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<html>
<head>
<title>
FEM2D_HEAT - Finite Element Solution of the Heat Equation on Arbitrary 2D Region
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
FEM2D_HEAT <br>
Finite Element Solution of the Heat Equation <br>
on a Triangulated Region
</h1>
<hr>
<p>
<b>FEM2D_HEAT</b>
is a C++ program which
applies the finite element method to solve
a form of the time-dependent heat equation over an arbitrary
triangulated region.
</p>
<p>
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.
</p>
<p>
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 <a href = "../../f_src/triangulation_refine/triangulation_refine.html">
TRIANGULATION_REFINE</a> refine it to something more reasonable.
</p>
<p>
For the following ridiculously small example:
<pre>
10-11-12
|\ |\
| \ | \
6 7 8 9
| \| \
1-2--3--4-5
</pre>
the node file would be:
<pre>
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
</pre>
and the triangle file would be
<pre>
1 3 10 2 7 6
3 5 12 4 9 8
12 10 3 11 7 8
</pre>
</p>
<p>
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:
<pre>
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.
</pre>
</p>
<p>
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 <i>myprog.C</i>,
containing several functions:
<ul>
<li>
<b>double rhs ( int node_num, double node_xy[], double time )</b>
evaluates the right hand side forcing term F(x,y,t).
</li>
<li>
<b>double k_coef ( int node_num, double node_xy[], double time )</b>
evaluates K(x,y,t);
</li>
<li>
<b>double *dirichlet_condition ( int node_num, double node_xy[], double time )</b>
evaluates G(x,y,t) for all nodes on the boundary;
</li>
<li>
<b>double *initial_condition ( int node_num, double node_xy[], double time )</b>
evaluates H(x,y,t) for all nodes at the initial time.
</li>
</ul>
</p>
<p>
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.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
g++ fem2d_heat.cpp <i>myprog.cpp</i> <br>
mv a.out fem2d_heat<br>
fem2d_heat <i>prefix</i>
</blockquote>
where prefix is the common file prefix:
<ul>
<li>
"prefix"_nodes.txt, contains the node coordinates.
</li>
<li>
"prefix"_elements.txt, contains the indices of nodes that form elements.
</li>
</ul>
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>FEM2D_HEAT</b> is available in
<a href = "../../cpp_src/fem2d_heat/fem2d_heat.html">a C++ version</a> and
<a href = "../../f_src/fem2d_heat/fem2d_heat.html">a FORTRAN90 version</a> and
<a href = "../../m_src/fem2d_heat/fem2d_heat.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Programs:
</h3>
<p>
<a href = "../../cpp_src/fd2d_heat_steady/fd2d_heat_steady.html">
FD2D_HEAT_STEADY</a>,
a C++ program which
uses the finite difference method (FDM) to solve the
steady (time independent) heat equation in 2D.
</p>
<p>
<a href = "../../cpp_src/fem1d_heat_steady/fem1d_heat_steady.html">
FEM1D_HEAT_STEADY</a>,
a C++ program which
uses the finite element method to solve the steady (time independent)
heat equation in 1D.
</p>
<p>
<a href = "../../cpp_src/fem2d_heat_square/fem2d_heat_square.html">
FEM2D_HEAT_SQUARE</a>,
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.
</p>
<p>
<a href = "../../cpp_src/stochastic_heat2d/stochastic_heat2d.html">
STOCHASTIC_HEAT2D</a>,
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.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Hans Rudolf Schwarz,<br>
Finite Element Methods,<br>
Academic Press, 1988,<br>
ISBN: 0126330107,<br>
LC: TA347.F5.S3313.
</li>
<li>
Gilbert Strang, George Fix,<br>
An Analysis of the Finite Element Method,<br>
Cambridge, 1973,<br>
ISBN: 096140888X,<br>
LC: TA335.S77.
</li>
<li>
Olgierd Zienkiewicz,<br>
The Finite Element Method,<br>
Sixth Edition,<br>
Butterworth-Heinemann, 2005,<br>
ISBN: 0750663200,<br>
LC: TA640.2.Z54
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "fem2d_heat.cpp">fem2d_heat.cpp</a>,
the source code;
</li>
<li>
<a href = "fem2d_heat.sh">fem2d_heat.sh</a>,
commands to compile the partial program;
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main routine of FEM2D_HEAT.
</li>
<li>
<b>ASSEMBLE_BACKWARD_EULER</b> adjusts the system for the backward Euler term.
</li>
<li>
<b>ASSEMBLE_BOUNDARY</b> modifies the linear system for the boundary conditions.
</li>
<li>
<b>ASSEMBLE_FEM</b> assembles the finite element system for the heat equation.
</li>
<li>
<b>BANDWIDTH</b> determines the bandwidth of the coefficient matrix.
</li>
<li>
<b>BASIS_11_T6:</b> one basis at one point for the T6 element.
</li>
<li>
<b>CH_CAP</b> capitalizes a single character.
</li>
<li>
<b>CH_EQI</b> is true if two characters are equal, disregarding case.
</li>
<li>
<b>CH_TO_DIGIT</b> returns the integer value of a base 10 digit.
</li>
<li>
<b>DGB_FA</b> performs a LINPACK-style PLU factorization of a DGB matrix.
</li>
<li>
<b>DGB_PRINT_SOME</b> prints some of a DGB matrix.
</li>
<li>
<b>DGB_SL</b> solves a system factored by DGB_FA.
</li>
<li>
<b>DTABLE_DATA_READ</b> reads the data from a real TABLE file.
</li>
<li>
<b>DTABLE_HEADER_READ</b> reads the header from a real TABLE file.
</li>
<li>
<b>FILE_COLUMN_COUNT</b> counts the number of columns in the first line of a file.
</li>
<li>
<b>FILE_NAME_INC</b> increments a partially numeric file name.
</li>
<li>
<b>FILE_NAME_SPECIFICATION</b> determines the names of the input files.
</li>
<li>
<b>FILE_ROW_COUNT</b> counts the number of row records in a file.
</li>
<li>
<b>I4_MAX</b> returns the maximum of two I4's.
</li>
<li>
<b>I4_MIN</b> returns the smaller of two I4's.
</li>
<li>
<b>I4_MODP</b> returns the nonnegative remainder of integer division.
</li>
<li>
<b>I4_WRAP</b> forces an integer to lie between given limits by wrapping.
</li>
<li>
<b>I4COL_COMPARE</b> compares columns I and J of an I4COL.
</li>
<li>
<b>I4COL_SORT_A</b> ascending sorts the columns of an I4COL.
</li>
<li>
<b>I4COL_SWAP</b> swaps two columns of an integer array.
</li>
<li>
<b>I4MAT_TRANSPOSE_PRINT_SOME</b> prints some of an I4MAT, transposed.
</li>
<li>
<b>ITABLE_DATA_READ</b> reads data from an integer TABLE file.
</li>
<li>
<b>ITABLE_HEADER_READ</b> reads the header from an integer TABLE file.
</li>
<li>
<b>I4VEC_PRINT_SOME</b> prints "some" of an integer vector.
</li>
<li>
<b>LVEC_PRINT</b> prints a logical vector.
</li>
<li>
<b>POINTS_PLOT</b> plots a pointset.
</li>
<li>
<b>QUAD_RULE</b> sets the quadrature rule for assembly.
</li>
<li>
<b>R8_ABS</b> returns the absolute value of an R8.
</li>
<li>
<b>R8_HUGE</b> returns a "huge" R8.
</li>
<li>
<b>R8_NINT</b> returns the nearest integer to an R8.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT_SOME</b> prints some of an R8MAT, transposed.
</li>
<li>
<b>R8VEC_PRINT_SOME</b> prints "some" of an R8VEC.
</li>
<li>
<b>REFERENCE_TO_PHYSICAL_T3</b> maps reference points to physical points.
</li>
<li>
<b>S_LEN_TRIM</b> returns the length of a string to the last nonblank.
</li>
<li>
<b>S_TO_I4</b> reads an I4 from a string.
</li>
<li>
<b>S_TO_I4VEC</b> reads an I4VEC from a string.
</li>
<li>
<b>S_TO_R8</b> reads an R8 from a string.
</li>
<li>
<b>S_TO_R8VEC</b> reads an R8VEC from a string.
</li>
<li>
<b>S_WORD_COUNT</b> counts the number of "words" in a string.
</li>
<li>
<b>SOLUTION_WRITE</b> writes the solution to a file.
</li>
<li>
<b>SORT_HEAP_EXTERNAL</b> externally sorts a list of items into ascending order.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>TIMESTRING</b> returns the current YMDHMS date as a string.
</li>
<li>
<b>TRIANGLE_AREA_2D</b> computes the area of a triangle in 2D.
</li>
<li>
<b>TRIANGULATION_ORDER6_BOUNDARY_NODE</b> indicates nodes on the boundary.
</li>
<li>
<b>TRIANGULATION_ORDER6_PLOT</b> plots a 6-node triangulation of a set of nodes.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 13 November 2006.
</i>
<!-- John Burkardt -->
</body>
</html>