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<html>
<head>
<title>
TCELL_ROM - TCELL Reduced Order Model Calculation.
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TCELL_ROM <br> Reduced Order Model <br> For Flow in TCELL
</h1>
<hr>
<p>
<b>TCELL_ROM</b>
is a MATLAB program which
handles reduced order
model computations based on a finite element approximation of
the fluid velocity in the time-dependent Navier Stokes equations
in a region called the "T-Cell",
by Professor Hyung-Chun Lee.
</p>
<p>
The reduced model was created by saving 500 "snapshots" of the computed
flow field during a time evolution with varying boundary conditions,
using the TCELL program.
</p>
<p>
Then a small basis set (typically of order 8) was extracted from these
snapshots, using the CVT_BASIS program or
the POD_BASIS program.
</p>
<p>
Some additional information, including the mass matrix and the
initial value of the reduced order coefficients, was then computed
using the TCELL_MASS program.
</p>
<p>
Finally, the evolution of the coefficient vector of the reduced order
model basis set could be computed. This is done by a simple MATLAB
program that calls an ODE solver.
</p>
<p>
The "T-Cell" region is a T-shaped region contained in a subset
of the unit square [01]x[01]. The top half of the region is a
channel open to the left and right, with a rectangular recess in
the bottom. The left channel opening extends from [0.0,0.5] to
[0.0,1.0], and the right channel from [1.0,0.5] to [1.0,1.0].
The recess is a square, whose bottom extends from [0.25,0.0] to
[0.75,0.0], and which extends upward to meet the channel.
<pre>
+--------------------+
+--------------------+
-> ->
Inlet ---> Channel ---> Outlet
-> ->
+-----+ +---+
+----+| Recess |+--+
|| ||
Void || || Void
|+----------+|
+------------+
</pre>
The grid can be thought of as being generated by a uniform 81 by 81
uniformly spaced grid of nodes, from which the nodes corresponding to
the two voids have been deleted. This leaves 4,961 nodes.
These nodes are organized into 2400 6-node triangular elements
which are quadratic in velocity and linear in pressure.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<dl>
<dt>
<b>tcell_rom</b>
</dt>
<dd>
</dd>
</dl>
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>TCELL_ROM</b> is available in
<a href = "../../m_src/tcell_rom/tcell_rom.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/cvt_basis_flow/cvt_basis_flow.html">
CVT_BASIS_FLOW</a>,
a FORTRAN90 program which
extracts dominant solution modes from a set of flow solutions
using CVT methods.
</p>
<p>
<a href = "../../f_src/pod_basis_flow/pod_basis_flow.html">
POD_BASIS_FLOW</a>,
a FORTRAN90 program which
extracts dominant solution modes from a set of flow solutions
using POD methods.
</p>
<p>
<a href = "../../f77_src/tcell/tcell.html">
TCELL</a>,
a FORTRAN77 program which
solves the time-dependent Navier Stokes fluid flow equations
in a 2D T-shaped region.
</p>
<p>
<a href = "../../m_src/tcell_flow_movie/tcell_flow_movie.html">
TCELL_FLOW_MOVIE</a>,
a MATLAB library which
creates an animation of the velocity solutions for the T cell;
</p>
<p>
<a href = "../../f77_src/tcell_mass/tcell_mass.html">
TCELL_MASS</a>,
a FORTRAN77 program which
computes the mass matrix for the T-Cell problem,
which is needed in order to for TCELL_ROM to run the
reduced order model (ROM) of the problem.
</p>
<p>
<a href = "../../m_src/velocity_arrows_grid/velocity_arrows_grid.html">
VELOCITY_ARROWS_GRID</a>,
a MATLAB program which
reads files of node and velocity data, and, using interpolation, creates a
vector plot with arrows
place on a uniform grid of the user's specification.
</p>
<h3 align = "center">
Author:
</h3>
<p>
Hyung-Chun Lee, <br>
Ajou University, Korea,<br>
in collaboration
with Max Gunzburger and John Burkardt.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<dd>
Max Gunzburger,<br>
Finite Element Methods for Viscous Incompressible Flows,<br>
A Guide to Theory, Practice, and Algorithms,<br>
Academic Press, 1989.
</dd>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "tcell_rom.m">tcell_rom.m</a>,
a MATLAB script which reads the data files, calls the ODE
solver, makes plots, and writes the solution to a file.
</li>
<li>
<a href = "tcell_rhs.m">tcell_rhs.m</a>,
a MATLAB function which evaluates the right hand side of
the system of ODE's that defines the time evolution of
the coefficients of the reduced order model.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
A run of the program requires two input files. A sample
run is available, in which 8 basis functions were generated
by the <b>CVT_BASIS</b> program.
<ul>
<li>
<a href = "init_output.txt">init_output.txt</a>,
the initial value of the coefficient vector.
</li>
<li>
<a href = "matrix_output.txt">matrix_output.txt</a>,
the mass matrix and other data needed to define the ODE.
</li>
<li>
<a href = "odesol.txt">odesol.txt</a>,
the value of the coefficient vector at each timestep.
</li>
<li>
<a href = "tcell_rom.png">tcell_rom.png</a>,
a <a href = "../../data/png/png.html">PNG</a> image of
the evolution of the coefficient vector values
over time;
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 01 September 2005.
</i>
<!-- John Burkardt -->
</body>
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</html>