A ROBUST ISOGEOMETRIC MESHFREE COLLOCATION METHOD FOR TARGETED REFINEMENT ANALYSIS

The reproducing kernel meshfree representation of isogeometric basis functions provides a meaningful way to link the meshfree methods and isogeometric analysis, while the costly gradient computation of isogeometric basis functions has not been well addressed. In this work, based upon the reproducing kernel gradient formulation of meshfree shape functions, a mixed gradient reproducing basis vector is defined by the reproducing points of isogeometric basis functions. Subsequently, a meshfree framework is developed to construct the gradients of isogeometric basis functions in a meshfree fashion, which further unifies the meshfree methods and isogeometric analysis. Under this framework, a robust isogeometric meshfree collocation method is proposed in this study, which ensures the consistency and efficiency of the approximation function in the whole domain and the flexibility of the local model refinement. Unlike the recursive construction process of isogeometric basis functions, the proposed approach is a direct and one-step procedure for the gradient evaluation, which is trivial for numerical implementation. It turns out that the proposed meshfree formulation of isogeometric gradients yields the same gradients as the standard gradients of isogeometric basis functions under certain conditions regarding the influence domains of meshfree shape functions. Meanwhile, the reproducing conditions of the direct gradients of the isogeometric basis functions are unstable, but the reconstructed meshfree reproducing gradients meet complete the gradient reproducing conditions, which ensures the convergence of the collocation methods. Numerical results demonstrate that the proposed collocation methodology exhibits much higher computational accuracy than the conventional isogeometric collocation in the analysis of local refinement problems.