumed natural strain, ANS)法缓解横向剪切闭锁. 通过5个静力学算例验证了CB壳单元的收敛精度, 通过一个动力学算例验证了CB壳单元可消除刚体运动带来的几何非线性.

     

    Abstract: The modeling method based on the local frame of Lie group (LFLG) for flexible multibody dynamics can naturally eliminate the geometric nonlinearity of the overall rigid motion, so that the generalized internal forces and inertial forces as well as their Jacobian matrices are invariant under the arbitrary rigid body motion. In this paper, a novel 5 Dofs continuum-based (CB) shell element based on the LFLG is proposed by integrating the idea of the LFLG and the CB shell theory. Compared with the geometrically exact shell element based on the LFLG, the proposed shell element greatly simplifies the complexity caused by the interpolation, and the discretized strain tensors naturally satisfy the objectivity. At the same time, the finite element discretization and variational operations are commutative, which further simplifies the computation of the generalized internal forces and their Jacobian matrices. To deal with the composite structure conveniently, the relationship between the mid-surface motion and the drilling Dofs is established by the polar decomposition of the mid-surface deformation gradient tensor on the basis of the 5 Dofs CB shell element. Then, a 6 Dofs CB shell element based on the LFLG is proposed. To improve the convergence accuracy of the proposed elements, the two-field Hellinger-Reissner variational principle and the assumed natural strain (ANS) method are used to alleviate the in-plane and transverse shear locking, respectively. Several static examples are presented to verify the convergence accuracy of the 6 Dofs CB shell element. A dynamic example is presented to demonstrate that the 6 Dofs CB shell element can eliminate the geometric nonlinearity of the overall rigid motion.

     

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