Finite Element Solution of the Heat Equation
on a Triangulated Region
is a C++ program which
applies the finite element method to solve
a form of the time-dependent heat equation over an arbitrary
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
TRIANGULATION_REFINE refine it to something more reasonable.
For the following ridiculously small example:
| \ | \
6 7 8 9
| \| \
the node file would be:
and the triangle file would be
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, perhaps called myprog.C,
containing several functions:
double rhs ( int node_num, double node_xy, double time )
evaluates the right hand side forcing term F(x,y,t).
double k_coef ( int node_num, double node_xy, double time )
double *dirichlet_condition ( int node_num, double node_xy, double time )
evaluates G(x,y,t) for all nodes on the boundary;
double *initial_condition ( int node_num, double node_xy, double time )
evaluates H(x,y,t) for all nodes at the initial time.
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.
g++ fem2d_heat.cpp myprog.cpp
where prefix is the common file prefix:
mv a.out fem2d_heat
"prefix"_nodes.txt, contains the node coordinates.
"prefix"_elements.txt, contains the indices of nodes that form elements.
The computer code and data files described and made available on this web page
are distributed under
the GNU LGPL license.
FEM2D_HEAT is available in
a C++ version and
a FORTRAN90 version and
a MATLAB version.
a C++ program which
uses the finite difference method (FDM) to solve the
steady (time independent) heat equation in 2D.
a C++ program which
uses the finite element method to solve the steady (time independent)
heat equation in 1D.
a C++ 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.
a C++ program which
implements a finite difference method (FDM) for the steady
(time independent) 2D heat equation,
with a stochastic heat diffusivity coefficient,
using gnuplot to illustrate the results.
Hans Rudolf Schwarz,
Finite Element Methods,
Academic Press, 1988,
Gilbert Strang, George Fix,
An Analysis of the Finite Element Method,
The Finite Element Method,
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 element.
CH_CAP capitalizes a single character.
CH_EQI is true if two characters are equal, disregarding case.
CH_TO_DIGIT returns the integer value of a base 10 digit.
DGB_FA performs a LINPACK-style PLU factorization of a DGB matrix.
DGB_PRINT_SOME prints some of a DGB matrix.
DGB_SL solves a system factored by DGB_FA.
DTABLE_DATA_READ reads the data from a real TABLE file.
DTABLE_HEADER_READ reads the header from a real TABLE file.
FILE_COLUMN_COUNT counts the number of columns in the first line of a file.
FILE_NAME_INC increments a partially numeric file name.
FILE_NAME_SPECIFICATION determines the names of the input files.
FILE_ROW_COUNT counts the number of row records in a file.
I4_MAX returns the maximum of two I4's.
I4_MIN returns the smaller of two I4's.
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 an I4COL.
I4COL_SORT_A ascending sorts the columns of an I4COL.
I4COL_SWAP swaps two columns of an integer array.
I4MAT_TRANSPOSE_PRINT_SOME prints some of an I4MAT, transposed.
ITABLE_DATA_READ reads data from an integer TABLE file.
ITABLE_HEADER_READ reads the header from an integer TABLE file.
I4VEC_PRINT_SOME prints "some" of an integer vector.
LVEC_PRINT prints a logical vector.
POINTS_PLOT plots a pointset.
QUAD_RULE sets the quadrature rule for assembly.
R8_ABS returns the absolute value of an R8.
R8_HUGE returns a "huge" R8.
R8_NINT returns the nearest integer to an R8.
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_LEN_TRIM returns the length of a string to the last nonblank.
S_TO_I4 reads an I4 from a string.
S_TO_I4VEC reads an I4VEC 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 returns the current YMDHMS date as a string.
TRIANGLE_AREA_2D computes the area of a triangle in 2D.
TRIANGULATION_ORDER6_BOUNDARY_NODE indicates nodes on the boundary.
TRIANGULATION_ORDER6_PLOT plots a 6-node triangulation of a set of nodes.
You can go up one level to
the C++ source codes.
Last revised on 13 November 2006.