A consistently coupled isogeometric–meshfree method

A consistently coupled isogeometric–meshfree method is presented. This method takes advantage of the geometry exactness of isogeometric analysis and the refinement flexibility of meshfree method. The coupling of isogeometric approximation and meshfree approximation is based upon the reproducing or consistency conditions which are crucial for the coupled method to achieve the expected optimal convergence rates. It is shown that unlike the reproducing kernel meshfree shape functions which satisfy the reproducing conditions with the nodal points as the reproducing locations, the monomial reproducing points for different orders of B-spline basis functions in isogeometric analysis are different and consequently a rational method is proposed to compute these reproducing points. Both theoretical proof and computational justification of the reproducing conditions for B-spline basis functions are given. Subsequently within the framework of reproducing conditions, a mixed reproducing point vector is proposed to ensure arbitrary order monomial reproducibility for both B-spline basis functions and reproducing kernel meshfree shape functions, which leads to a consistently coupled approximation with smoothing transition between B-spline basis functions and reproducing kernel meshfree shape functions. Consequently a coupled isogeometric–meshfree method is established with the Galerkin formulation. The effectiveness of the proposed coupled isogeometric–meshfree method is demonstrated through a series of benchmark numerical examples.