Commit d76ced51 authored by Turnhout, M.C. van's avatar Turnhout, M.C. van
Browse files

update/fix elup description (close issue #5)

parent fa6d0886
......@@ -563,24 +563,30 @@ This section intentionally left blank.
\section{2D pressure-velocity element: \kwc{elup}}\label{elup}
This element is used with \kwc{femlin\_e} and \kwc{femnlt} and solves the stationary Stokes equation (\kwc{femlin\_e}, section \ref{enginefemlin_ef})
This element is used with \kwc{femlin\_e}, \kwc{femnl}, and \kwc{femnlt}.
In combination with \kwc{femlin\_e} (section \ref{enginefemlin_ef}), \kwc{elup} solves the stationary Stokes equation:
\begin{equation}
-\grad p\ten{I} + \grad\cdot 2\eta\ten{D} = \vec{0}\label{Seq}
\end{equation}
or the instationary Navier-Stokes equation (\kwc{femnlt}, section \ref{enginefemnlt}).
In combination with \kwc{femnl} (section \ref{enginefemnl}), \kwc{elup} solves the stationary Navier-Stokes equation:
\begin{equation}
\rho\left(\vec{v}\cdot\grad\vec{v}\right) = -\grad p \ten{I} + \grad\cdot 2\eta\ten{D} \label{NSseq}
\end{equation}
And in combination with \kwc{femnlt} (section \ref{enginefemnlt}), \kwc{elup} solves the instationary Navier-Stokes equation:
\begin{equation}
\rho\left(\pderiv{\vec{v}}{t}+\vec{v}\cdot\grad\vec{v}\right) = -\grad \ten{I}p + \grad\cdot 2\eta\ten{D} \label{NSeq}
\rho\left(\pderiv{\vec{v}}{t}+\vec{v}\cdot\grad\vec{v}\right) = -\grad p\ten{I} + \grad\cdot 2\eta\ten{D} \label{NSeq}
\end{equation}
for incompressible fluids \cite[section 2.2]{Oomens2019}:
These equations are all solved for incompressible (Newtonian) fluids \cite[section 2.2]{Oomens2019}:
\begin{equation}
\grad\cdot\vec{v} = 0 \label{NSincmp}
\end{equation}
The tensor $\ten{D}$ is the rate of deformation tensor in these equations:
And the tensor $\ten{D}$ is the rate of deformation tensor in these equations:
\begin{equation}
\ten{D} = \frac{1}{2}\left( \grad\vec{v} + \left(\grad\vec{v}\right)^\mathrm{T}\right)
\end{equation}
~\\ Mandatory input consists of the parameter $\eta$ in equations \ref{Seq}--\ref{NSeq}, the parameter $\rho$ for equation \ref{NSeq} (i.e.\ when used with \kwc{femnlt}), a parameter \kwo{axi} related to the problem geometry, and a parameter \kwo{ietype} related to the topology of the element (table \ref{inputelup}). \\
~\\ Mandatory input consists of the parameter $\eta$ in equations \ref{Seq}--\ref{NSeq}, the parameter $\rho$ for equations \ref{NSseq}--\ref{NSeq} (i.e.\ when used with \kwc{femnl} or \kwc{femnlt}), a parameter \kwo{axi} related to the problem geometry, and a parameter \kwo{ietype} related to the topology of the element (table \ref{inputelup}). \\
\begin{table}[h]
\center
......@@ -589,9 +595,9 @@ or the instationary Navier-Stokes equation (\kwc{femnlt}, section \ref{enginef
\hline\noalign{\smallskip}
parameter & description & comment\\
\noalign{\smallskip}\hline\noalign{\bigskip}
`engine' & \texttt{\kwc{femlin\_e}} or \texttt{\kwc{femnlt}} & sections \ref{enginefemlin_e} \& \ref{enginefemnlt}\\
`engine' & \texttt{\kwc{femlin\_e}}, \texttt{\kwc{femnl}} or \texttt{\kwc{femnlt}} & sections \ref{enginefemlin_e}--\ref{enginefemnlt}\\
\texttt{\kwo{mat.mat}(\kwo{iimat}, 1)} & viscosity $\eta$ for equations \ref{Seq} --\ref{NSeq} & \\
\texttt{\kwo{mat.mat}(\kwo{iimat}, 2)} & density $\rho$ for equation \ref{NSeq} & for \kwc{femnlt}\\
\texttt{\kwo{mat.mat}(\kwo{iimat}, 2)} & density $\rho$ for equations \ref{NSseq}--\ref{NSeq} & for \kwc{femnl}, \kwc{femnlt}\\
\texttt{\kwo{mat.mat}(\kwo{iimat}, 3)} & \texttt{\kwo{axi}} $= 0$ for a planar geometry & `default' \\
& \texttt{\kwo{axi}} $= 1$ for an axi-symmetric geometry & \\
\texttt{\kwo{mat.mat}(\kwo{iimat}, 9)} & \texttt{\kwo{dt}} & for \kwc{femnlt}\\
......@@ -603,7 +609,7 @@ parameter & description & comment\\
& \texttt{\kwo{ietype}} $= 5$ for Crouzeix-Raviart triangles & figure \ref{eluptriCR} \\
& \texttt{\kwo{ietype}} $= 6$ for linear quads with mid-point & figure \ref{elupbilin} \\
& \texttt{\kwo{ietype}} $= 7$ for linear quads with mid-point without bubble & figure \ref{elupbilin} \\
\texttt{\kwo{mat.types}(\kwo{iitype}, :)} & \texttt{'}\kwc{elup}\texttt{'} & for \kwc{femnl}/\kwc{femnlt}\\
\texttt{\kwo{mat.types}(\kwo{iitype}, :)} & \texttt{'}\kwc{elup}\texttt{'} & \\
\noalign{\smallskip}\hline
\end{tabularx}
\end{table}
......@@ -640,7 +646,7 @@ Taylor-Hood quad & \ref{elupquadTH} & quadratic & linear & $Q_2Q_1$ & 2.4a\\
\item this element adds extra degrees of freedom for interpolation of the pressure to the elements as they produced by \kwc{crmesh} (figure \ref{elelup}, table \ref{elupref}).
\item so each node has two degrees of freedom for velocity, an each element has at least one node with additional degrees of freedom for the pressure (figure \ref{elelup}, table \ref{elupref}).
\item the parameters \kwo{dt} and \kwo{theta} (but not \kwo{ntime}) for time-dependent solutions (\kwo{istat} $ =2$) with \kwc{femnlt} are stored in \kwo{mat.mat} (section \ref{enginefemnlt}).
\item this element uses the updated Lagrange method.
\item this element uses the updated Lagrange method when it is used for incompressible solids.
\item the linear quads with mid-point (\kwo{ietype} $> 5$, figure \ref{elupbilin}) do not satisfy the \textsl{inf-sup} condition \cite[section 2.3.2]{Oomens2019}.
\item this element is used in:
\begin{itemize}
......
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