自适应自然单元法研究——误差估计

实现自然单元法自适应分析的首要步骤是对NEM求解误差进行有效估计,鉴于Z-Z方法在有限元自适应分析中的成功应用以及近年来在无网格伽辽金方法误差估计中的高效实用,尝试将其应用于自然单元法的求解误差估计。应用自然单元法和位移模型求解固体力学问题时,由于NEM形函数在离散节点处的导数不存在以及在边界节点间满足线性插值,以致自然单元法求解不能直接给出节点上的应力、应变值,而且应力、应变值在边界上为分段常数。因此,有必要根据求解得到的位移场进行应力恢复以提取节点处的应力、应变值并构造全域光滑的应力场。针对自然单元法的特点,建议先利用自然单元法求解得到的全域光滑位移场,在节点处应用移动最小二乘方法提取应变、应力值;然后利用节点上恢复的应力、应变值,采用自然相邻点插值构造全域光滑的应力、应变场。大量数值实例表明,构造的光滑应力场具有较原始解更高的数值精度和收敛性。基于Z-Z方法将原始应力解和应力恢复解之间的差值作为误差的近似估计是可行的、简单高效的。

STUDY ON ADAPTIVE NATURAL ELEMENT METHOD ERROR ESTIMATION

Valid estimation of the posterior error is the first step to the implementation of adaptive analysis with the natural element method(NEM). For the successful applications of Z-Z error estimation in the adaptive finite element analysis and the element-free Galerkin method and its simplicity and high efficiency,the implementation of Z-Z method in NEM is attempted. When the NEM is used to solve the solid mechanics problems employing the displacement model,the strain and stress on the scattered nodes cannot be directly acquired form the displacement field because the derivatives on the nodes of the shape function do not exist. What is more,the linear interpolation property of the shape function between the adjacent boundary nodes results shows that the stress/strain is piecewise constant on the boundary. So recovery scheme is performed to extract strain/stress on the nodes and construct a smooth strain/stress field. Moving least squares method is used to extract strain/stress on the nodes from the displacement field acquired from the NEM solution. Then a smoothing stress/strain field can be attained through natural neighbor interpolation employing the recovered strain or stress on the nodes. Numerical examples show that the recovery scheme is feasible;both the accuracy and the convergence of the smoothing stress field are higher. Then difference of original stress field and the recovered stress field is calculated as the error estimationbased on the Z-Z method. Numerical example shows that the Z-Z error estimator is feasible,simple and efficient.