FEM2D_HEAT_SQUARE
FEM2D_HEAT applied to a SQUARE region

FEM2D_HEAT_SQUARE is a FORTRAN90 library which defines the geometry of a square region, as well as boundary and initial conditions for a given heat problem, and is called by FEM2D_HEAT as part of a solution procedure.

The 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

K = 0,

F = (2*pi*pi-1)*sin(pi*x)*sin(pi*y)*exp(-t).

U = sin(pi*x) * sin(pi*y) * exp(-t).

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 FORTRAN90 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 FORTRAN90 program which uses the finite element method and the backward Euler method to solve the 2D time-dependent heat equation on an arbitrary triangulated region.

Source Code:

• square.f90, the user-supplied routines to evaluate the right hand side, linear coefficient, initial and boundary conditions;

Examples and Tests:

• square.txt, the output file.
• 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;

List of Routines:

• DIRICHLET_CONDITION sets the value of a Dirichlet boundary condition.
• INITIAL_CONDITION sets the initial condition.
• K_COEF evaluates the coefficient K(X,Y,T) function.
• RHS gives the right-hand side of the differential equation.

You can go up one level to the FORTRAN90 source codes.