## 2 Methods
### Elemental Strain-Rate Calculation  
Having defined the continuous space of velocities, we can calculate the derivatives along each of the spatial directions and derive the strain-rate tensor field. Following the continuum mechanics concepts in [23] and assuming small deformations, the _strain-rate tensor_ is calculated as  
\[\mathbf{E}\left(\mathbf{x},t^{n}\right):= \frac{1}{2}\left(\
abla\mathbf{v}\left(\mathbf{x},t^{n}\right)+\left( \
abla\mathbf{v}\left(\mathbf{x},t^{n}\right)\right)^{\top}\right),\]  
with \(\
abla\mathbf{v}\) the gradient of velocity. Each component of the \(3\times 3\)-tensor is developed in Cartesian coordinates as  
\[ \begin{bmatrix}E_{11}&E_{12}&E_{13}\\ E_{21}&E_{22}&E_{23}\\ E_{31}&E_{32}&E_{33}\end{bmatrix} = \begin{bmatrix}\frac{\partial\gamma_{1} \left(\mathbf{x},t^{n}\right)}{\partial x_{1}}&\frac{1}{2}\left(\frac{\partial \gamma_{1}\left(\mathbf{x},t^{n}\right)}{\partial x_{2}}+\frac{\partial\gamma_{2} \left(\mathbf{x},t^{n}\right)}{\partial x_{1}}\right)&\frac{1}{2}\left(\frac{ \partial\gamma_{1}\left(\mathbf{x},t^{n}\right)}{\partial x_{3}}+\frac{\partial \gamma_{2}\left(\mathbf{x},t^{n}\right)}{\partial x_{1}}\right)\\ \frac{1}{2}\left(\frac{\partial\gamma_{2}\left(\mathbf{x},t^{n}\right)}{\partial x _{1}}+\frac{\partial\gamma_{1}\left(\mathbf{x},t^{n}\right)}{\partial x_{2}}\right) &\frac{\partial\gamma_{2}\left(\mathbf{x},t^{n}\right)}{\partial x_{2}}&\frac{1}{2 }\left(\frac{\partial\gamma_{2}\left(\mathbf{x},t^{n}\right)}{\partial x_{3}}+ \frac{\partial\gamma_{2}\left(\mathbf{x},t^{n}\right)}{\partial x_{2}}\right)\\ \frac{1}{2}\left(\frac{\partial\gamma_{3}\left(\mathbf{x},t^{n}\right)}{\partial x _{1}}+\frac{\partial\gamma_{1}\left(\mathbf{x},t^{n}\right)}{\partial x_{3}} \right)&\frac{1}{2}\left(\frac{\partial\gamma_{3}\left(\mathbf{x},t^{n}\right)}{ \partial x_{2}}+\frac{\partial\gamma_{2}\left(\mathbf{x},t^{n}\right)}{\partial x _{3}}\right)&\frac{\partial\gamma_{3}\left(\mathbf{x},t^{n}\right)}{\partial x_{3}} \end{bmatrix} .\]  
Moreover, the six independent components of the strain-rate tensor can be arranged using _Voigt's_ notation into a six-component strain-rate vector as follows:  
\[\mathbf{E}\left(\mathbf{x},t^{n}\right)= \begin{bmatrix}E_{11}\left(\mathbf{x},t^{n}\right)& E_{22}\left(\mathbf{x},t^{n}\right)&E_{33}\left(\mathbf{x},t^{n}\right)&\gamma_{12} \left(\mathbf{x},t^{n}\right)&\gamma_{23}\left(\mathbf{x},t^{n}\right)&\gamma_{31} \left(\mathbf{x},t^{n}\right)\end{bmatrix} ^{\top}, \tag{7}\]  
where \(\gamma_{12}\left(\mathbf{x},t\right)=2E_{12}\left(\mathbf{x},t\right),\gamma_{23} \left(\mathbf{x},t\right)=2E_{23}\left(\mathbf{x},t\right)\) and \(\gamma_{13}\left(\mathbf{x},t\right)=2E_{13}\left(\mathbf{x},t\right)\) are the _shear-rate strains_. With this notation in hand, the strain-rate tensor can be calculated as  
\[\mathbf{E}\left(\mathbf{x},t^{n}\right)= \begin{bmatrix}E_{11}\left(\mathbf{x},t^{n}\right) \\ E_{22}\left(\mathbf{x},t^{n}\right)\\ E_{33}\left(\mathbf{x},t^{n}\right)\\ \gamma_{12}\left(\mathbf{x},t^{n}\right)\\ \gamma_{23}\left(\mathbf{x},t^{n}\right)\\ \gamma_{31}\left(\mathbf{x},t^{n}\right)\end{bmatrix} = \begin{bmatrix}\frac{\partial} {\partial x_{1}}&0&0\\ 0&\frac{\partial}{\partial x_{2}}&0\\ 0&0&\frac{\partial}{\partial x_{3}}\\ \frac{\partial}{\partial x_{2}}&\frac{\partial}{\partial x_{1}}&0\\ 0&\frac{\partial}{\partial x_{3}}&\frac{\partial}{\partial x_{2}}\\ \frac{\partial}{\partial x_{3}}&0&\frac{\partial}{\partial x_{1}}\end{bmatrix} \begin{bmatrix}v_{1}\left(\mathbf{x},t^{n}\right)\\ v_{2}\left(\mathbf{x},t^{n}\right)\\ v_{3}\left(\mathbf{x},t^{n}\right)\end{bmatrix} , \tag{8}\]  
by defining the matrix operator of derivatives over the velocity field. In the case of the right hand side velocities, we can arrange a node-wise vector of discrete velocities in the tetrahedron \(K\), as  
\[\mathbf{V}\left(K,t^{n}\right)=