TCELL_MASS
TCELL Mass Matrix Calculation.

TCELL_MASS is a FORTRAN77 program which computes and saves mass matrix associated with a finite element approximation of the fluid velocity in the time-dependent Navier Stokes equations in a region called the "T-Cell", by Hyung-Chun Lee.

The actual simulation of successive solutions to the flow problem can be done by the TCELL program, which automatically builds and uses the mass matrix. However, what TCELL_MASS does is allow us to "export" the information about the mass matrix (and several other important quantities) so that a reduced order model of the TCELL flow problem can be solved.

In fact, enough information about the reduced order model will be saved so that the resulting system can be solved by a simple program that invokes a MATLAB 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
                  |+----------+|
                  +------------+
      

Usage:

tcell_mass

runs the program

Author:

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

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

TCELL_MASS is available in a FORTRAN77 version.

Related Data and Programs:

BUMP, a FORTRAN90 program which solves a fluid flow problem in a channel including a bump which obstructs and redirects the flow.

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, a dataset directory which contains 500 successive flow solutions of the T-Cell problem.

Reference:

1. Max Gunzburger,
   Finite Element Methods for Viscous Incompressible Flows,
   A Guide to Theory, Practice, and Algorithms,
   Academic Press, 1989.
2. John Burkardt, Max Gunzburger, Hyung-Chun Lee,
   Centroidal Voronoi Tessellation-Based Reduced-Order Modelling of Complex Systems,
   SIAM Journal on Scientific Computing,
   Volume 28, Number 2, 2006, pages 459-484.

Source Code:

• tcell_mass.f, the source code.
• tcell_mass.sh, commands to compile, link and load the source code.

Examples and Tests:

A typical run of the program requires several input files. Here is a set of command files and input appropriate for a particular run. In this case, the 8 base vectors come from run 56 of CVT_BASIS_FLOW.

• tcell_mass_run1.csh, commands to run the program.
• tcell_mass_output.txt, an output file containing printed messages from the program.
• elenode.dat, an input file containing the number of nodes and elements, the node coordinates, and the node indices that form each element.
• up1st.dat, the solution coefficients for steady state flow with parameter ALPHA = 1.
• up3st.dat, the solution coefficients for steady state flow with parameter ALPHA = 3.
• gen_001.txt, basis vector #1
• gen_002.txt, basis vector #2
• gen_003.txt, basis vector #3
• gen_004.txt, basis vector #4
• gen_005.txt, basis vector #5
• gen_006.txt, basis vector #6
• gen_007.txt, basis vector #7
• gen_008.txt, basis vector #8
• init_output.txt, the first output file, containing one set of ODE coefficients.
• matrix_output.txt, the second output file, containing a second set of ODE coefficients.

List of Routines:

• MAIN is the main program for TCELL_MASS.
• ASSEM1 determines the initial value of the coefficient vector.
• ASSEM2 determines the mass matrix and other coefficients in the ODE.
• GRID_READ reads the node and element information and computes element areas.
• QBF evaluates the quadratic basic functions.
• QUAD13 sets the nodes and weights for 13 point Gauss quadrature.

You can go up one level to the FORTRAN77 source codes.