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

FREE_FEM_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 FREE_FEM_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.

Related Programs:

CVT_TRIANGULATION is a FORTRAN90 program which constructs a CVT triangulation for certain regions.

FEM is a data directory which contains a description of the data files that can be used to describe a finite element model.

FEM_50 is a MATLAB program which implements the finite element method in just 50 lines of code.

FEM_50_HEAT is a modified version of FEM_50 suitable for solving the heat equation.

FEM_BASIS_T3_DISPLAY is a MATLAB program which displays a basis function associated with a linear triangle ("T3") mesh.

FEM_BASIS_T6_DISPLAY is a MATLAB program which displays a basis function associated with a quadratic triangle ("T6") mesh.

FEM_IO is a FORTRAN90 library for reading or writing the node, element and data files that define a finite element model.

FEM_TO_TEC is a MATLAB program which can convert an FEM model into a TEC graphics file.

FEM1D, is a FORTRAN90 program which applies the finite element method, with piecewise linear basis functions, to a linear two point boundary value problem;

FEM1D_ADAPTIVE is a FORTRAN90 program which applies the finite element method to a linear two point boundary value problem in a 1D region, using adaptive refinement to improve the solution.

FEM1D_NONLINEAR is a FORTRAN90 program which applies the finite element method to a nonlinear two point boundary value problem in a 1D region.

FEM1D_PMETHOD is a FORTRAN90 program which applies the p-method version of the finite element method to a linear two point boundary value problem in a 1D region.

FEM2D_HEAT is a FORTRAN90 program which solves the heat equation on the unit square, using the finite element method.

FEM2D_PACK is a FORTRAN90 library which contains utilities for 2D finite element calculations.

FEM2D_POISSON is a FORTRAN90 program which solves Poisson's equation on the unit square, using the finite element method.

FEM2D_SAMPLE is a FORTRAN90 library which evaluates a finite element function defined on an order 3 or order 6 triangulation.

FFH_SPARSE is a MATLAB program which solves the time dependent heat equation in an arbitrary triangulated 2D region, using MATLAB's sparse matrix storage format and solver.

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

FREE_FEM_NAVIER_STOKES is a FORTRAN90 program which solves the 2D incompressible Navier Stokes equations in an arbitrary triangulated region.

FREE_FEM_POISSON is a FORTRAN90 program which solves the steady Poisson equation on a triangulated 2D region.

FREE_FEM_STOKES is a FORTRAN90 program which solves the steady Stokes flow equations on a triangulated 2D region.

HOT_PIPE is a MATLAB program which can be run with FEM_50_HEAT.

HOT_POINT is a MATLAB program which can be run with FEM_50_HEAT.

PLOT_POINTS is a FORTRAN90 program which can make a plot of the nodes that define the region.

TABLE is a data file format which is used to store the input and output files used by the program.

TABLE_DELAUNAY is a FORTRAN90 program which constructs a Delaunay triangulation of a region which is described by a set of points.

TABLE_IO is a FORTRAN90 library which supplies the routines used to read the data files.

TRIANGULATION_ORDER6 is a data directory which contains a description of the format for the two files needed to describe an order 6 triangulation.

TRIANGULATION_ORDER6_CONTOUR is a MATLAB program which can make contour plots of the computed solution.

TRIANGULATION_PLOT is a FORTRAN90 program which can display an image of the triangulation used by the program.

TRIANGULATION_REFINE is a FORTRAN90 program which can refine a triangulation.

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:

• free_fem_heat.f90, the source code;
• free_fem_heat.csh, commands to compile the partial program;

Examples and Tests:

• square is a simple square region used for testing.

List of Routines:

• MAIN is the main routine of FREE_FEM_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.
• TIMESTRING writes the current YMDHMS date into a string.
• 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.