FEM2D_HEAT_RECTANGLE
Finite Element Solution in 2D
Time Dependent Heat Equation

FEM2D_HEAT_RECTANGLE is a C++ program which solves the time-dependent 2D heat equation using the finite element method in space, and a method of lines in time with the backward Euler approximation for the time derivative.

The computational region is a rectangle, with homogenous Dirichlet boundary conditions applied along the boundary. The state variable U(X,Y,T) is then constrained by:

        Ut - ( Uxx + Uyy ) = F(x,y,t)  in the box;
                  U(x,y,t) = G(x,y,t) for (x,y) on the boundary;
                  U(x,y,t) = H(x,y,t) for t = t_init.
      

The computational region is first covered with an NX by NY rectangular array of points, creating (NX-1)*(NY-1) subrectangles. Each subrectangle is divided into two triangles, creating a total of 2*(NX-1)*(NY-1) geometric "elements". Because quadratic basis functions are to be used, each triangle will be associated not only with the three corner nodes that defined it, but with three extra midside nodes. If we include these additional nodes, there are now a total of (2*NX-1)*(2*NY-1) nodes in the region.

We now assume that, at any fixed time b, the unknown function U(x,y,t) can be represented as a linear combination of the basis functions associated with each node. The value of U at the boundary nodes is obvious, so we concentrate on the NUNK interior nodes where U(x,y,t) is unknown. For each node I, we determine a basis function PHI(I)(x,y), and evaluate the following finite element integral:

        Integral ( Ux(x,y,t) * dPHIdx(I)(x,y) + dUdy(x,y,t) * dPHIdy(I)(x,y) ) =
        Integral ( F(x,y,t) * PHI(I)(x,y)
      

The time derivative is handled by the backward Euler approximation.

The program allows the user to supply two routines:

• rhs ( x, y, time ) returns the right hand side F(x,y,time) of the heat equation.
• exact_u ( node_num, node_xy, time, u_exact ) returns the exact solution U_EXACT evaluated at each of the NODE_NUM points whose coordinates are stored in NODE_XY(1:2,1:NODE_NUM), at time TIME.

There are a few variables that are easy to manipulate. In particular, the user can change the variables NX and NY in the main program, to change the number of nodes and elements. The variables (XL,YB) and (XR,YT) define the location of the lower left and upper right corners of the rectangular region, and these can also be changed in a single place in the main program.

The program writes out a file containing an Encapsulated PostScript image of the nodes and elements, with numbers. Unfortunately, for values of NX and NY over 10, the plot is too cluttered to read. For lower values, however, it is a valuable map of what is going on in the geometry.

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

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

Related Data and Programs:

FEM2D_HEAT, a C++ program which solves the time dependent heat equation on an arbitrary triangulated region in 2D.

Author:

Janet Peterson.

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_rectangle.cpp, the source code;

Examples and Tests:

Data files created by the program:

• rectangle_output.txt, the printed output from a run;
• fem2d_heat_rectangle_nodes.png, a PNG image of the nodes, for NX = NY = 7 (the picture is hard to read for larger values of NX and NY);
• rectangle_nodes.txt, a text file containing a list, for each node, of its X and Y coordinates;
• rectangle_elements.png, a PNG image of the elements, for NX = NY = 7 (the picture is hard to read for larger values of NX and NY);
• rectangle_elements.txt, a text file containing a list, for each element, of the six nodes that compose it;
• rectangle_time.txt, a text file containing the solution times;
• rectangle_u0000.txt, the solution U at time step 0;
• rectangle_u0001.txt, the solution U at time step 1;
• rectangle_u0002.txt, the solution U at time step 2;
• rectangle_u0003.txt, the solution U at time step 3;
• rectangle_u0004.txt, the solution U at time step 4;
• rectangle_u0005.txt, the solution U at time step 5;
• rectangle_u0006.txt, the solution U at time step 6;
• rectangle_u0007.txt, the solution U at time step 7;
• rectangle_u0008.txt, the solution U at time step 8;
• rectangle_u0009.txt, the solution U at time step 9;
• rectangle_u0010.txt, the solution U at time step 10;

The MATLAB program CONTOUR_SEQUENCE4 can make contour plots from the sequence of solutions:

• rectangle_u0000.png, the solution U at time step 0;
• rectangle_u0001.png, the solution U at time step 1;
• rectangle_u0002.png, the solution U at time step 2;
• rectangle_u0003.png, the solution U at time step 3;
• rectangle_u0004.png, the solution U at time step 4;
• rectangle_u0005.png, the solution U at time step 5;
• rectangle_u0006.png, the solution U at time step 6;
• rectangle_u0007.png, the solution U at time step 7;
• rectangle_u0008.png, the solution U at time step 8;
• rectangle_u0009.png, the solution U at time step 9;
• rectangle_u0010.png, the solution U at time step 10;

List of Routines:

• MAIN is the main routine of the finite element program FEM2D_HEAT_RECTANGLE.
• ADJUST_BACKWARD_EULER adjusts the system for the backward Euler term.
• ADJUST_BOUNDARY modifies the linear system for boundary conditions.
• AREA_SET sets the area of each element.
• ASSEMBLE assembles the matrix and right-hand side using piecewise quadratics.
• BANDWIDTH determines the bandwidth of the coefficient matrix.
• COMPARE compares the exact and computed solution at the nodes.
• DGB_FA performs a LINPACK-style PLU factorization of an DGB matrix.
• DGB_PRINT_SOME prints some of a DGB matrix.
• DGB_SL solves a system factored by DGB_FA.
• ELEMENT_WRITE writes the elements to a file.
• ERRORS calculates the error in the L2 and H1-seminorm.
• EXACT_U calculates the exact solution and its first derivatives.
• FILE_NAME_INC increments a partially numeric file name.
• GRID_T6 produces a grid of pairs of 6 node triangles.
• I4_MAX returns the maximum of two ints.
• I4_MIN returns the smaller of two ints.
• I4VEC_PRINT_SOME prints "some" of an I4VEC.
• NODE_BOUNDARY_SET assigns an unknown value index at each node.
• NODES_PLOT plots a pointset.
• NODES_WRITE writes the nodes to a file.
• QBF evaluates the quadratic basis functions.
• QUAD_A sets the quadrature rule for assembly.
• QUAD_E sets a quadrature rule for the error calculation.
• R8_HUGE returns a "huge" R8.
• R8_MAX returns the maximum of two R8's.
• R8_MIN returns the minimum of two R8's.
• R8_NINT returns the nearest integer to an R8.
• R8VEC_PRINT_SOME prints "some" of an R8VEC.
• RHS gives the right-hand side of the differential equation.
• S_LEN_TRIM returns the length of a string to the last nonblank.
• SOLUTION_WRITE writes the solution to a file.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TRIANGULATION_ORDER6_PLOT plots a 6-node triangulation of a pointset.
• XY_SET sets the XY coordinates of the nodes.

You can go up one level to the C++ source codes.