Towards a general interpolation scheme

Abstract

This paper presents an interpolation scheme applicable to meshless/mesh-based methods. The formulation is capable of generating ordinary as well as non-negative shape functions in a unified style. The flexibility of producing either strong Kronecker-delta in the domain or weak Kronecker-delta property at the boundaries is one of the method’s prominent features. Both convex and non-convex boundaries are treated, even when non-negative shape functions are of concern. It is shown that a variety of interpolant/non-interpolant functions are achievable. The existence of weak/strong Kronecker-delta property at the boundaries allows for using the interpolants in the Galerkin-weak type of formulation. In this context, it is shown that some alternatives are far preferable to the others. For instance comparison of the performances of non-negative interpolants with weak and strong Kronecker-delta shows that the former interpolants are superior to the latter ones. All the interpolants introduced may seamlessly convert to the finite element ones, and thus, they may be employed in solutions using a dual discretization. Due to the continuity of the shape functions (up to the desired order), one may use them in a collocation approach. Several examples are solved to demonstrate the capabilities/flexibility of the interpolants.