LOCKING ALLEVIATION TECHNIQUES OF TWO TYPES OF BEAM ELEMENTS BASED ON THE LOCAL FRAME FORMULATION
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摘要: 对于大转动、大变形柔性体的刚柔耦合动力学问题,基于李群SE(3)局部标架(local frame formulation, LFF)的建模方法能够规避刚体运动带来的几何非线性问题,离散数值模型中广义质量矩阵与切线刚度矩阵满足刚体变换的不变性,可明显地提高柔性多体系统动力学问题的计算效率. 有限元方法中,闭锁问题是导致单元收敛性能低下的主要原因, 例如梁单元的剪切以及泊松闭锁.多变量变分原理是缓解梁、板/壳单元闭锁的有效手段. 该方法不仅离散位移场,同时离散应力场或应变场, 可提高应力与应变的计算精度. 本文基于上述局部标架,研究几类梁单元的闭锁处理方法, 包括几何精确梁(geometrically exact beam formulation, GEBF)与绝对节点坐标(absolute nodal coordinate formulation, ANCF)梁单元. 其中, 采用Hu-Washizu三场变分原理缓解几何精确梁单元中的剪切闭锁,采用应变分解法缓解基于局部标架的ANCF全参数梁单元中的泊松闭锁. 数值算例表明,局部标架的梁单元在描述高转速或大变形柔性多体系统时,可消除刚体运动带来的几何非线性, 极大地减少系统质量矩阵和刚度矩阵的更新次数.缓解闭锁后的几类局部标架梁单元收敛性均得到了明显提升.
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关键词:
- 局部标架方法 /
- 几何精确梁 /
- 绝对节点坐标梁单元 /
- 闭锁 /
- Hu-Washizu三场变分原理
Abstract: For rigid-flexible coupling dynamic problems with large rotation and large deformation, the modeling method based on the local frame formulation (LFF) of SE(3) group can avoid geometrically nonlinear problem caused by the rigid-body motion. In discretized flexible multibody systems, the generalized mass matrix and the tangent stiffness matrix are invariant under the arbitrary rigid-body motion, which can improve computational efficiency significantly. In the finite element method, locking is the main reason for low convergence rate of elements, such as shear and Poisson locking in beam elements. Mixed methods are effective strategies to alleviate locking in beam and plate/shell elements. In these methods, not only the displacement field but also the stress field and the strain field are discretized, which can increase the accuracy of stress and strain. Based on the local frame formulation, the paper studies locking alleviation techniques of several beam elements, including geometrically exact beam formulation (GEBF) and absolute nodal coordinate formulation (ANCF) beam elements. The Hu-Washizu variational principle is used to alleviate shear locking in the geometrically exact beam, while the strain split method is used to eliminate Poisson locking in the fully parameterized ANCF beam. Numerical examples show that the proposed beam elements based on the local frame formulation can eliminate geometrically nonlinearity caused by the rigid-body motion and can minimize the updating times of mass matrices and tangent stiffness matrices when modeling flexible multibody systems with high rotational speed or large deformation. After locking alleviation, the convergence rate of the above beam elements improves significantly. -
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