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<html>
<head>
<title>
STOCHASTIC_HEAT2D - 2D Steady Heat Equation with Stochastic Diffusivity
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
STOCHASTIC_HEAT2D <br> 2D Steady Heat Equation with Stochastic Diffusivity
</h1>
<hr>
<p>
<b>STOCHASTIC_HEAT2D</b>
is a C++ program which
solves the steady state heat equation in a 2D
rectangular region with a stochastic heat diffusivity,
using the finite difference method (FDM), and stochastic model
from Babuska, Nobile and Tempone, using GNUPLOT to illustrate
the results.
</p>
<p>
The physical region, and the boundary conditions, are suggested
by this diagram:
<pre>
U = 0, Y = 1.0
+-------------------+
| ... |
| .:*:. |
U = 0 | ::: | U = 0
X = 0.0 | | X = 1.0
| |
+-------------------+
U = 0, Y = 0.0
</pre>
</p>
<p>
A heat source is applied, which is concentrated at the point
X = 0.6, Y = 0.8, whose center is suggested by the "*" in the diagram.
</p>
<p>
The region is covered with a grid of NX by NY nodes, and an NX by NY
array U is used to record the temperature. The correspondence between
array indices and locations in the region is suggested by giving the
indices of the four corners:
<pre>
I = NY
+------------------+
| |
J = 1 | | J = NX
| |
+------------------+
I = 1
</pre>
</p>
<p>
The form of the steady heat equation is
<pre>
- d/dx K(x,y) du/dx - d/dy K(x,y) du/dy = F(x,y)
</pre>
where K(x,y) is the heat conductivity, and F(x,y) is a
heat source term. For this program, the heat conductivity
function is assumed to be a function of four stochastic parameters
OMEGA(1) through OMEGA(4); the form of the functional dependence
is described in a paper by Babusk, Nobile and Tempone.
</p>
<p>
By using a simple finite difference approximation, this single equation
can be replaced by NX * NY linear equations in NX * NY variables;
each equation is associated with one of the nodes in the mesh. Nodes
long the boundary generate boundary condition equations, while interior
nodes generate equations that approximate the steady heat equation.
</p>
<p>
Choosing a particular set of values for OMEGA produces a single "realization"
of the temperature field. However, an interesting question is to observe
the influence of the OMEGA's on the solution U. As an example, the program
fixes the values of OMEGA(3) and OMEGA(4), evaluates the solution U over
a grid of values for OMEGA(1) and OMEGA(2), and plots the mean 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>STOCHASTIC_HEAT2D</b> is available in
<a href = "../../c_src/stochastic_heat2d/stochastic_heat2d.html">a C version</a> and
<a href = "../../cpp_src/stochastic_heat2d/stochastic_heat2d.html">a C++ version</a> and
<a href = "../../f77_src/stochastic_heat2d/stochastic_heat2d.html">a FORTRAN77 version</a> and
<a href = "../../f_src/stochastic_heat2d/stochastic_heat2d.html">a FORTRAN90 version</a> and
<a href = "../../m_src/stochastic_heat2d/stochastic_heat2d.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<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 1D Time Independent
Heat Equations.
</p>
<p>
<a href = "../../cpp_src/fem2d_heat/fem2d_heat.html">
FEM2D_HEAT</a>,
a C++ program which
solves the 2D time dependent heat equation on the unit square.
</p>
<p>
<a href = "../../cpp_src/gnuplot/gnuplot.html">
GNUPLOT</a>,
C++ programs which illustrate how a program can write
data and command files so that gnuplot can create plots of the program results.
</p>
<p>
<a href = "../../cpp_src/heated_plate/heated_plate.html">
HEATED_PLATE</a>,
a C++ program which
solves the steady state heat equation in a 2D
rectangular region, and is intended as
a starting point for a parallel version.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Ivo Babuska, Fabio Nobile, Raul Tempone,<br>
A Stochastic Collocation Method for Elliptic Partial Differential Equations
with Random Input Data,<br>
SIAM Journal on Numerical Analysis,<br>
Volume 45, Number 3, 2007, pages 1005-1034.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "stochastic_heat2d.cpp">stochastic_heat2d.cpp</a>, the source code.
</li>
<li>
<a href = "stochastic_heat2d.hpp">stochastic_heat2d.hpp</a>, the include file.
</li>
<li>
<a href = "stochastic_heat2d.sh">stochastic_heat2d.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "stochastic_heat2d_prb.cpp">stochastic_heat2d_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "stochastic_heat2d_prb.sh">stochastic_heat2d_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "stochastic_heat2d_prb_output.txt">stochastic_heat2d_prb_output.txt</a>,
the output file.
</li>
<li>
<a href = "solution_data.txt">solution_data.txt</a>
graphics data for an image of the solution corresponding to an example selection of OMEGA.
</li>
<li>
<a href = "solution_commands.txt">solution_commands.txt</a>
GNUPLOT commands to create an image of the solution corresponding
to an example selection of OMEGA.
</li>
<li>
<a href = "solution.png">solution.png</a>
an image of the solution corresponding to an example selection of OMEGA.
</li>
<li>
<a href = "umean_data.txt">umean_data.txt</a>
graphics data to create an image of mean temperature as a function of OMEGA(1) and OMEGA(2).
</li>
<li>
<a href = "umean_commands.txt">umean_commands.txt</a>
GNUPLOT commands to create an image of mean temperature as a function of OMEGA(1) and OMEGA(2).
</li>
<li>
<a href = "umean.png">umean.png</a>
an image of mean temperature as a function of OMEGA(1) and OMEGA(2).
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>DIFFUSIVITY_2D_BNT</b> evaluates a 2D stochastic diffusivity function.
</li>
<li>
<b>INTERIOR</b> sets up the matrix and right hand side at interior nodes.
</li>
<li>
<b>R8_UNIFORM_01</b> returns a unit pseudorandom R8.
</li>
<li>
<b>R8MAT_FS</b> factors and solves a system with one right hand side.
</li>
<li>
<b>R8VEC_LINSPACE</b> creates a vector of linearly spaced values.
</li>
<li>
<b>R8VEC_MESH_2D</b> creates a 2D mesh from X and Y vectors.
</li>
<li>
<b>R8VEC_NORMAL_01</b> returns a unit pseudonormal R8VEC.
</li>
<li>
<b>R8VEC_PRINT</b> prints an R8VEC.
</li>
<li>
<b>R8VEC_UNIFORM_01</b> returns a unit pseudorandom R8VEC.
</li>
<li>
<b>STOCHASTIC_HEAT2D</b> solves the steady 2D heat equation.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</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 04 September 2013.
</i>
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