Natural element approximations involving bubbles for treating mechanical models in incompressible media

In this paper, a new approach is proposed to address issues associated with incompressibility in the context of the meshfree natural element method (NEM). The NEM possesses attractive features such as interpolant shape functions or auto-adaptive domain of influence, which alleviates some of the most common difficulties in meshless methods. Nevertheless, the shape functions can only reproduce linear polynomials, and in contrast to moving least squares methods, it is not easy to define interpolations with arbitrary approximation consistency. In order to treat mechanical models involving incompressible media in the framework of mixed formulations, the associated functional approximations must satisfy the well-known inf–sup, or LBB condition. In the proposed approach, additional degrees of freedom are associated with some topological entities of the underlying Delaunay tessellation, i.e. edges, triangles and tetrahedrons. The associated shape functions are computed from the product of the NEM shape functions related to the original nodes. Different combinations can be used to construct new families of NEM approximations. As these new approximations functions are not related to any node, as they vanish at the nodes, from now on we refer these shape functions as bubbles. The shape functions can be corrected enforcing different reproducing conditions, when they are used as weights in the moving least square (MLS) framework. In this manner, the effects of the obtained higher approximation consistency can be evaluated. In this work, we restrict our attention to the 2D case, and the following constructions will be considered: (a) bubble functions associated with the Delaunay triangles, called b1-NEM and (b) bubble functions associated with the Delaunay edges, called b2-NEM. We prove that all these approximation schemes allow direct enforcement of essential boundary conditions. The bubble-NEM schemes are then used to approximate the displacements in the linear elasticity mixed formulation, the pressure being approximated by the standard NEM. The numerical LBB test is passed for all the bubble-NEM approximations, and pressure oscillations are removed in the incompressible li