Quasi-convex reproducing kernel meshfree method

A quasi-convex reproducing kernel approximation is presented for Galerkin meshfree analysis. In the proposed meshfree scheme, the monomial reproducing conditions are relaxed to maximizing the positivity of the meshfree shape functions and the resulting shape functions are referred as the quasi-convex reproducing kernel shape functions. These quasi-convex meshfree shape functions are still established within the framework of the classical reproducing or consistency conditions, namely the shape functions have similar form as that of the conventional reproducing kernel shape functions. Thus this approach can be conveniently implemented in the standard reproducing kernel meshfree formulation without an overmuch increase of computational effort. Meanwhile, the present formulation enables a straightforward construction of arbitrary higher order shape functions. It is shown that the proposed method yields nearly positive shape functions in the interior problem domain, while in the boundary region the negative effect of the shape functions are also reduced compared with the original meshfree shape functions. Subsequently a Galerkin meshfree analysis is carried out by employing the proposed quasi-convex reproducing kernel shape functions. Numerical results reveal that the proposed method has more favorable accuracy than the conventional reproducing kernel meshfree method, especially for structural vibration analysis.