## 2 Radial Point Interpolation Method
\Bigg{\}}=\boldsymbol{\varphi}(\mathbf{x}_{I})\mathbf{u} _{\mathbf{s}}, \tag{7}\]  
being the shape function defined as  
\[\boldsymbol{\varphi}(\mathbf{x}_{I})=\Big{\{}\mathbf{R}^{T}(\mathbf{x}_{I}), \mathbf{p}^{T}(\mathbf{x}_{I})\Big{\}}\mathbf{G}^{-1}=\{\varphi_{1}(\mathbf{x }_{I}),\varphi_{2}(\mathbf{x}_{I}),\cdots,\varphi_{n}(\mathbf{x}_{I})\}. \tag{8}\]  
In elasticity, the equilibrium conditions can be represented by the following system of partial differential equations:  
\[\triangledown\boldsymbol{\sigma}+\mathbf{b}=0, \tag{9}\]  
where \(\triangledown\) represents the nabla operator, \(\boldsymbol{\sigma}\) the Cauchy stress tensor, and \(\mathbf{b}\) the body force vector. The boundary surface can be divided into two types of boundaries:  
* Natural boundaries (\(\Gamma_{t}\)), where \(\sigma\mathbf{n}=\bar{\mathbf{t}}\). * Essential boundaries (\(\Gamma_{u}\)), where \(\mathbf{u}=\bar{\mathbf{u}}\). where \(\bar{\mathbf{u}}\) represents the prescribed displacement vector on the essential boundary, \(\Gamma_{u}\). Similarly, \(\bar{\mathbf{t}}\) denotes the traction force applied on the natural boundary, \(\Gamma_{t}\), and \(\mathbf{n}\) represents its outward normal unit vector. The Cauchy stress tensor \(\boldsymbol{\sigma}\), usually represented in the Voigt notation (Equation (10)), and the strain \(\boldsymbol{\varepsilon}\) (Equation (11)) can be related with [PERSON]'s law: \(\boldsymbol{\sigma}=\mathbf{c}\cdot\boldsymbol{\varepsilon}\), being the strain state obtained from the displacement field: \(\boldsymbol{\varepsilon}=\mathbf{L}\cdot\mathbf{u}\). In a full 3D problem, the differential operator \(\mathbf{L}\) and the material constitutive matrix \(\mathbf{c}\) are represented as shown in Equations (12) and (13), respectively:  
\[\boldsymbol{\sigma}=\Big{\{}\sigma_{xx}\quad\sigma_{yy}\quad\sigma_{zz}\quad \tau_{xy}\quad\tau_{yz}\quad\tau_{zx}\Big{\}}^{T}, \tag{10}\]  
\[\boldsymbol{\varepsilon}=\Big{\{}\varepsilon_{xx}\quad\epsilon_{yy}\quad \epsilon_{zz}\quad\gamma_{xy}\quad\gamma_{yz}\quad\tau_{zx}\Big{\}}^{T}, \tag{11}\]  
\[\mathbf{L}= \begin{bmatrix}\frac{\partial}{\partial x}&0&0&\frac{\partial}{ \partial y}&0&\frac{\partial}{\partial x}\\ 0&\frac{\partial}{\partial y}&0&\frac{\partial}{\partial x}&\frac{\partial}{ \partial x}&0\\ 0&0&\frac{\partial}{\partial x}&0&\frac{\partial}{\partial y}&\frac{\partial}{ \partial x}\end{bmatrix} ^{T}, \tag{12}\]  
\[\mathbf{c}=\mu_{1} \begin{bmatrix}1&v&v&0&0&0\\ v&1&v&0&0&0\\ v&v&1&0&0&0\\ 0&0&0&\mu_{2}&0&0\\ 0&0&0&0&0&\mu_{2}\end{bmatrix} , \tag{13}\]  
where \(\mu_{1}=\frac{E}{1-\triangledown^{2}}\) and \(\mu_{2}=\frac{1-\triangledown}{2}\). The selection of a methodology for approximating solutions to partial differential equations hinges on the intended application. Within computational mechanics, both strong and weak formulations are widely used approximation methods. While the strong form offers a potentially more direct and accurate approach in certain cases, it can become challenging to implement for problems of greater complexity. In these scenarios, due to its ability to generate a more robust system of equations, using the weak form is more advantageous. Thus, assuming the virtual work principle, the energy conservation is imposed:  
\[\Psi=\int_{\Omega}\delta\boldsymbol{\varepsilon}^{T}\boldsymbol{\sigma}\;d \Omega-\int_{\Omega}\delta\mathbf{u}^{T}\boldsymbol{\sigma}\;b\Omega-\int_{ \Gamma_{t}}\delta\mathbf{u}^{T}\mathbf{t}\;d\Gamma=0, \tag{14}\]which allows, after standard manipulation [7], to obtain the following expression:  
\[\mathbf{\Psi}=\delta\mathbf{u}^{T}(\mathbf{K}\cdot\mathbf{u}-\mathbf{f_{b}}-\mathbf{f_ {t}})=0, \tag{15}\]  
leading to the simplified expression: \(\mathbf{K}\cdot\mathbf{u}=\mathbf{f_{b}}+\mathbf{f_{t}}\).