A Meshfree Cell-based Smoothed Point Interpolation Method for Solid Mechanics Problems

Abstract

In the framework of a weakened weak (W{sup 2}) formulation using a generalized gradient smoothing operation, this paper introduces a novel meshfree cell-based smoothed point interpolation method (CS-PIM) for solid mechanics problems. The W{sup 2} formulation seeks solutions from a normed G space which includes both continuous and discontinuous functions and allows the use of much more types of methods to create shape functions for numerical methods. When PIM shape functions are used, the functions constructed are in general not continuous over the entire problem domain and hence are not compatible. Such an interpolation is not in a traditional H{sup 1} space, but in a G{sup 1} space. By introducing the generalized gradient smoothing operation properly, the requirement on function is now further weakened upon the already weakened requirement for functions in a H{sup 1} space and G{sup 1} space can be viewed as a space of functions with weakened weak (W{sup 2}) requirement on continuity. The cell-based smoothed point interpolation method (CS-PIM) is formulated based on the W{sup 2} formulation, in which displacement field is approximated using the PIM shape functions, which possess the Kronecker delta property facilitating the enforcement of essential boundary conditions [3]. The gradient (strain) field is constructed by the generalized gradient smoothing operation within the cell-based smoothing domains, which are exactly the triangular background cells. A W{sup 2} formulation of generalized smoothed Galerkin (GS-Galerkin) weak form is used to derive the discretized system equations. It was found that the CS-PIM possesses the following attractive properties: (1) It is very easy to implement and works well with the simplest linear triangular mesh without introducing additional degrees of freedom; (2) it is at least linearly conforming; (3) this method is temporally stable and works well for dynamic analysis; (4) it possesses a close-to-exact stiffness, which is much softer than the overly-stiff FEM model and much stiffer than the overly-soft node-based smoothed point interpolation method (NS-PIM); (5) the results of the present method are of better accuracy and higher convergence rate than the linear FEM model using the same set of triangular meshes.