Commit 928687f8 authored by Martin Schneider's avatar Martin Schneider

[handbook][disc] Update tpfa description and general statements

parent 8ac05303
......@@ -173,27 +173,11 @@ flow and transport equations:
Equation \eqref{eq:discfin} has to be fulfilled for each box $B_j$.
\subsection{Cell Centered Finite Volume Methods -- A Short Introduction}\label{cc}
Cell-centered finite volume methods use the elements of the grid as control volumes.
For each control volume the discrete values are determined at the element/control
volume center (not required to be the barycenters).
The cell-centered finite volume method uses the elements of the grid as control volumes.
For each control volume all discrete values are determined at the element/control
volume center (see Figure~\ref{pc:cc}).
The mass or energy fluxes are evaluated at the integration points ($x_{ij}$),
which are located at the midpoints of the control
volume faces. This is a two point flux approximation since the flux between
the element/control volume centers $i$ and $j$ is calculated
only with information from these two points. In contrast the box method uses
a multi-point flux approximation where all nodes of the
element influence the flux between two specific nodes. \par
Neumann boundary conditions are applied at the boundary control volume faces
and Dirichlet boundary conditions at the boundary control volumes. \par
The cell centered finite volume method is robust and mass conservative but
should only be applied for structured grids
(the control volume face normal vector ($n_{ij}$) should be parallel to the
direction of the gradient between the two element/control
volume centers).
We consider a domain $\Omega \in \mathbb{R}^d$, $d \in \{ 2, 3 \}$ with boundary $\Gamma = \bar{\Omega} / \Omega$ and the following elliptic model problem:
We consider a domain $\Omega \subset \mathbb{R}^d$, $d \in \{ 2, 3 \}$ with boundary $\Gamma = \partial \Omega$. Within this section, we consider the following elliptic problem
\begin{equation}
\begin{aligned}
\nabla \cdot \left( - \mathbf{\Lambda} \nabla u \right) &= q &&\mathrm{in} \, \Omega \\
......@@ -203,26 +187,29 @@ We consider a domain $\Omega \in \mathbb{R}^d$, $d \in \{ 2, 3 \}$ with boundary
\end{aligned}
\end{equation}
Here, $\mathbf{\Lambda} = \mathbf{\Lambda}(\mathbf{x}, \mathbf{u})$ is a symmetric and positive definite tensor of second rank, $u = u (\mathbf{x})$ is unknown and $q = q(\mathbf{x}, \mathbf{u})$ is a source/sink. For the derivation of the finite-volume formulation we integrate the first equation of \eqref{eq:elliptic} over $\Omega$ and apply the Gauss divergence theorem:
Here, $\mathbf{\Lambda} = \mathbf{\Lambda}(\mathbf{x}, \mathbf{u})$ is a symmetric and positive definite tensor of second rank (e.g. permeability, diffusivity, etc.), $u = u (\mathbf{x})$ is unknown and $q = q(\mathbf{x}, \mathbf{u})$ is a source/sink.
We denote by $\mathcal{M}$ the mesh that results from the division of the domain $\Omega$ into $n_e$ control volumes $K \subset \Omega$. Each $K$ is a polygonal open set such that $K \cap L = \emptyset, \forall{K \neq L}$ and $\overline{\Omega} = \cup_{K \in \mathcal{M}} \overline{K}$.
For the derivation of the finite-volume formulation we integrate the first equation of \eqref{eq:elliptic} over a control volume $K$ and apply the Gauss divergence theorem:
\begin{equation}
\int_{\Gamma} \left( - \mathbf{\Lambda} \nabla u \right) \cdot \mathbf{n} \mathrm{d} \Gamma = \int_\Omega q \mathrm{d}\Omega.
\int_{\partial K} \left( - \mathbf{\Lambda} \nabla u \right) \cdot \mathbf{n} \, \mathrm{d} \Gamma = \int_K q \, \mathrm{d}\Omega.
\label{eq:ellipticIntegrated}
\end{equation}
We denote by $\mathcal{M}$ the mesh that results from the division of the domain $\Omega$ into $n_e$ control volumes $K_i$. Each $K_i$ is a polygonal open set and $\mathring{K_i} \mathring{\cap K_j} = \emptyset, \forall{i \neq j}$ and $\Omega = \cup_i^{n_e} K_i$. We then enforce equation \eqref{eq:ellipticIntegrated} to be fulfilled in each control volume, which leads to
Splitting the control volume boundary $\partial K$ into a finite number of faces $\sigma \subset \partial K$ (such that $\sigma = \overline{K} \cap \overline{L}$ for some neighboring control volume $L$) and replacing the exact fluxes by an approximation, i.e. $F_{K, \sigma} \approx \int_{\sigma} \left( - \mathbf{\Lambda} \nabla u \right) \cdot \mathbf{n} \mathrm{d} \Gamma$, yield
\begin{equation}
\sum_{\sigma \in K} F_{K, \sigma} = Q_K, \forall_{K \in \mathcal{M}}
\sum_{\sigma \subset \partial K} F_{K, \sigma} = Q_K, \quad \forall \, {K \in \mathcal{M}},
\label{eq:ccdisc}
\end{equation}
where $F_{K, \sigma} \approx \int_{\sigma} \left( - \mathbf{\Lambda} \nabla u \right) \cdot \mathbf{n} \mathrm{d} \Gamma$ is the discrete flux through a face $\sigma$ of cell $K$ and $Q_k = \int_K q \mathrm{d}x$ is the integrated source/sink term.
Cell-centered finite-volume schemes differ in the way how the term
$(\mathbf{\Lambda} \nabla u ) \cdot \mathbf{n} $ is approximated. Using the symmetry of the tensor $\mathbf{\Lambda}$, this term can be rewritten as
where $F_{K, \sigma}$ is the discrete flux through face $\sigma$ flowing out of cell $K$ and $Q_K := \int_K q \, \mathrm{d}x$ is the integrated source/sink term. Equation \eqref{eq:ccdisc} is the typical cell-centered finite-volume formulation.
Finite-volume schemes differ in the way how the term
$(\mathbf{\Lambda} \nabla u ) \cdot \mathbf{n} $ is approximated (i.e. the choice of the fluxes $F_{K, \sigma}$). Using the symmetry of the tensor $\mathbf{\Lambda}$, this term can be rewritten as
$\nabla u \cdot \mathbf{\Lambda}\mathbf{n}$, which corresponds to the directional derivative of $u$ in co-normal direction $\mathbf{\Lambda}\mathbf{n}$.
In the following, the main ideas of the two-point flux approximation and the multi-point flux approximation methods are briefly described. Hereby, we restrict the discussion to the two-dimensional case.
Please also note that other types of equations, e.g. instationary parabolic problems, can be discretized by applying some time discretization scheme to the time derivatives and by using the finite-volume scheme for the flux discretization. For simplicity the discussion is restricted to the elliptic problem \eqref{eq:elliptic}.
\subsubsection{Tpfa Method}\label{cc_tpfa}
The linear two-point flux approximation is a simple but robust cell-centered finite-volume scheme, which is commonly used in commercial software.
This scheme can be derived by using the conormal decomposition, which reads
......@@ -230,7 +217,15 @@ This scheme can be derived by using the conormal decomposition, which reads
\mathbf{\Lambda}_K \mathbf{n}_{K, \sigma} = t_{K,\sigma} \mathbf{d}_{K,\sigma} + \mathbf{d}^{\bot}_{K,\sigma}, \quad t_{K,\sigma} = \frac{\mathbf{n}_{K, \sigma}^T \mathbf{\Lambda}_K \mathbf{d}_{K,\sigma} }{\mathbf{d}_{K,\sigma}^T \mathbf{d}_{K,\sigma}}, \; \mathbf{d}^{\bot}_{K,\sigma} = \mathbf{\Lambda}_K \mathbf{n}_{K, \sigma} - t_{K,\sigma} \mathbf{d}_{K,\sigma},
\label{eq:conormalDecTpfa}
\end{equation}
with the distance vector $\mathbf{d}_{K,\sigma} := \mathbf{x}_\sigma - \mathbf{x}_K$ and $\mathbf{d}_{K,\sigma}^T \mathbf{d}^{\bot}_{K,\sigma} = 0$. The same can be done for the conormal $\mathbf{\Lambda}_L \mathbf{n}_{L, \sigma}$. The $t_{K,\sigma}$ and $t_{L,\sigma}$ are the transmissibilities associated with the face $\sigma$. These transmissibilities are calculated in \Dumux by using the function \texttt{computeTpfaTransmissibility}.
with the distance vector $\mathbf{d}_{K,\sigma} := \mathbf{x}_\sigma - \mathbf{x}_K$ and $\mathbf{d}_{K,\sigma}^T \mathbf{d}^{\bot}_{K,\sigma} = 0$, see Figure \ref{pc:cctpfa} for the used notations. The same can be done for the conormal $\mathbf{\Lambda}_L \mathbf{n}_{L, \sigma}$. The $t_{K,\sigma}$ and $t_{L,\sigma}$ are the transmissibilities associated with the face $\sigma$. These transmissibilities are calculated in \Dumux by using the function \texttt{computeTpfaTransmissibility}.
\begin{figure} [ht]
\centering
\includegraphics[width=0.4\linewidth,keepaspectratio]{PNG/cctpfa.png}
\caption{Two neighboring control volumes sharing the face $\sigma$.}
\label{pc:cctpfa}
\end{figure}
With these notations, it follows that for each cell $K$ and face $\sigma$
\begin{equation}
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