# Elastoplastic Analysis of Plates with Radial Point Interpolation Meshless Methods
## 2 Meshless Methods
### The Radial Point Interpolation Method
(2020), for a 2D problem, each influence domain should possess between \(n=9\) and \(n=16\) nodes. Thus, respectively, and in accordance with ([PERSON] et al., 2020), for a 3D problem, each influence domain should possess between \(n=27\) and \(n=64\) nodes. Respecting the same number of nodes inside each influence domain allows the construction of shape functions with the same level of complexity for each integration point \(\mathbf{x}_{I}\). Regarding the construction of the shape functions, RPIM uses the radial point interpolating (RPI) technique. Thus, consider a function \(\mathbf{u}(\mathbf{x})\) defined in the domain \(\Omega\in\mathbb{R}^{3}\) discretized by a set of \(N\) nodes: \(X=\{\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{N}\}\in\mathbb{R}^{3}\). Assuming an integration point, \(\mathbf{x}_{I}\), possessing an influence domain with \(n\) nodes inside, it is possible to obtain the function's interpolated value for \(\mathbf{x}_{I}\) by using the following construction:  
\[ \begin{split} u(\mathbf{x}_{I})&=\sum_{i=1}^{n}r_{ i}(\mathbf{x}_{I})\cdot a_{i}(\mathbf{x}_{I})+\sum_{j=1}^{m}p_{j}(\mathbf{x}_{I}) \cdot b_{j}(\mathbf{x}_{I})=\\ &=\mathbf{r}(\mathbf{x}_{I})^{T}\cdot\mathbf{a}(\mathbf{x}_{I})+ \mathbf{p}(\mathbf{x}_{I})^{T}\cdot\mathbf{b}^{T}(\mathbf{x}_{I})=\{\mathbf{r }(\mathbf{x}_{I})^{T},\mathbf{p}(\mathbf{x}_{I})^{T}\}\bigg{\{}\begin{matrix} \mathbf{a}\\ \mathbf{b}\end{matrix}\bigg{\}},\end{split}  \tag{9}\]  
where \(\mathbf{r}(\mathbf{x}_{I})\) is a radial basis function, \(\mathbf{p}(\mathbf{x}_{I})\) is a polynomial basis function, with \(m\) monomials, and \(a_{i}(\mathbf{x}_{I})\) and \(b_{j}(\mathbf{x}_{I})\) are nonconstant coefficients of \(\mathbf{r}(\mathbf{x}_{I})\) and \(\mathbf{p}(\mathbf{x}_{I})\), respectively. The vectors present in Equation (9) are defined by  
\[ \begin{split}\mathbf{r}(\mathbf{x}_{I})&=\{r_{1}( \mathbf{x}_{I}),r_{2}(\mathbf{x}_{I}),\ldots,r_{n}(\mathbf{x}_{I})\}^{T}\\ \mathbf{p}(\mathbf{x}_{I})&=\{p_{1}(\mathbf{x}_{I}), p_{2}(\mathbf{x}_{I}),\ldots,p_{m}(\mathbf{x}_{I})\}^{T}\\ \mathbf{a}(\mathbf{x}_{I})&=\{a_{1}(\mathbf{x}_{I}), a_{2}(\mathbf{x}_{I}),\ldots,a_{n}(\mathbf{x}_{I})\}^{T}\\ \mathbf{b}(\mathbf{x}_{I})&=\{b_{1}(\mathbf{x}_{I}), b_{2}(\mathbf{x}_{I}),\ldots,b_{m}(\mathbf{x}_{I})\}^{T}.\end{split}  \tag{10}\]  
Complete polynomial basis functions are added to the basis function to ensure the consistency of the RPI functions, e.g., adding to the RBF a linear polynomial ensures \(\mathrm{C}^{1}\) consistency and helps the RPI pass the standard patch test. Generally, low-order polynomial basis are considered, such as  
\[ \begin{split}\mathbf{p}(\mathbf{x}_{I})&=\{1\},m=1 \\ \mathbf{p}(\mathbf{x}_{I})&=\{1,x_{I},y_{I},z_{I} \}^{T},m=4.\\ \mathbf{p}(\mathbf{x}_{I})&=\{1,x_{I},y_{I},z_{I}, x_{I}^{2},x_{I}\cdot y_{I},y_{I}^{2},y_{I}\cdot z_{I},z_{I}^{2},z_{I}\cdot x_{I} \}^{T},m=10.\end{split}  \tag{11}\]  
As recommended in the literature ([PERSON] et al., 2020), for efficiency purposes, in this work, a constant polynomial basis function is applied, \(m=1\). Figure 1: Integration scheme for the RPIM, using rectangular parallelepiped integration cells. Regarding the radial basis function, this work assumes the multiquadratic radial basis function (MQ-RBF), proposed initially by [PERSON] [33],  
\[r_{j}(\mathbf{x}_{i})=r_{ij}=(d_{ij}^{2}+c^{2})^{p}=\left(\left(\sqrt{(x_{j}-x_{i })^{2}+(y_{j}-y_{i})^{2}+(z_{j}-z_{i})^{2}}\right)^{2}+c^{2}\right)^{p}, \tag{12}\]  
in which \(c\) and \(p\) are the MQ-RBF shape parameters. For 3D analyses, the literature [1] shows that \(c\) should be close to zero, but not be zero, and \(p\) should be close to one, but not be one. Therefore, following the literature recommendation, in this work the following MQ-RBF shape parameters are considered: \(c=10^{-4}\) and \(p=1-10^{-4}\). The MQ-RBF shape parameters were obtained by optimizing the Kronecker delta function of the 3D RPI shape functions. The results shown in [1] proved to be accurate and stable, allowing us to reproduce highly complex variable fields.