# NURBS-Enhanced Meshfree Method with an Integration Subtraction Technique for Complex Topology
## 1 Introduction  
As isogeometric analysis (IGA) has offered the possibility of integrating methods for analysis and computer-aided design (CAD) into a single, unified process, the time taken from design to analysis is greatly reduced [1]. IGA has been a topic of great interest in the area of recently developed technology in computational mechanics. When non-uniform rational B-spline (NURBS) has been used to describe complex geometry precisely and interactively in analysis, more physically accurate results and greater convenience in solving adaptive problems have been obtained. In conventional isogeometric analysis, as proposed by [PERSON], NURBS is used to integrate both of geometric and analysis information in one model [2], which leads to various advantages over the standard finite element method (FEM) [3; 4; 5; 6; 7]. However, the need for tensor product meshes has been a limitation in the analysis of arbitrary complex topology. To increase the efficiency of generating tensor product meshes for arbitrary complex topology, many methods have been developed, such as trimmed surfaces, patching operations, subdivision surfaces, and skeleton partitions [8; 9; 10; 11]. In addition, other kinds of NURBS-enhanced FEM methods have been developed [12; 13] at the same time. Furthermore, many other spline methods are promoted to overcome the drawback of tensor product requirement of NURBS, i.e., T-splines, PHT-splines, LHB-splines, etc. [14; 15; 16]. However, once the shape boundary or topology of an analysis domain is changed, the complicated mesh processneeds to be performed again, and the global stiffness matrix needs to be partially corrected both in value and in size. These measurements are still somewhat tedious and time consuming in interactive design, topology optimization, crack growth, and other adaptive problems. As the tensor product mesh has been a limitation in conventional isogeometric analysis, combinations of various meshfree methods and IGA have been developed by many researchers and applied successfully in many fields [17; 18; 19; 20]. In meshfree methods, the domain of a problem is represented, ideally, only by a set of arbitrarily distributed nodes. The first meshfree method is invented by [PERSON] [21] and [PERSON] [22] for astrophysical problems without boundaries which is known as smooth particle hydrodynamics (SPH). To overcome the consistency and stability problem in SPH, reproducing kernel particle method (RKPM) and element free galerkin (EFG) are developed by [PERSON] [23] and [PERSON] [24]. As for the advantage of fast convergence ability and simplicity, radial basis function (RBF) is introduced to solve partial differential equations (PDEs) by [PERSON] in 1991 [25]. When a problem is solved in PDEs of integrated weak form in meshfree method, background cells or similar subdomains are always needed to achieve a consecutive integration. Triangular background cells, which can be easily created automatically for complicated domains, are sufficient for numerical integration operations in meshfree methods [26]. With the advantage of no special requirement for the mesh quality in meshfree methods, the mesh process in the integration region becomes easier. However, once the structure topology is changed, remeshing work for the new integration region or refinement along the inner boundary is needed. As new nodes and elements are added and the unnecessary ones are deleted in the process of refinement, the global stiffness matrix still needs to be partially corrected both in value and in size. An attractive and competitive way to solve this problem is to couple a boundary element method (BEM) with IGA. A recent framework of the combination is proposed by [PERSON] [27], in which the NURBS functions were used to approximate the unknown fields in BEM. This IGABEM method circumvents the need to discretize the domain, as required by the IGAFEM [28]. However, the requirement for so-called fundamental solutions has been a limit for its application [29].