[cig-commits] r7943 - doc/CitcomS/manual
tan2 at geodynamics.org
tan2 at geodynamics.org
Tue Sep 11 17:14:47 PDT 2007
Author: tan2
Date: 2007-09-11 17:14:46 -0700 (Tue, 11 Sep 2007)
New Revision: 7943
Modified:
doc/CitcomS/manual/citcoms.lyx
Log:
Update the numercial method
Modified: doc/CitcomS/manual/citcoms.lyx
===================================================================
--- doc/CitcomS/manual/citcoms.lyx 2007-09-11 18:03:49 UTC (rev 7942)
+++ doc/CitcomS/manual/citcoms.lyx 2007-09-12 00:14:46 UTC (rev 7943)
@@ -553,10 +553,7 @@
.cfg
\family default
files on input, which are easier to create and read.
-\end_layout
-
-\begin_layout Standard
-Other improvements included the incorporation of geoid calculations that
+ Other improvements included the incorporation of geoid calculations that
had been left out of earlier releases, as well as new scripts to allow
results to be visualized with
\begin_inset LatexCommand \htmlurl[MayaVi2]{svn.enthought.com/enthought/wiki/MayaVi}
@@ -763,11 +760,11 @@
\begin_layout Standard
\align right
\begin_inset Formula \begin{equation}
-\rho C_{P}\left(T_{,t}+u_{i}T_{,i}\right)=\rho C_{P}\kappa T_{,ii}+latent+\rho\alpha gu_{r}\left(T+T_{0}\right)+\Phi+\rho H\label{eq:conservation of energy}\end{equation}
+\rho c_{P}\left(T_{,t}+u_{i}T_{,i}\right)=\rho c_{P}\kappa T_{,ii}+latentheat+\rho\alpha gu_{r}\left(T+T_{0}\right)+\Phi+\rho H\label{eq:conservation of energy}\end{equation}
\end_inset
-
+h
\end_layout
\begin_layout Standard
@@ -813,7 +810,7 @@
\end_inset
is the temperature at the surface,
-\begin_inset Formula $C_{P}$
+\begin_inset Formula $c_{P}$
\end_inset
is the heat capacity,
@@ -832,7 +829,7 @@
\begin_inset Formula $\Phi$
\end_inset
- is the viscous dissipation (TODO: unfinised) and
+ is the viscous dissipation (TODO: latent heat unfinished) and
\emph on
H
\emph default
@@ -866,8 +863,8 @@
\end_inset
is time.
- With phase transitions and temperature and composition variation, the density
- anomalies are:
+ With phase transitions and temperature and composition variations, the
+ density anomalies are:
\end_layout
\begin_layout Standard
@@ -1009,7 +1006,7 @@
\begin_layout Standard
\align right
\begin_inset Formula \begin{equation}
-H=C_{P0}\Delta TH^{'}\label{eq:9}\end{equation}
+H=c_{P0}\Delta TH^{'}\label{eq:9}\end{equation}
\end_inset
@@ -1051,7 +1048,7 @@
\end_inset
is a reference viscosity,
-\begin_inset Formula $C_{P0}$
+\begin_inset Formula $c_{P0}$
\end_inset
is the reference heat capacity, and
@@ -1074,7 +1071,7 @@
\begin_layout Standard
\begin_inset Formula \begin{equation}
--P_{,i}+\left(\eta(u_{i,j}+u_{j,i}-\frac{2}{3}u_{k,k}\delta_{ij})\right)_{,i}+(RaT+Rab\Gamma-RacC)\delta_{ir}=0\label{eq:13}\end{equation}
+-P_{,i}+\left(\eta(u_{i,j}+u_{j,i}-\frac{2}{3}u_{k,k}\delta_{ij})\right)_{,i}+(Ra\bar{\rho}T+Rab\Gamma-RacC)\delta_{ir}=0\label{eq:13}\end{equation}
\end_inset
@@ -1283,13 +1280,22 @@
\end_inset
for the explicit entries.
+
+\end_layout
+
+\begin_layout Standard
+In the incompressible case,
+\begin_inset Formula $\mathbf{C}$
+\end_inset
+
+ is zero.
Equation
\begin_inset LatexCommand \ref{eq:20}
\end_inset
can be transformed by premultiplying by
-\begin_inset Formula $\mathrm{\left(\mathbf{B}^{\mathit{T}}+\mathbf{C}\right)\mathit{\mathbf{A}^{\mathbf{\mathit{-1}}}}}$
+\begin_inset Formula $\mathrm{\mathbf{B}^{\mathit{T}}\mathit{\mathbf{A}^{\mathbf{\mathit{-1}}}}}$
\end_inset
and using equation
@@ -1303,7 +1309,7 @@
\begin_layout Standard
\align right
\begin_inset Formula \begin{equation}
-\left(\mathbf{B}^{\mathit{T}}+\mathbf{C}\right)\mathit{\mathbf{A}^{\mathbf{\mathit{-1}}}\mathbf{B}\mathit{p=\mathbf{\left(\mathbf{B}^{\mathit{T}}+\mathbf{C}\right)A^{\mathit{-1}}\mathit{f}}}}\label{eq:21}\end{equation}
+\mathbf{\mathbf{B}^{\mathit{T}}}\mathbf{A}^{\mathbf{\mathit{-1}}}\mathbf{B}\mathit{p=\mathbf{\mathbf{B}^{\mathit{T}}A^{\mathit{-1}}\mathit{f}}}\label{eq:21}\end{equation}
\end_inset
@@ -1311,6 +1317,7 @@
\end_layout
\begin_layout Standard
+\noindent
This equation is solved using the Uzawa algorithm, an established method
for solving the minimization of a dual function
\begin_inset LatexCommand \cite{Cahouet/Chabard Some fast 3D}
@@ -1323,10 +1330,69 @@
\end_inset
- or a bi-conjugate gradient stablized scheme is used for this iteration
- and forms the basis for the technique used in CitComS.
+ is used for this iteration and forms the basis for the technique used in
+ CitComS.
\end_layout
+\begin_layout Standard
+In the incompressible case, there are two different stratgies to solve equation
+
+\begin_inset LatexCommand \ref{eq:19}
+
+\end_inset
+
+and
+\begin_inset LatexCommand \ref{eq:20}
+
+\end_inset
+
+.
+ The first strategy is to add another iteration when solving equation
+\begin_inset LatexCommand \ref{eq:21}
+
+\end_inset
+
+.
+ The right-hand-side vector is updated by the velocity solution of previous
+ iteraion.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{equation}
+\mathbf{\mathbf{B}^{\mathit{T}}}\mathbf{A}^{\mathbf{\mathit{-1}}}\mathbf{B}\mathit{p^{(i)}=\mathbf{\mathbf{B}^{\mathit{T}}A^{\mathit{-1}}\mathit{\left(f+\mathbf{C}u^{(i-1)}\right)}}}\label{eq:21}\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\noindent
+This equation can be solved using the same conjugate gradient scheme as
+ the incompressible case.
+ The second startegy is to transfrom equation
+\begin_inset LatexCommand \ref{eq:20}
+
+\end_inset
+
+ by premultiplying by
+\begin_inset Formula $\mathrm{\left(\mathbf{B}^{\mathit{T}}+\mathbf{C}\right)\mathit{\mathbf{A}^{\mathbf{\mathit{-1}}}}}$
+\end_inset
+
+ and using equation
+\begin_inset LatexCommand \ref{eq:19}
+
+\end_inset
+
+ to eliminate the velocity unknowns:
+\begin_inset Formula \begin{equation}
+\left(\mathbf{B}^{\mathit{T}}+\mathbf{C}\right)\mathit{\mathbf{A}^{\mathbf{\mathit{-1}}}\mathbf{B}\mathit{p=\mathbf{\left(\mathbf{B}^{\mathit{T}}+\mathbf{C}\right)A^{\mathit{-1}}\mathit{f}}}}\label{eq:21}\end{equation}
+
+\end_inset
+
+This equation is solved using a bi-conjugate gradient stablized scheme.
+\end_layout
+
\begin_layout Section
Meshes and Geometry
\end_layout
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