Commit d76ced51 by Turnhout, M.C. van

### 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: -\grad p\ten{I} + \grad\cdot 2\eta\ten{D} = \vec{0}\label{Seq} 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: \rho\left(\vec{v}\cdot\grad\vec{v}\right) = -\grad p \ten{I} + \grad\cdot 2\eta\ten{D} \label{NSseq} And in combination with \kwc{femnlt} (section \ref{enginefemnlt}), \kwc{elup} solves the instationary Navier-Stokes 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} for incompressible fluids \cite[section 2.2]{Oomens2019}: These equations are all solved for incompressible (Newtonian) fluids \cite[section 2.2]{Oomens2019}: \grad\cdot\vec{v} = 0 \label{NSincmp} 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: \ten{D} = \frac{1}{2}\left( \grad\vec{v} + \left(\grad\vec{v}\right)^\mathrm{T}\right) ~\\ 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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