基于自然单元法的极限上限分析

自然单元法是一种基于离散点集的Voronoi图和Delaunay三角化几何信息,以自然邻近插值为试函数的新型数值方法.相对于一般无网格法中常采用的移动最小二乘近似而言,自然邻近插值不涉及到复杂的矩阵求逆运算,更不需要任何人为的参数,可以提高计算效率.采用该方法构造的形函数满足Delta函数的性质,可以像有限元一样准确地施加边界条件,可以方便处理场函数及其导数的不连续性的问题.论文将自然单元法应用到极限上限分析中,编制了相应的计算程序,通过极限分析的几个经典算例进行了验证,同时采用类似于分片应力磨平的方式,编制相应的磨平程序,由计算点上的塑性耗散功外推得到了节点上的塑性耗散功的值,从而画出了极限状态下结构的塑性耗散功的分布云图.计算结果表明采用自然单元法求解极限上限分析具有稳定性好,精度高,收敛快等优点.

UPPER-BOUND LIMIT ANALYSES BASED ON NATURAL ELEMENT METHOD

The natural element method(NEM) is a novel numerical method based on voronoi diagram and delaunay triangulation of the scattered points in problem domain,and its shape function is built upon the notion of natural neighbor interpolation.Compared with the moving least square(MLS) approximation used widely in many meshless methods,natural neighbor interpolation does not involve the complex matrix inversion,needs no artificial parameter but can improve the computational efficiency.The obtained shape function satisfies the property of Kronecker delta function,and this character results in the consequence that the essential boundary condition can be easily imposed as in finite element method(FEM).Meanwhile,the problems with discontinuous field functions likewise their discontinuous derivatives can be conveniently treated.According to the kinematic theorem of plastic limit analysis,this article applies the natural element method to upper bound limit analysis.The corresponding mathematical programming formulations are established and the computational codes are implemented to solve them.Several classical examples of limit analysis are adopted to verify the performance of these codes.Furthermore,the smoothing operations are also provided to calculate plastic dissipation work on the nodes by the similar treatment of stress smoothing operation,and the contours of plastic dissipation work at the limit states are displayed.The computational results show that utilizing natural element method to solve upper bound limit analysis problems possesses the advantage of good stability,high accuracy and fast convergence.