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<html>
<head>
<title>
FEM1D_HEAT_STEADY - Finite Element Method, 1D, Steady State Heat Equation
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
FEM1D_HEAT_STEADY <br> Finite Element Method, 1D, Steady State Heat Equation
</h1>
<hr>
<p>
<b>FEM1D_HEAT_STEADY</b>
is a C++ program which
applies the finite element method, with piecewise linear elements,
to solve the steady state heat equation in one spatial dimension.
</p>
<p>
The finite element method is used with piecewise linear elements.
</p>
<p>
The steady state heat equation that is to be solved has the form:
<pre>
- d/dx ( k(x) * du/dx ) = f(x)
</pre>
in the interval A < x < B. The functions k(x) and f(x) are
given.
</p>
<p>
Boundary conditions are applied at the endpoints, and in this case,
these are assumed to have the form:
<pre>
u(A) = UA;
u(B) = UB.
</pre>
</p>
<p>
To compute a finite element approximation, a set of N equally spaced
nodes is defined from A to B, a set of piecewise linear basis functions
is set up, with one basis function associated with each node,
and then the differential
equation is multiplied by each basis function, and integration by parts is
used to simplify the integrand.
</p>
<p>
A simple two point Gauss quadrature formula is used to estimate the
resulting integrals over each interval.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
call <b>fem1d_heat_steady</b> ( <i>n</i>, <i>a</i>, <i>b</i>, <i>ua</i>, <i>ub</i>,
<i>k</i>, <i>f</i>, <i>x</i>, <i>u</i> )
</blockquote>
where
<ul>
<li>
<i>n</i> is the number of equally spaced nodes.
</li>
<li>
<i>a, b</i> are the left and right endpoints.
</li>
<li>
<i>ua, ub</i> are the prescribed values of U at the endpoints;
</li>
<li>
<i>k</i> is the function which evaluates k(x);
</li>
<li>
<i>f</i> is the function which evaluates f(x).
</li>
<li>
<i>x</i> is the input vector of <i>n</i> nodes, with x(1) = a, x(n) = b.
</li>
<li>
<i>u</i> is the output vector containing the estimated solution
value at the nodes.
</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>FEM1D_HEAT_STEADY</b> is available in
<a href = "../../c_src/fem1d_heat_steady/fem1d_heat_steady.html">a C version</a> and
<a href = "../../cpp_src/fem1d_heat_steady/fem1d_heat_steady.html">a C++ version</a> and
<a href = "../../f77_src/fem1d_heat_steady/fem1d_heat_steady.html">a FORTRAN77 version</a> and
<a href = "../../f_src/fem1d_heat_steady/fem1d_heat_steady.html">a FORTRAN90 version</a> and
<a href = "../../m_src/fem1d_heat_steady/fem1d_heat_steady.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/fd1d_heat_steady/fd1d_heat_steady.html">
FD1D_HEAT_STEADY</a>,
a C++ program which
applies the finite difference method
to the steady state heat equation in one spatial dimension.
</p>
<p>
<a href = "../../data/fem1d/fem1d.html">
FEM1D</a>,
a data directory which
contains examples of 1D FEM files,
three text files that describe a 1D finite element model;
</p>
<p>
<a href = "../../cpp_src/fem1d/fem1d.html">
FEM1D</a>,
a C++ program which
applies the finite element method to a linear two point boundary value problem
in a 1D region.
</p>
<p>
<a href = "../../cpp_src/fem1d_adaptive/fem1d_adaptive.html">
FEM1D_ADAPTIVE</a>,
a C++ program which
applies the finite
element method to a linear two point boundary value problem
in a 1D region, using adaptive refinement to improve the solution.
</p>
<p>
<a href = "../../cpp_src/fem1d_bvp_linear/fem1d_bvp_linear.html">
FEM1D_BVP_LINEAR</a>,
a C++ program which
applies the finite
element method to a linear two point boundary value problem
in a 1D region, using piecewise linear elements.
</p>
<p>
<a href = "../../cpp_src/fem1d_nonlinear/fem1d_nonlinear.html">
FEM1D_NONLINEAR</a>,
a C++ program which
applies the finite element method to a nonlinear two point boundary value problem
in a 1D region.
</p>
<p>
<a href = "../../cpp_src/fem1d_pack/fem1d_pack.html">
FEM1D_PACK</a>,
a C++ library which
contains utilities for 1D finite element calculations.
</p>
<p>
<a href = "../../cpp_src/fem1d_pmethod/fem1d_pmethod.html">
FEM1D_PMETHOD</a>,
a C++ program which
applies the p-method version of the finite element method to a linear
two point boundary value problem in a 1D region.
</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>
Dianne O'Leary,<br>
Finite Differences and Finite Elements: Getting to Know You,<br>
Computing in Science and Engineering,<br>
Volume 7, Number 3, May/June 2005.
</li>
<li>
Dianne O'Leary,<br>
Scientific Computing with Case Studies,<br>
SIAM, 2008,<br>
ISBN13: 978-0-898716-66-5,<br>
LC: QA401.O44.
</li>
<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 = "fem1d_heat_steady.cpp">fem1d_heat_steady.cpp</a>, the source code.
</li>
<li>
<a href = "fem1d_heat_steady.hpp">fem1d_heat_steady.hpp</a>, the include file.
</li>
<li>
<a href = "fem1d_heat_steady.sh">fem1d_heat_steady.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "fem1d_heat_steady_prb.cpp">fem1d_heat_steady_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "fem1d_heat_steady_prb.sh">fem1d_heat_steady_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "fem1d_heat_steady_prb_output.txt">fem1d_heat_steady_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>FEM1D_HEAT_STEADY</b> solves the steady 1D heat equation with finite elements.
</li>
<li>
<b>I4VEC_ZERO_NEW</b> creates and zeroes an I4VEC.
</li>
<li>
<b>R8_ABS</b> returns the absolute value of an R8.
</li>
<li>
<b>R8MAT_SOLVE2</b> computes the solution of an N by N linear system.
</li>
<li>
<b>R8MAT_ZERO_NEW</b> returns a new zeroed R8MAT.
</li>
<li>
<b>R8VEC_EVEN</b> returns an R8VEC of values evenly spaced between ALO and AHI.
</li>
<li>
<b>R8VEC_ZERO_NEW</b> creates and zeroes an R8VEC.
</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 09 April 2011.
</i>
<!-- John Burkardt -->
</body>
</html>