# NURBS-Enhanced Meshfree Method with an Integration Subtraction Technique for Complex Topology
## 2 Basic Theory for the NURBS and RPIM Meshfree Methods
### RPIM Meshfree Method  
Meshfree methods are developed to eliminate several key drawbacks of the finite element method [38]:  
1. It is time consuming to generate a quality mesh in an arbitrary geometry with the desired accuracy.
2. It is difficult to construct approximations with an arbitrary order of continuity, making PDEs with higher-order differentiation or problems with discontinuities difficult to solve.
3. Performing h- or \(p\)-adaptive refinement is tedious.
4. The finite element method is ineffective in dealing with mesh entanglement-related difficulties (such as those in large deformation and fragment-impact problems).  
As shown in Figure 3, the approximation function of unknowns in the meshfree method is constructed by scattered nodes, which are called field nodes. Thus, the information for approximation has been separated from the integration cells, which means that the mesh quality associated with approximation has been weakened, thereby giving us more freedom in discretizing the material domain.  
The RPIM approximation basic function is constructed by the radial basis function and point interpolation method on each field node within the supported domain. The known function values vector \(\mathbf{U}_{s}\) of inner nodes in the supported domain can be expressed as [26]  
\[\mathbf{U}_{s}=\mathbf{R}_{0}\mathbf{a}+\mathbf{P}_{m}\mathbf{b} \tag{7}\]  
where \(\mathbf{R}_{0}\) is the radial basis matrix, \(\mathbf{P}_{m}\) is the polynomial matrix, and \(\mathbf{a}\) and \(\mathbf{b}\) form the coefficient vector, which needs to be calculated. The polynomial term must satisfy the following conditional positive-definite constraint [39]:  
\[\mathbf{P}_{m}\mathbf{a}=0\]  
Figure 3: (**a**) Finite element method (FEM) with a local element; (**b**) meshfree approximation function constructed by field nodes within a support domain.  
Combining the above equations and rewriting them in matrix form, we have  
\[\widetilde{\mathbf{U}}_{s}=\left\{\begin{array}{c}\mathbf{U}_{s}\\ 0\end{array}\right\}=\underbrace{\left[\begin{array}{cc}\mathbf{R}_{0}& \mathbf{P}_{m}\\ \mathbf{P}_{m}&0\end{array}\right]}_{G}\left\{\begin{array}{c}\mathbf{a}\\ \mathbf{b}\end{array}\right\}=\mathbf{Ga}_{0} \tag{9}\]  
Then, the function values of other unknown points (red triangles in Figure 3) in the supported domain can be approximated by  
\[\mathbf{u}(\mathbf{x})=\mathbf{R}(\mathbf{x})\mathbf{a}+\mathbf{P}(\mathbf{x })\mathbf{b}=\left\{\begin{array}{cc}\mathbf{R}(\mathbf{x})&\mathbf{P}( \mathbf{x})\end{array}\right\}\mathbf{G}^{-1}\widetilde{\mathbf{U}}_{s}= \mathbf{\Phi}(\mathbf{x})\widetilde{\mathbf{U}}_{s} \tag{10}\]  
where \(\mathbf{\Phi}(\mathbf{x})\) is the shape function of the RPIM, which represents each node's contribution to the unknown points in the supported domain.  
The effects of different radial basis functions and shape parameters have been discussed in [26]. Here, in this presentation, the multiquadric radial basis function (MQ-RBF) is chosen for static elastic problems with shape parameters \(q=1.03\) and \(\alpha_{c}=3.0\). The MQ-RBF \(R(x,y)\) is defined by  
\[R(x,y)=\left(r_{i}^{2}+\left(\alpha_{c}d_{c}\right)^{2}\right)^{q},\quad r_{i} ^{2}=\left(x-x_{  
It follows from the Kronecker \(\delta\) function property of the RPIM that boundary constraints can be directly applied to the field nodes. Then, Equation (16) can be rewritten as  
\[\mathbf{KU}=\mathbf{F} \tag{17}\]  
which is the final discretization equation shaped by the meshfree RPIM. When the linear equation in Equation (17) is solved, the displacement \(\mathbf{U}\) of all field nodes will be gained, and the stress \(\boldsymbol{\sigma}\) can be recovered from:  
\[\boldsymbol{\sigma}=\mathbf{D}\boldsymbol{\varepsilon}=\mathbf{D}\mathbf{B}_ {d}\mathbf{u}_{d} \tag{18}\]