Euler梁的无网格求解方法探讨

基于再生条件建立了一种用于Euler梁(薄梁)分析,同时考虑挠度和转角影响的双变量无网格计算方法.与现有采用固定基的双变量无网格近似相比,此方法采用移动基函数,有更小的数值再生误差;与只考虑挠度的单变量无网格近似相比,此方法有更高的插值精度.这些特性在文中得到了数值验证.此外,通过推广位移边界条件处理的变换法,进一步把双变量无网格近似中广义节点挠度和转角系数与相对应的真实挠度和转角节点值联系起来,使得Galerkin无网格法求解Euler梁问题中挠度和转角边界条件的处理变得与有限元类似,较为便利.Euler梁算例表明,具有移动基的单变量与双变量两种无网格算法收敛速度相当,但采用移动基的双变量无网格法有更高的计算精度.

On the Meshfree Analysis of Euler Beam

Based on the reproducing conditions(or consistency conditions),a new meshfree double-variable approximation is proposed for Euler beam analysis.This approximation employs a shifted basis and is built upon both the defection and rotation nodal variables.Compared with the existing double-variable approximation with a fixed basis,numerically the current approximation satisfies the consistency conditions more accurately,although they are equivalent in the mathematical sense.In contrast to the single-variable approximation which is constructed only using the nodal coefficients of deflection,the present approximation has smaller interpolation error.These characteristics are verified through numerical tests.To treat the essential boundary conditions in the Galerkin meshfree formulation,the transformation method is generalized here to relate the nodal coefficients and their corresponding physical nodal values.Through this transformation the essential boundary treatment can be taken care easily by the similar manners used in FEM.Numerical results of an Euler beam example demonstrate the convergence rates for the single-variable and double-variable approximations are relatively close,while the numerical accuracy for the double-variable approach is much higher.