# A Co-Rotational Meshfree Method for the Geometrically Nonlinear Analysis of Structures
## 3 Differential Reproducing Kernel Approximation Collocation Method (DRK)  
### Reproducing Kernel Approximation  
Most numerical methods transform a differential equation into the integration form while solving it, such as the Galerkin method [23; 27]. However, the DRK approximations adopted in this paper can circumvent the numerical mesh for numerical integration. The DRK collocation method is developed based on the DRK approximation [32; 33; 34]. The DRK approximation can be derived from the kernel function with a reproducing condition or the moving least square method. Two kinds of derivations represent different characteristics of the DRK approximation. In this section, a brief introduction is given with regard to the moving least square method. Consider a set of \(N\) nodal points scattered in a domain \(\Omega\) and let \(x^{*}\) be the reference point. The DRK approximation of a variable \(u(x)\) in the neighborhood of the reference point \(x^{*}\), denoted by \(u^{th}(x)\), is  
\[u(x)\approx u^{h}(x)=\sum\limits_{i=1}^{M}p_{i}(x^{*}-x)a_{i}=\mathbf{p}^{T}(x^{*} -x)\mathbf{a} \tag{45}\]  
where \(\mathbf{p}^{T}=\{p_{1}(x),p_{2}(x),\dots,p_{M}(x)\}\) is a linear independent basis; \(\mathbf{a}^{T}=\{a_{1},\ a_{2},\dots,\ a_{M}\}\) is a vector of undetermined coefficients. Hence, the error in the weighted least-squares of \(N\) nodes at neighborhood with respect to the reference point \(x^{*}\) can be expressed as follows:  
\[\begin{array}{ll}J&=\sum\limits_{i=1}^{Ne}w_{a}(x^{*}-x_{j})[\mathbf{p}^{T}(x^{ *}-x_{j})\mathbf{a}-u(x_{j})]^{2}\\ &=(\mathbf{a}^{T}\mathbf{P}(x^{*})-\mathbf{u}^{T})\mathbf{W}(x^{*})(\mathbf{P}^{T}(x^{*})\mathbf{a}- \mathbf{u})\end{array} \tag{46}\]  
where \(w_{a}(x^{*}-x)\) is the local weight function and \(Ne\) is the number of neighboring nodes of \(x^{*}\). The index \(a\) is the radius of support and \(u(x_{j})\) is the approximation of \(u(x)\) at node \(x_{j}\). By minimizing the error of the weighted least-square, the approximation \(u^{th}(x)\) can be obtained as follows:  
\[u^{h}(x)=\mathbf{p}^{T}(x^{*}-\mathbf{x})\mathbf{M}^{-1}(x^{*})\mathbf{B}(x^{*})\mathbf{u}=\mathbf{N} ^{T}(\mathbf{x};\mathbf{x}^{*})\mathbf{u}, \tag{47}\]  
where \(\mathbf{M}(\mathbf{x}^{*})=\mathbf{P}(\mathbf{x}^{*})\mathbf{W}(\mathbf{x}^{*})\mathbf{P}^{T}(\mathbf{x}^{*})\), \(\mathbf{B}(\mathbf{x}^{*})=\mathbf{P}(\mathbf{x}^{*})\mathbf{W}(\mathbf{x}^{*})\), \(\mathbf{u}^{T}=\{u(\mathbf{x}_{1}),u(\mathbf{x}_{2}),\dots u(\mathbf{x}_{N})\}\) and \(\mathbf{N}^{T}(\mathbf{x};\mathbf{x}^{*})\) is the shape function that can be defined as follows:  
\[\mathbf{N}^{T}(\mathbf{x};\mathbf{x}^{*})=\mathbf{p}^{T}(\mathbf{x}^{*}-\mathbf{x})\mathbf{M}^{-1}(\mathbf{x} ^{*})\mathbf{B}(\mathbf{x}^{*}) \tag{48}\]  
The approximation \(u^{th}(x)\) in Equation (47) is a local approximation in the neighborhood with respect to the reference point \(x^{*}\). However, the global approximation can be fulfilled by the moving least-square method by imposing \(x=x^{*}\) such that point \(x^{*}\) will move with the point that has to be computed. In this way, the global approximation can be expressed as follows:  
\[u^{h}(\mathbf{x})=\mathbf{p}^{T}(0)\mathbf{M}^{-1}(\mathbf{x})\mathbf{B}(\mathbf{x})\mathbf{u}=\mathbf{N}^{T} (\mathbf{x})\mathbf{u} \tag{49}\]  
where \(\mathbf{N}^{T}(\mathbf{x})=\mathbf{p}^{T}(0)\mathbf{M}^{-1}(\mathbf{x})\mathbf{B}(\mathbf{x})\).