A NURBS-based Parametric Method Bridging Mesh-free and Finite Element Formulations

Abstract

A generalization of a NURBS based parametric mesh-free method (NPMM), recently proposed by Shaw and Roy (2008), is considered. A key feature of this parametric formulation is a geometric map that provides a local bijection between the physical domain and a rectangular parametric domain. This enables constructions of shape functions and their derivatives over the parametric domain whilst satisfying polynomial reproduction and interpolation properties over the (non-rectangular) physical domain. Hence the NPMM enables higher-dimensional B-spline based functional approximations over non-rectangular domains even as the NURBS basis functions are constructed via the usual tensor products of their one-dimensional counterparts. Nevertheless the method still lacks the universality that the FEM enjoys. In particular, for many non-simply connected domains, the geometric map may not be locally bijective everywhere and this severely restricts the general applicability of the NPMM. In this paper, a piecewise form of the NPMM is proposed, wherein the domain is decomposed into a collection of simply connected sub-domains or element patches (analogous to the FEM). The NPMM is then employed over each sub-domain without affecting the continuity of approximated functions across inter-sub-domain boundaries. This is quite unlike the usual FEM. The proposed procedure not only possesses the generality of the FEM, it is also equipped with higher order, globally smooth and interpolating basis functions. It may thus be interpreted as a seamless bridge between the FEM and mesh-free methods. In the context of weak implementations of the piecewise NPMM, we propose a conformal knot-grid integration scheme. Finally, we illustrate these schemes for weak numerical solutions of a few linear and nonlinear boundary value problems of engineering interest.