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

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.