FEM2D_HEAT_SQUARE
A Problem in a Square Region for FEM2D_HEAT
FEM2D_HEAT_SQUARE is a MATLAB library which
defines the geometry and other data for the the heat equation
over a square region, for solution by FEM2D_HEAT.
This mesh uses 81 nodes and forms 32 quadratic triangular elements.
We assume that the equation to be solved is
dUdT  Laplacian U + K * U = F
with
K = 0,
and
F = (2*pi*pi1)*sin(pi*x)*sin(pi*y)*exp(t).
The exact solution is:
U = sin(pi*x) * sin(pi*y) * exp(t).
which is the formula we use to define our functions G
(for the boundary conditions) and H (for the initial
conditions).
Licensing:
The computer code and data files described and made available on this web page
are distributed under
the GNU LGPL license.
Languages:
FEM2D_HEAT_SQUARE is available in
a MATLAB version.
Related Data and Programs:
CONTOUR_SEQUENCE4,
a MATLAB program which
makes a sequence of contour plots based on a single XY coordinate file,
and a sequence of Z(X,Y) data files.
FEM2D_HEAT,
a MATLAB program which
uses the finite element method and the backward Euler method to solve the
2D timedependent heat equation on an arbitrary triangulated region.
In order to run, it requires
usersupplied routines that define problem data.
TRIANGULATION_DISPLAY,
a MATLAB program which
displays the nodes and elements of a triangulation on the MATLAB graphics screen;
Reference:

Hans Rudolf Schwarz,
Finite Element Methods,
Academic Press, 1988,
ISBN: 0126330107,
LC: TA347.F5.S3313.

Gilbert Strang, George Fix,
An Analysis of the Finite Element Method,
Cambridge, 1973,
ISBN: 096140888X,
LC: TA335.S77.

Olgierd Zienkiewicz,
The Finite Element Method,
Sixth Edition,
ButterworthHeinemann, 2005,
ISBN: 0750663200,
LC: TA640.2.Z54
Source Code:

dirichlet_condition.m,
the usersupplied routine to evaluate the boundary conditions;

initial_condition.m,
the usersupplied routines to evaluate the initial conditions;

k_coef.m,
the usersupplied routines to evaluate the linear coefficient.

rhs.m,
the usersupplied routines to evaluate the right hand side;

square.out,
output from a run of the program;

square_nodes.png,
a PNG image of
the 49 nodes;

square_nodes.txt,
a text file containing a list, for each node, of its X and Y
coordinates;

square_elements.png,
a PNG image of
the 32 elements;

square_elements.txt,
a text file containing a list, for each element, of the six
nodes that compose it;

square_time.txt,
a text file containing the solution times;

u0000.txt,
the solution U at time step 0;

u0001.txt,
the solution U at time step 1;

u0002.txt,
the solution U at time step 2;

u0003.txt,
the solution U at time step 3;

u0004.txt,
the solution U at time step 4;

u0005.txt,
the solution U at time step 5;

u0006.txt,
the solution U at time step 6;

u0007.txt,
the solution U at time step 7;

u0008.txt,
the solution U at time step 8;

u0009.txt,
the solution U at time step 9;

u0010.txt,
the solution U at time step 10;
The MATLAB program CONTOUR_SEQUENCE4 can make contour
plots from the sequence of solutions:

u0000.png,
the solution U at time step 0;

u0001.png,
the solution U at time step 1;

u0002.png,
the solution U at time step 2;

u0003.png,
the solution U at time step 3;

u0004.png,
the solution U at time step 4;

u0005.png,
the solution U at time step 5;

u0006.png,
the solution U at time step 6;

u0007.png,
the solution U at time step 7;

u0008.png,
the solution U at time step 8;

u0009.png,
the solution U at time step 9;

u0010.png,
the solution U at time step 10;
You can go up one level to
the MATLAB source code page.
Last revised on 11 January 2011.