FEM2D_HEAT
Finite Element Solution of the Heat Equation
on a Triangulated Region
FEM2D_HEAT
is a C++ program which
applies the finite element method to solve
a form of the timedependent 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:
101112
\ \
 \  \
6 7 8 9
 \ \
12345
the node file would be:
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
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 )
evaluates K(x,y,t);

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.
Usage:
g++ fem2d_heat.cpp myprog.cpp
mv a.out fem2d_heat
fem2d_heat prefix
where prefix is the common file prefix:

"prefix"_nodes.txt, contains the node coordinates.

"prefix"_elements.txt, contains the indices of nodes that form elements.
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 C++ program which
uses the finite difference method (FDM) to solve the
steady (time independent) heat equation in 2D.
FEM1D_HEAT_STEADY,
a C++ program which
uses the finite element method to solve the steady (time independent)
heat equation in 1D.
FEM2D_HEAT_SQUARE,
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.
STOCHASTIC_HEAT2D,
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.
Reference:

Hans Rudolf Schwarz,
Finite Element Methods,
Academic Press, 1988,
ISBN: 0126330107,
LC: TA347.F5.S3313.

Gilbert Strang, George Fix,
An Analysis of the Finite Element Method,
Cambridge, 1973,
ISBN: 096140888X,
LC: TA335.S77.

Olgierd Zienkiewicz,
The Finite Element Method,
Sixth Edition,
ButterworthHeinemann, 2005,
ISBN: 0750663200,
LC: TA640.2.Z54
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 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 LINPACKstyle 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 6node triangulation of a set of nodes.
You can go up one level to
the C++ source codes.
Last revised on 13 November 2006.