SHAPE RECONSTRUCTION OF PLANE BEAM WITH FINITE DEFORMATION BASED ON ABSOLUTE NODAL COORDINATE FORMULATION
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摘要: 现有的对有限变形条件下柔性结构变形重构的研究往往单纯基于曲率与应变间的几何关系, 同时忽略了被测体的纵向变形及其与弯曲变形的耦合效应. 为得到一种更加精确且能借助现有的力学工具进行应用方向扩展的变形重构方法, 以平面梁为对象, 借鉴变形重构逆有限元法的思想, 将平面梁的变形重构问题视作一类最优化问题. 首先, 通过引入绝对节点坐标法(absolute nodal coordinate formulation, ANCF)对柔性结构大变形下非线性的平面梁应变−位移关系进行精确描述, 构造了一种逆梯度缩减ANCF平面索梁单元. 然后, 对此逆ANCF单元进行改进, 在简化节点自由度的同时通过引入罚函数确保单元节点处的曲率连续性, 既保证了本问题的适定性, 也提升了最终解的精确性. 最后, 基于该单元利用Newton法构造了平面梁有限变形下变形重构问题的两种求解算法, 即逐单元算法和多单元整体算法, 以实现不同需求下的稳定求解. 数值仿真结果表明, 本方法在大变形条件下的变形重构误差小于1%, 而且在测点较少的情况下依然保持较高的精度, 同时验证了本方法的收敛性与计算效率.Abstract: Most of the existing researches on deformation reconstruction of flexible structures with finite deformation are only based on the geometric relationship between curvature and strain, which ignores the longitudinal deformation and the coupling effect of the longitudinal deformation and the bending deformation. In order to construct a more accurate deformation reconstruction method which can be extended with the help of existing mechanical tools, this paper takes the plane beam as the object, partially inherits inverse finite element method developed by Tessler A, and regards the deformation reconstruction problem of plane beam as a kind of numerical optimization problem. Firstly, by introducing the absolute nodal coordinate formulation (ANCF) into the description of mapping relationship between strain and displacement, an inverse gradient reduced ANCF plane beam element is derived. Secondly, the inverse ANCF element is modified to simplify the degree of freedom of nodes and ensure the C2 continuity at nodes by introducing the penalty function, which not only ensures the problem is well-posed, but also improves the accuracy of the final result. Finally, based on the inverse ANCF element, the Newton method is used to develop two types of algorithms for deformation reconstruction under different working conditions, one is the element-by-element algorithm and the other is the multi-element algorithm. The numerical simulation results show that the reconstruction relative error of this method is less than 1% under the condition of large deformation, and it still maintains high accuracy under the condition of few measuring points. The convergence and computational efficiency of the method are verified by numerical simulation example.
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图 1 基于应变信息的平面梁变形重构模型
Figure 1. Shape reconstruction model of plane beam based on strain information
图 2 基于线性应变−位移关系的变形重构
Figure 2. Deformation reconstruction based on linear strain displacement relationship
图 3 梯度缩减ANCF梁单元内部的虚假轴向应变
Figure 3. False axial strain in gradient reduction ANCF beam element
图 4 边界约束及曲率连续性约束示意图
Figure 4. Schematic diagram of boundary constraints and curvature continuity constraints
图 9 大变形条件下不同测点数的重构效果
Figure 9. Reconstruction performance of different number of measuring points under large deformation
图 11 不同方法的悬臂梁变形重构仿真结果
Figure 11. Reconstruction performance of cantilever beam in different methods
图 12 轴向大变形条件下的简支梁示意图
Figure 12. Schematic of simply supported beam under large axial deformation
图 13 轴向大变形条件下曲率连续化方法和iANCF方法的对比
Figure 13. Comparison of curvature-based method and iANCF under large axial deformation
表 1 20个测点下的相对误差及计算用时
Table 1. Reconstruction error and calculation time under 20 measuring points condition
$u$/mm ${x_ {\rm{err} } }$ $ {y_{{\rm{err}}}} $ $ {\theta _{{\rm{err}}}} $ t/s 50 6.28×10−4 3.68×10−4 7.42×10−8 0.137 100 6.20×10−4 3.82×10−4 2.88×10−7 0.138 200 7.96×10−4 6.86×10−4 1.10×10−5 0.145 
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表 2 10个测点下的相对误差及计算用时
Table 2. Reconstruction error and calculation time under 10 measuring points condition
$u$/mm ${x_ {\rm{err} } }$ ${y_ {\rm{err} } }$ ${\theta _ {\rm{err} } }$ t/s 50 8.67×10−4 3.75×10−4 9.12×10−7 0.118 100 8.31×10−4 3.69×10−4 4.12×10−6 0.123 200 9.18×10−4 6.04×10−4 2.56×10−4 0.124
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表 3 4个测点下的相对误差及计算用时
Table 3. Reconstruction error and calculation time under 4 measuring points condition
$u$/mm ${x_ {\rm{err} } }$ ${y_ {\rm{err} } }$ ${\theta _ {\rm{err} } }$ t/s 50 2.38×10−3 1.53×10−3 5.34×10−5 0.113 100 2.68×10−3 1.64×10−3 2.60×10−4 0.113 200 5.29×10−2 2.49×10−2 1.88×10−2 0.118
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