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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 FORTRAN90 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 containing four subroutines,
<ul>
<li>
<b>SUBROUTINE RHS ( N, X, Y, TIME, U )</b>
evaluates the right hand side forcing term F(x,y,t);
</li>
<li>
<b>SUBROUTINE K_COEF ( N, X, Y, TIME, U )</b> evaluates Kx,y,t);
</li>
<li>
<b>SUBROUTINE DIRICHLET_CONDITION ( N, X, Y, TIME, U )</b>
evaluates G(x,y,t), and is only called at nodes on the boundary;
</li>
<li>
<b>SUBROUTINE INITIAL_CONDITION ( N, X, Y, TIME, U )</b>
evaluates H(x,y,t), and is only called for TIME = TINIT.
</li>
</ul>
</p>
<p>
To run the program, the user compiles the user routines,
links them with <b>FEM2D_HEAT</b>, and runs the executable.
</p>
<p>
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.
</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">
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 = "../../f_src/fem1d_heat_steady/fem1d_heat_steady.html">
FEM1D_HEAT_STEADY</a>,
a FORTRAN90 program which
uses the finite element method to solve the steady (time independent)
heat equation in 1D.
</p>
<p>
<a href = "../../f_src/fem2d_heat_rectangle/fem2d_heat_rectangle.html">
FEM2D_HEAT_RECTANGLE</a>,
a FORTRAN90 program which
solves the 2D time dependent heat equation on the unit square,
using a uniform grid of triangular elements.
</p>
<p>
<a href = "../../f_src/fem2d_heat_square/fem2d_heat_square.html">
FEM2D_HEAT_SQUARE</a>,
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.
</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.f90">fem2d_heat.f90</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 6 node triangular element.
</li>
<li>
<b>CH_CAP</b> capitalizes a single character.
</li>
<li>
<b>CH_EQI</b> is a case insensitive comparison of two characters for equality.
</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 an DGB matrix.
</li>
<li>
<b>DGB_MXV</b> multiplies a DGB matrix times a vector.
</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 data from a double precision table file.
</li>
<li>
<b>DTABLE_HEADER_READ</b> reads the header from a double precision 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 filename.
</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>GET_UNIT</b> returns a free FORTRAN unit number.
</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 a integer array.
</li>
<li>
<b>I4COL_SORT_A</b> ascending sorts an integer array of columns.
</li>
<li>
<b>I4COL_SWAP</b> swaps columns I and J of a integer array of column data.
</li>
<li>
<b>I4MAT_TRANSPOSE_PRINT_SOME</b> prints some of the transpose of an I4MAT.
</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>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>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_TO_I4</b> reads an I4 from a string.
</li>
<li>
<b>S_TO_I4VEC</b> reads an integer vector 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>TRIANGLE_AREA_2D</b> computes the area of a triangle in 2D.
</li>
<li>
<b>TRIANGULATION_ORDER6_BOUNDARY_NODE</b> indicates which nodes are 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 = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 29 December 2010.
</i>
<!-- John Burkardt -->
</body>
</html>