# Eigendegradation Algorithm Applied to Visco-Plastic Weak Layers
## 3 Time and Spatial Discretization
### Spatial Discretization  
The dynamic problem of a dry soil (mono-phase material) is studied in this research, in which the time is an important issue in the following analyses. The formulation of the dynamic problem can be defined by the governing equation of the balance of the linear momentum:  
\[\text{div}\ \mathbf{\sigma}-\rho\mathbf{a}+\rho\mathbf{g}=\mathbf{0}. \tag{23}\]  
The weak form derivation, following the [PERSON] procedure, needs to multiply (23) by the virtual displacement (a test function) \(\delta\mathbf{u}\), and the integration over the domain. After applying [PERSON]'s theorem, Equation (23) reads as follows:  
\[-\int_{\Omega}\mathbf{\sigma}\,:\,\text{grad}\delta\mathbf{u}\ \text{d}\Omega+\int_{ \Gamma_{t}}\delta\mathbf{u}\cdot\overline{\mathbf{t}}\,\text{d}\gamma-\int_{\Omega} \delta\mathbf{u}\cdot\rho\mathbf{a}\ \text{d}\Omega+\int_{\Omega}\delta\mathbf{u}\cdot\rho\mathbf{g}\ \text{d}\Omega=0 \tag{24}\]  
where \(\Omega\) is the volume of the body, and the boundary where tractions are applied is depicted by \(\Gamma\). The internal forces of the body are represented by the first term of the equation; meanwhile, the external forces are represented by the second and fourth terms. The third one refers to the inertial terms. The next step is the interpolation through the Optimal Transportation Meshfree. As we mentioned, this numerical method is based on a calculation framework of nodes and material points. The shape functions are based on the work of [PERSON] and [PERSON] [57], where the Local Max-Ent shape function (LME) of the material point (\(x\)) with respect to the neighborhood (\(x_{a}\)) is defined as follows:  
\[N_{a}(\mathbf{x})=\frac{\exp\bigl{[}-\beta_{LME}\ |\mathbf{x}-\mathbf{x_{a}}|^{2 }+\mathbf{\lambda}^{*}\cdot(\mathbf{x}-\mathbf{x_{a}})\bigr{]}}{Z(\mathbf{x},\bm {\lambda}^{*}(\mathbf{x}))}, \tag{25}\]  
in which \(Z(\mathbf{x},\mathbf{\lambda})\) is computed along a neighborhood \(N_{b}\) as:  
\[Z(\mathbf{x},\mathbf{\lambda})=\sum_{a=1}^{Nb}\exp\Bigl{[}-\beta_{LME}\ |\mathbf{x}-\mathbf{x_{a}}|^{2}+\mathbf{\lambda}\cdot(\mathbf{x}-\mathbf{x_{a}}) \Bigr{]}. \tag{26}\]  
The derivatives of this shape function can be obtained from the Hessian matrix \(\mathbf{J}\) as follows:  
\[\
abla N_{a}^{*} = -N_{a}^{*}\left(\mathbf{J}^{*}\right)^{-1}(\mathbf{x}-\mathbf{x_ {a}}), \tag{27}\]  
The value of parameter \(\beta_{LME}\) is associated to the shape of the neighborhood as well as the discretization size (or nodal spacing). \(h\). Both parameters are related through parameter \(\gamma_{LME}\), which controls the locality of the shape functions, as it is observed following:  
\[\beta=\frac{\gamma_{LME}}{h^{2}}. \tag{28}\]  
It bears pointing out that \(\mathbf{\lambda}^{*}(\mathbf{x})\) is obtained through the minimization of the function \(g(\mathbf{\lambda})=\log Z(\mathbf{x},\mathbf{\lambda})\) to guarantee the maximum entropy.  
By employing the outlined shape functions and applying Galerkin procedure to the weak form, \(\mathbf{u}\) can be interpolated by employing:  
\[\mathbf{u}\approx\mathbf{u}^{h}=\mathbf{N}_{u}\cdot\tilde{\mathbf{u}} \tag{29}\]where \(\square^{h}\) represents the OTM approximation of the field \(\square\) and \(\square\) represents the nodal values. \(\mathbf{N}_{u}=[N_{1}\mathbf{I},N_{2}\mathbf{I},...,N_{m}\mathbf{I}]\) is the shape function, in which \(m\) is the number of neighbor nodes. The shape functions are defined in the updated configuration, \(\mathbf{N}=\mathbf{N}(\mathbf{x})\). Moreover, the following property helps to calculate time variations: \(\square^{h}=\mathbf{N}\cdot\square\).