Quasi-Convex Reproducing Kernel Meshfree Method

Meshfree shape functions are constructed entirely in terms of a set of nodes and enjoy the higher order smoothness and conforming approximation properties, and there is no requirement for the topologically connected elements which are necessary for the conventional finite element method. Consequently meshfree methods can effectively relieve the mesh-related issues in the traditional finite element analysis, at least to some extent. However, the commonly used moving least square or reproducing kernel meshfree shape functions are not convex approximations. Here, a convex approximation means that the shape functions are non-negative. It has been shown that convex approximations yield non-negative mass matrices and do show better solution accuracy and frequency spectrum compared with non-convex approximations. The max-entropy meshfree approximation has convex meshfree shape functions by introducing exponential basis function. Nonetheless, the construction of max-entropy meshfree shape functions requires very costly iterative computations, and moreover it is very hard to generalize this approach to arbitrary higher order convex shape functions.In this thesis, a novel quasi-convex reproducing kernel approximation is presented for Galerkin meshfree analysis. The quasi-convex reproducing kernel shape functions are established within the framework of the relaxed reproducing or consistency conditions which are similar to the classical consistency conditions, thus the present shape functions have similar form as their reproducing kernel counterparts. Consequently this approach can be conveniently implemented in the standard reproducing kernel meshfree formulation without an overmuch increase of computational effort. In the proposed meshfree scheme, the monomial reproducing conditions are relaxed by introducing nodal gaps and the resulting shape functions are referred as the quasi-convex reproducing kernel shape functions. 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. It is also shown that the derivatives of the proposed quasi-convex reproducing kernel meshfree shape functions are more smoothing than those of the traditional reproducing kernel shape functions Subsequently a Galerkin meshfree analysis is carried out by employing the proposed quasi-convex reproducing kernel shape functions. Both the2nd order problems such as1D rod,2D membrane and elasticity problems, and the4th order problems of thin beams and plates are studied to investigate the performance of the present method. The numerical results uniformly reveal that the proposed method have more favorable accuracy than the conventional reproducing kernel meshfree method, especially for structural vibration analysis.