Reproducing kernel formulation of B-spline and NURBS basis functions: A meshfree local refinement strategy for isogeometric analysis

A reproducing kernel meshfree formulation is presented for the B-spline and non-uniform rational B-spline (NURBS) basis functions used in isogeometric analysis. It is shown that after properly introducing meshfree nodes, support size and consistency conditions, the reproducing kernel meshfree shape functions are capable of exactly representing the isogeometric B-spline and NURBS basis functions. Consequently, the proposed formulation successfully establishes a correspondence or close link between the meshfree methods and the isogeometric analysis. More importantly, the proposed reproducing kernel meshfree representation of isogeometric basis functions provides a reliable meshfree strategy to the local model refinement in isogeometric analysis. This strategy inherits the strength of meshfree methods and gives considerable easiness for the local refinement, i.e., the shape functions in the refined regions can be naturally constructed in a straightforward meshfree manner. Meanwhile, the consistency and independence of the shape functions required by the subsequent computational analysis are ensured by the consistency conditions of reproducing kernel meshfree formulation. A detailed illustration of the proposed approach for isogeometric local model refinement is presented. The effectiveness of the proposed meshfree local refinement strategy for isogeometric analysis is demonstrated through numerical examples.