FEM2D_HEAT
Finite Element Solution of the Heat Equation
on a Triangulated Region

FEM2D_HEAT is a MATLAB program which applies the finite element method to solve a form of the time-dependent heat equation over an arbitrary triangulated region.

The computational region is initially unknown by the program. The user specifies it by preparing a file containing the coordinates of the nodes, and a file containing the indices of nodes that make up triangles that form a triangulation of the region.

Normally, the user does not type in this information by hand, but has a program fill in the nodes, and perhaps another program that constructs the triangulation. However, in the simplest case, the user might construct a very crude triangulation by hand, and have TRIANGULATION_REFINE refine it to something more reasonable.

For the following ridiculously small example:

       10-11-12
        |\   |\
        | \  | \
        6  7 8  9
        |   \|   \
        1-2--3--4-5
      

         0.0  0.0
         1.0  0.0
         2.0  0.0
         3.0  0.0
         4.0  0.0
         0.0  1.0
         1.0  1.0
         2.0  1.0
         3.0  1.0
         0.0  2.0
         1.0  2.0
         2.0  2.0
      

         1  3 10  2  7  6
         3  5 12  4  9  8
        12 10  3 11  7  8
      

The program is set up to handle the time dependent heat equation with a right hand side function, and nonhomogeneous Dirichlet boundary conditions. The state variable U(T,X,Y) is then constrained by:

        Ut - ( Uxx + Uyy ) + K(x,y,t) * U = F(x,y,t)  in the region
                                        U = G(x,y,t)  on the boundary
                                        U = H(x,y,t)  at initial time TINIT.
      

To specify the right hand side function F(x,y,t), the linear coefficient K(x,y,t), the boundary condition function G(x,y,t), and the initial condition H(x,y,t), the user has to supply the functions,

• FUNCTION F = RHS ( NODE_NUM, NODE_XY, TIME ) evaluates the right hand side forcing term F(x,y,t);
• FUNCTION K = K_COEF ( NODE_NUM, NODE_XY, TIME ) evaluates K(x,y,t);
• FUNCTION G = DIRICHLET_CONDITION ( NODE_NUM, NODE_XY, TIME ) evaluates G(x,y,t) for nodes on the boundary;
• FUNCTION H = INITIAL_CONDITION ( NODE_NUM, NODE_XY, TIME ) evaluates H(x,y,t), and is only called for TIME = TINIT.

The program writes out a file containing the solution value at every node. This file may be used to create contour plots of the solution.

Program Usage:

fem2d_heat ( 'prefix' )

• 'prefix_nodes.txt' is the name of the node file;
• 'prefix_elements.txt' is the name of the element file;

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 is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

FEM2D_HEAT_SQUARE is a MATLAB 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.

TRIANGULATION_DISPLAY, a MATLAB program which displays the nodes and elements of a triangulation on the MATLAB graphics screen;

Reference:

1. Hans Rudolf Schwarz,
   Finite Element Methods,
   Academic Press, 1988,
   ISBN: 0126330107,
   LC: TA347.F5.S3313.
2. Gilbert Strang, George Fix,
   An Analysis of the Finite Element Method,
   Cambridge, 1973,
   ISBN: 096140888X,
   LC: TA335.S77.
3. Olgierd Zienkiewicz,
   The Finite Element Method,
   Sixth Edition,
   Butterworth-Heinemann, 2005,
   ISBN: 0750663200,
   LC: TA640.2.Z54

Source Code:

• fem2d_heat.m is the main routine.

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