薄板分析的线性基梯度光滑伽辽金无网格法

薄板问题的控制方程为四阶微分方程,因而当采用伽辽金法进行分析时,形函数需要满足C$^{1}$连续性要求,且至少使用二次基函数才能保证方法的收敛性.无网格形函数虽然易于满足C$^{1}$连续性要求,但由于不是多项式,其二阶导数的计算较为复杂耗时,同时也对刚度矩阵的数值积分提出了更高的要求.本文提出了一种薄板分析的线性基梯度光滑伽辽金无网格法,该方法的基础是线性基无网格形函数的光滑梯度.在梯度光滑构造的理论框架内,无网格形函数的二阶光滑梯度可以表示为形函数一阶梯度的线性组合,因而可以提高形函数二阶梯度的计算效率.分析表明,线性基无网格形函数的光滑梯度不仅满足其固有的线性梯度一致性条件,还满足本属于二次基函数对应的额外高阶一致性条件,因此能够恰当地运用到薄板结构的伽辽金分析.此外,插值误差分析也很好地验证了线性基无网格光滑梯度的收敛特性.算例结果进一步表明,线性基梯度光滑伽辽金无网格法的收敛率与传统二次基伽辽金无网格法相当,但精度更高,同时刚度矩阵所需的高斯积分点数明显减少. 

A GRADIENT SMOOTHING GALERKIN MESHFREE METHOD FOR THIN PLATE ANALYSIS WITH LINEAR BASIS FUNCTION1)

The fourth order governing equation of thin plate necessitates the employment of C$^{1}$ continuous shape functions with a minimum degree of two in a Galerkin formulation. Thus at least a quadratic basis function should be utilized in meshfree approximation to enable the Galerkin meshfree thin plate analysis. However, due to the rational nature of reproducing kernel meshfree shape functions, the computation of the second order derivatives of meshfree shape functions is quite complex and costly, which also requires expensive high order Gauss quadrature rules to properly integrate the stiffness matrix. In this work, a gradient smoothing Galerkin meshfree method with particular reference to the linear basis function is proposed for thin plate analysis. The foundation of the present development is the construction of smoothed meshfree gradients with linear basis function, where the second order smoothed gradients are expressed as combinations of standard first order gradients and the computational burden is remarkably reduced. Furthermore, it is shown that the smoothed meshfree gradients with linear basis function satisfy both the linear and quadratic gradient consistency conditions and consequently they are adequate for thin plate analysis in the context of Galerkin formulation. An interpolation error study is given as well to validate the higher order consistency conditions and applicability of smoothed meshfree gradients for Galerkin analysis of thin plates. It turns out that efficient lower order Gauss integration rules now work well for the proposed method. Numerical results demonstrate that compared with the conventional Galerkin meshfree method with quadratic basis function, the proposed gradient smoothing Galerkin meshfree method with linear basis function yields similar convergence rates, but with better accuracy and less integration points for stiffness computation.