### [handbook] rename domain G to Omega

parent 12b07f0b
 \section{Spatial Discretization Schemes} \label{spatialdiscretization} We discretize space with the cell-centered finite volume method (\ref{box} ), the box method (\ref{cc}) We discretize space with the cell-centered finite volume method (\ref{cc} ), the box method (\ref{box}) or a staggered grid scheme. Grid adaption is available for both box and cell-centered finite volume method. Note that the current implementation only ensures mass conservation for incompressible fluids. In general, the spatial parameters, especially the porosity, have to be assigned on the coarsest level of discretization. ... ... @@ -13,7 +12,7 @@ the coarsest level of discretization. The so called box method unites the advantages of the finite-volume (FV) and finite-element (FE) methods. First, the model domain $G$ is discretized with a FE mesh consisting of nodes First, the model domain $\Omega$ is discretized with a FE mesh consisting of nodes $i$ and corresponding elements $E_k$. Then, a secondary FV mesh is constructed by connecting the midpoints and barycenters of the elements surrounding node $i$ creating a box $B_i$ around node $i$ (see Figure \ref{pc:box}a). ... ... @@ -46,17 +45,17 @@ In the following, the discretization of the balance equation is going to be deri From the \textsc{Reynolds} transport theorem follows the general balance equation: \begin{equation} \underbrace{\int_G \frac{\partial}{\partial t} \: u \: dG}_{1} + \underbrace{\int_{\partial G} (\mathbf{v} u + \mathbf w) \cdot \textbf n \: d\varGamma}_{2} = \underbrace{\int_G q \: dG}_{3} \underbrace{\int_\Omega \frac{\partial}{\partial t} \: u \: dx}_{1} + \underbrace{\int_{\partial\Omega} (\mathbf{v} u + \mathbf w) \cdot \textbf n \: d\varGamma}_{2} = \underbrace{\int_\Omega q \: dx}_{3} \end{equation} \begin{equation} f(u) = \int_G \frac{\partial u}{\partial t} \: dG + \int_{G} \nabla \cdot \underbrace{\left[ \mathbf{v} u + \mathbf w(u)\right] }_{F(u)} \: dG - \int_G q \: dG = 0 f(u) = \int_\Omega \frac{\partial u}{\partial t} \: dx + \int_{\Omega} \nabla \cdot \underbrace{\left[ \mathbf{v} u + \mathbf w(u)\right] }_{F(u)} \: dx - \int_\Omega q \: dx = 0 \end{equation} where term 1 describes the changes of entity $u$ within a control volume over time, term 2 the advective, diffusive and dispersive fluxes over the interfaces of the control volume and term 3 is the source and sink term. $G$ denotes the of the control volume and term 3 is the source and sink term. $\Omega$ denotes the model domain and $F(u) = F(\mathbf v, p) = F(\mathbf v(x,t), p(x,t))$. Like the FE method, the box method follows the principle of weighted residuals. ... ... @@ -107,43 +106,43 @@ of the residual $\varepsilon$ with a weighting function $W_j$ and claiming that this product has to vanish within the whole domain, \begin{equation} \int_G W_j \cdot \varepsilon \: \overset {!}{=} \: 0 \qquad \textrm{with} \qquad \sum_j W_j =1 \int_\Omega W_j \cdot \varepsilon \: \overset {!}{=} \: 0 \qquad \textrm{with} \qquad \sum_j W_j =1 \end{equation} yields the following equation: \begin{equation} \int_G W_j \frac{\partial \tilde u}{\partial t} \: dG + \int_G W_j \cdot \left[ \nabla \cdot F(\tilde u) \right] \: dG - \int_G W_j \cdot q \: dG = \int_G W_j \cdot \varepsilon \: dG \: \overset {!}{=} \: 0 . \int_\Omega W_j \frac{\partial \tilde u}{\partial t} \: dx + \int_\Omega W_j \cdot \left[ \nabla \cdot F(\tilde u) \right] \: dx - \int_\Omega W_j \cdot q \: dx = \int_\Omega W_j \cdot \varepsilon \: dx \: \overset {!}{=} \: 0 . \end{equation} Then, the chain rule and the \textsc{Green-Gaussian} integral theorem are applied. \begin{equation} \int_G W_j \frac{\partial \sum_i N_i \hat u_i}{\partial t} \: dG + \int_{\partial G} \left[ W_j \cdot F(\tilde u)\right] \cdot \mathbf n \: d\varGamma_G + \int_G \nabla W_j \cdot F(\tilde u) \: dG - \int_G W_j \cdot q \: dG = 0 \int_\Omega W_j \frac{\partial \sum_i N_i \hat u_i}{\partial t} \: dx + \int_{\partial\Omega} \left[ W_j \cdot F(\tilde u)\right] \cdot \mathbf n \: d\varGamma_\Omega + \int_\Omega \nabla W_j \cdot F(\tilde u) \: dx - \int_\Omega W_j \cdot q \: dx = 0 \end{equation} A mass lumping technique is applied by assuming that the storage capacity is reduced to the nodes. This means that the integrals $M_{i,j} = \int_G W_j \: N_i \: dG$ reduced to the nodes. This means that the integrals $M_{i,j} = \int_\Omega W_j \: N_i \: dx$ are replaced by the mass lumping term $M^{lump}_{i,j}$ which is defined as: \begin{equation} M^{lump}_{i,j} =\begin{cases} \int_G W_j \: dG = \int_G N_i \: dG = V_i &i = j\\ M^{lump}_{i,j} =\begin{cases} \int_\Omega W_j \: dx = \int_\Omega N_i \: dx = V_i &i = j\\ 0 &i \neq j\\ \end{cases} \end{equation} where $V_i$ is the volume of the FV box $B_i$ associated with node $i$. The application of this assumption in combination with $\int_G W_j \:q \: dG = V_i \: q$ yields $\int_\Omega W_j \:q \: dx = V_i \: q$ yields \begin{equation} V_i \frac{\partial \hat u_i}{\partial t} + \int_{\partial G} \left[ W_j \cdot F(\tilde u)\right] \cdot \mathbf n \: d\varGamma_G + \int_G \nabla W_j \cdot F(\tilde u) \: dG- V_i \cdot q = 0 \, . + \int_{\partial\Omega} \left[ W_j \cdot F(\tilde u)\right] \cdot \mathbf n \: d\varGamma_\Omega + \int_\Omega \nabla W_j \cdot F(\tilde u) \: dx- V_i \cdot q = 0 \, . \end{equation} Defining the weighting function $W_j$ to be piecewisely constant over a ... ...
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