AN ADAPTIVE ISOGEOMETRIC MESHFREE METHOD BASED ON THE GIFT FRAMEWORK

Although the meshfree representation theory of isogeometric basis functions provides a flexible meshfree scheme for adaptive isogeometric analysis, existing studies have been conducted within the isoparametric framework. In this framework, the same basis functions are employed to construct both the geometric mapping and the approximation of field variables, which limits the flexibility of adaptive analysis to some extent. To address this limitation, this paper introduces the geometry-independent field approximaTion (GIFT) framework and develops an adaptive isogeometric meshfree method in which the geometric description is decoupled from the approximation of field variables. Specifically, the proposed method constructs two separate function spaces, one for geometric representation and the other for the approximation of field variables. This separation allows different sets of basis functions and different discretization structures to be adopted for the geometric description and the field approximation. The non-uniform rational B-splines (NURBS) space is used to provide an exact representation of the geometry of the problem domain and to establish the geometric mapping from the parametric domain to the physical domain. Meanwhile, a function space for approximating the physical field is constructed in the parametric domain using meshfree particles. The NURBS geometric mapping is subsequently used to obtain the corresponding approximation of field variables in the physical domain. During the adaptive process, the NURBS structure defining the geometric mapping is kept unchanged, while local refinement is applied only to the meshfree particles. Within this solution framework, a series of numerical examples is used to systematically verify the effectiveness of the proposed method, with particular attention to its computational accuracy and convergence behavior in problems involving local singularities and high gradients. The numerical results demonstrate that the method effectively combines the geometric exactness offered by isogeometric analysis with the advantages of meshfree methods in adaptive refinement. It thus provides a computational approach with a high degree of accuracy and flexibility for solving problems involving complex boundaries and localized regions in which the physical fields exhibit high gradients.