TCELL_ROM Reduced Order Model For Flow in TCELL

TCELL_ROM 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.

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.

Then a small basis set (typically of order 8) was extracted from these snapshots, using the CVT_BASIS program or the POD_BASIS program.

Some additional information, including the mass matrix and the initial value of the reduced order coefficients, was then computed using the TCELL_MASS program.

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.

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.

```             +--------------------+
+--------------------+
->                      ->
Inlet --->      Channel       --->  Outlet
->                      ->
+-----+          +---+
+----+|  Recess  |+--+
||          ||
Void ||          || Void
|+----------+|
+------------+
```
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.

tcell_rom

Languages:

TCELL_ROM is available in a MATLAB version.

Related Data and Programs:

CVT_BASIS_FLOW, a FORTRAN90 program which extracts dominant solution modes from a set of flow solutions using CVT methods.

POD_BASIS_FLOW, a FORTRAN90 program which extracts dominant solution modes from a set of flow solutions using POD methods.

TCELL, a FORTRAN77 program which solves the time-dependent Navier Stokes fluid flow equations in a 2D T-shaped region.

TCELL_FLOW_MOVIE, a MATLAB library which creates an animation of the velocity solutions for the T cell;

TCELL_MASS, 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.

VELOCITY_ARROWS_GRID, 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.

Author:

Hyung-Chun Lee,
Ajou University, Korea,
in collaboration with Max Gunzburger and John Burkardt.

Reference:

Max Gunzburger,
Finite Element Methods for Viscous Incompressible Flows,
A Guide to Theory, Practice, and Algorithms,

Source Code:

• tcell_rom.m, a MATLAB script which reads the data files, calls the ODE solver, makes plots, and writes the solution to a file.
• tcell_rhs.m, 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.

Examples and Tests:

A run of the program requires two input files. A sample run is available, in which 8 basis functions were generated by the CVT_BASIS program.

• init_output.txt, the initial value of the coefficient vector.
• matrix_output.txt, the mass matrix and other data needed to define the ODE.
• odesol.txt, the value of the coefficient vector at each timestep.
• tcell_rom.png, a PNG image of the evolution of the coefficient vector values over time;

You can go up one level to the MATLAB source codes.

Last revised on 01 September 2005.