NURBS-based parametric mesh-free methods

A mesh-free error reproducing kernel method (ERKM) has recently been proposed by [A. Shaw, D. Roy, A NURBS-based error reproducing kernel method with applications in solid mechanics, Comput. Mech. 40(1) (2007) 127–148]. The ERKM is based on an initial approximation of the target function by non-uniform-rational-B-splines (NURBS) followed by reproduction of the error via a family of non-NURBS basis functions. However, specifications of the window supports of non-NURBS basis functions remain a tricky issue. Moreover, NURBS in higher dimensions (>1) is generally defined over rectangular (cuboidal in 3D) grid structures and thus, in many problems of practical interest, the geometric complexity of the domain would prevent making use of NURBS in the ERKM. Presently, we develop a parametric reformulation of the ERKM purely using NURBS. This reformulation allows the method to be applicable to non-rectangular (non-cuboidal in 3D) physical domains in two or still higher dimensions without a need to explicitly specify the support size of the window. A key feature of this development is a geometric map that provides a local bijection between the physical domain and rectangular (cuboidal) parametric domain. The shape functions and their derivatives are then constructed over the parametric domain so that polynomial reproduction and interpolation properties are satisfied over the physical domain with the geometric map being simultaneously preserved. A couple of new schemes are also proposed to empower the shape functions with the interpolation property that in turn enables a precise imposition of Dirichlet boundary conditions. We illustrate the parametric mesh-free method in the context of strong/weak solutions of a few linear and non-linear boundary and initial value problems of engineering interest.