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

FEM2D_HEAT is a FORTRAN90 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 a file containing four subroutines,

• SUBROUTINE RHS ( N, X, Y, TIME, U ) evaluates the right hand side forcing term F(x,y,t);
• SUBROUTINE K_COEF ( N, X, Y, TIME, U ) evaluates Kx,y,t);
• SUBROUTINE DIRICHLET_CONDITION ( N, X, Y, TIME, U ) evaluates G(x,y,t), and is only called at nodes on the boundary;
• SUBROUTINE INITIAL_CONDITION ( N, X, Y, TIME, U ) evaluates H(x,y,t), and is only called for TIME = TINIT.

To run the program, the user compiles the user routines, links them with FEM2D_HEAT, and runs the executable.

The program writes out a file containing an Encapsulated PostScript image of the nodes and elements, with numbers. If there are too many nodes, the plot may be 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 is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Programs:

FD2D_HEAT_STEADY, a FORTRAN90 program which uses the finite difference method (FDM) to solve the steady (time independent) heat equation in 2D.

FEM1D_HEAT_STEADY, a FORTRAN90 program which uses the finite element method to solve the steady (time independent) heat equation in 1D.

FEM2D_HEAT_RECTANGLE, a FORTRAN90 program which solves the 2D time dependent heat equation on the unit square, using a uniform grid of triangular elements.

FEM2D_HEAT_SQUARE, 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.

STOCHASTIC_HEAT2D, a FORTRAN90 program which implements a finite difference method (FDM) for the steady (time independent) 2D heat equation, with a stochastic heat diffusivity coefficient.

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.f90, the source code;

List of Routines:

• MAIN is the main routine of FEM2D_HEAT.
• ASSEMBLE_BACKWARD_EULER adjusts the system for the backward Euler term.
• ASSEMBLE_BOUNDARY modifies the linear system for the boundary conditions.
• ASSEMBLE_FEM assembles the finite element system for the heat equation.
• BANDWIDTH determines the bandwidth of the coefficient matrix.
• BASIS_11_T6: one basis at one point for the T6 6 node triangular element.
• CH_CAP capitalizes a single character.
• CH_EQI is a case insensitive comparison of two characters for equality.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• DGB_FA performs a LINPACK-style PLU factorization of an DGB matrix.
• DGB_MXV multiplies a DGB matrix times a vector.
• DGB_PRINT_SOME prints some of a DGB matrix.
• DGB_SL solves a system factored by DGB_FA.
• DTABLE_DATA_READ reads data from a double precision table file.
• DTABLE_HEADER_READ reads the header from a double precision table file.
• FILE_COLUMN_COUNT counts the number of columns in the first line of a file.
• FILE_NAME_INC increments a partially numeric filename.
• FILE_NAME_SPECIFICATION determines the names of the input files.
• FILE_ROW_COUNT counts the number of row records in a file.
• GET_UNIT returns a free FORTRAN unit number.
• I4_MODP returns the nonnegative remainder of integer division.
• I4_WRAP forces an integer to lie between given limits by wrapping.
• I4COL_COMPARE compares columns I and J of a integer array.
• I4COL_SORT_A ascending sorts an integer array of columns.
• I4COL_SWAP swaps columns I and J of a integer array of column data.
• I4MAT_TRANSPOSE_PRINT_SOME prints some of the transpose of an I4MAT.
• ITABLE_DATA_READ reads data from an integer table file.
• ITABLE_HEADER_READ reads the header from an integer table file.
• LVEC_PRINT prints a logical vector.
• POINTS_PLOT plots a pointset.
• QUAD_RULE sets the quadrature rule for assembly.
• R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT, transposed.
• R8VEC_PRINT_SOME prints "some" of an R8VEC.
• REFERENCE_TO_PHYSICAL_T3 maps reference points to physical points.
• S_TO_I4 reads an I4 from a string.
• S_TO_I4VEC reads an integer vector from a string.
• S_TO_R8 reads an R8 from a string.
• S_TO_R8VEC reads an R8VEC from a string.
• S_WORD_COUNT counts the number of "words" in a string.
• SOLUTION_WRITE writes the solution to a file.
• SORT_HEAP_EXTERNAL externally sorts a list of items into ascending order.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TRIANGLE_AREA_2D computes the area of a triangle in 2D.
• TRIANGULATION_ORDER6_BOUNDARY_NODE indicates which nodes are on the boundary.
• TRIANGULATION_ORDER6_PLOT plots a 6-node triangulation of a set of nodes.

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