# Elastoplastic Analysis of Plates with Radial Point Interpolation Meshless Methods
## 2 Meshless Methods
### The Radial Point Interpolation Method
The polynomial basis has to satisfy an extra requirement in order to obtain a unique solution,  
\[\sum_{i=1}^{n}p_{j}(\mathbf{x}_{i})a_{i}(\mathbf{x}_{i})=0 \tag{13}\]  
with \(j=\{1,2,\ldots,m\}\). Therefore, combining Equation (9) with Equation (13), a new equation matrix can be written as  
\[ \begin{bmatrix}\mathbf{R}&\mathbf{P}\\ \mathbf{P}^{T}&0\end{bmatrix}\begin{Bmatrix}\mathbf{a}\\ \mathbf{b}\end{Bmatrix} =\mathbf{G} \begin{Bmatrix}\mathbf{a}\\ \mathbf{b}\end{Bmatrix} = \begin{Bmatrix}\mathbf{u}_{s}\\ 0\end{Bmatrix}  \tag{14}\]  
where \(\mathbf{u}_{s}\) is given by  
\[\mathbf{u}_{s}=\{u_{1},u_{2},\ldots,u_{n}\}^{T} \tag{15}\]  
with the symmetric matrix \(\mathbf{R}\) obtained with  
\[\mathbf{R}= \begin{bmatrix}r_{11}&r_{12}&\ldots&r_{1n}\\ r_{21}&r_{22}&\ldots&r_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ r_{n1}&r_{n2}&\ldots&r_{mn}\end{bmatrix}  \tag{16}\]  
and matrix \(\mathbf{P}\), of size \([n\times m]\), of the polynomial basis being given by  
\[\mathbf{p}= \begin{bmatrix}p_{1}(\mathbf{x}_{1})&p_{2}(\mathbf{x}_{1})&\ldots &p_{m}(\mathbf{x}_{1})\\ p_{1}(\mathbf{x}_{2})&p_{2}(\mathbf{x}_{2})&\ldots&p_{m}(\mathbf{x}_{2})\\ \vdots&\vdots&\ddots&\vdots\\ p_{1}(\mathbf{x}_{n})&p_{2}(\mathbf{x}_{n})&\ldots&p_{m}(\mathbf{x}_{n})\end{bmatrix} . \tag{17}\]  
The moment matrix \(\mathbf{G}\) is symmetric because the distance is directionally independent. Solving Equation (14),  
\[ \begin{Bmatrix}\mathbf{a}\\ \mathbf{b}\end{Bmatrix} =\mathbf{G}^{-1}\cdot \begin{Bmatrix}\mathbf{u}_{s}\\ 0\end{Bmatrix}  \tag{18}\]  
and substituting Equation (18) into Equation (9), the interpolation is obtained:  
\[u(\mathbf{x}_{I})=\{\mathbf{r}(\mathbf{x}_{I})^{T},\mathbf{p}(\mathbf{x}_{I}) ^{T}\}\cdot\mathbf{G}^{-1}\cdot \begin{Bmatrix}\mathbf{u}_{s}\\ 0\end{Bmatrix} =\{\Phi(\mathbf{x}_{I}),\Psi(\mathbf{x}_{I})\}\cdot \begin{Bmatrix} \mathbf{u}_{s}\\ 0\end{Bmatrix} , \tag{19}\]  
where \(\Phi(\mathbf{x}_{I})\) is the interpolation of the integration point \(\mathbf{x}_{I}\),  
\[\{\Phi(\mathbf{x}_{I}),\Psi(\mathbf{x}_{I})\}=\{\mathbf{r}(\mathbf{x}_{I})^{T },\mathbf{p}(\mathbf{x}_{I})^{T}\}\cdot\mathbf{G}^{-1}=\{\phi_{1}(\mathbf{x}_{ I}),\ldots,\phi_{n}(\mathbf{x}_{I}),\psi_{1}(\mathbf{x}_{I}),\ldots,\psi_{m}( \mathbf{x}_{I})\} \tag{20}\]  
In order to achieve a well-conditioned moment matrix, \(\mathbf{G}\), the number of monomials, \(m\), of the polynomial basis function should be much lower than the number of nodes,\(n\), inside the influence domain of \(\mathbf{x}_{I}\): \(m\ll n\). Such a relation is assured because in 3D analyses, the polynomials have \(m=1\), \(m=4\) and \(m=10\) for constant, linear, and quadratic polynomial basis functions, respectively, and the number of nodes inside the influence domain is typically \(n\geq 27\). The Galerkin weak form depends on the partial derivative of \(\Phi(\mathbf{x}_{I})\). Thus, in regard to a general direction \(\xi\), the partial derivatives can be written as  
\[\Phi_{\xi}(\mathbf{x}_{I})=\{\mathbf{r}(\mathbf{x}_{I})_{\xi}^{T},\mathbf{p}( \mathbf{x}_{I})_{\xi}^{T}\}\cdot\mathbf{G}^{-1}, \tag{21}\]  
with the partial derivatives of the MQ-RBF in order to \(\xi\) being defined as  
\[(r_{ij})_{\xi}=2\cdot p\cdot(d_{ij}^{2}+c^{2})^{p-1}\cdot(\xi_{j}-\xi_{i}). \tag{22}\]  
As the literature shows, RPI functions possess several useful properties, such as the Kronecker delta property and satisfy the partition of unity. Being interpolating functions, they allow one to impose directly the essential and natural boundary conditions, reducing the overall computational cost of the analysis when compared with approximating meshless methods [1].