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<html>
<head>
<title>
FEM1D_PROJECT - Project Data onto a 1D Finite Element Mesh
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
FEM1D_PROJECT <br> Project Data onto a 1D Finite Element Mesh
</h1>
<hr>
<p>
<b>FEM1D_PROJECT</b>
is a C++ program which
computes the projection of sample data onto a given finite element mesh.
</p>
<p>
This procedure can be used to determine the least squares approximant to
data (actually, to the piecewise linear interpolant of that data) or to
determine the finite element coefficients appropriate when recomputing
a finite element solution from a fine mesh to a coarse mesh.
</p>
<p>
The sample data is given simply as two files, one containing the points,
the other, the corresponding data values.
</p>
<p>
The user must also supply two files defining the finite element mesh,
containing the points, and the pairs of points that form the elements.
</p>
<p>
The program takes the sample data, projects it onto the finite element mesh,
and determines a file of finite element coefficients that represent the
projection of the data onto the finite element space.
</p>
<p>
The current version of the program can only handle finite element
meshes which are made of piecewise linear basis functions.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>fem1d_project</b> <i>sample_prefix</i> <i>fem_prefix</i>
</blockquote>
where <i>sample_prefix</i> is the common prefix for the SAMPLE files:
<ul>
<li>
<i>sample_prefix</i><b>_nodes.txt</b>, the node coordinates.
</li>
<li>
<i>sample_prefix</i><b>_values.txt</b>, the data values;
</li>
</ul>
and <i>fem_prefix</i> is the common prefix for the FEM files:
<ul>
<li>
<i>fem_prefix</i><b>_nodes.txt</b>, the node coordinates.
</li>
<li>
<i>fem_prefix</i><b>_elements.txt</b>, the nodes that make up each element;
</li>
<li>
<i>fem_prefix</i><b>_values.txt</b>, created by the program, and contains
the projections of the data values onto the finite element space, that is,
these may be regarded as coefficients of finite element functions
representing the projections of the data.
</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_PROJECT</b> is available in
<a href = "../../c_src/fem1d_project/fem1d_project.html">a C version</a> and
<a href = "../../cpp_src/fem1d_project/fem1d_project.html">a C++ version</a> and
<a href = "../../f77_src/fem1d_project/fem1d_project.html">a FORTRAN77 version</a> and
<a href = "../../f_src/fem1d_project/fem1d_project.html">a FORTRAN90 version</a> and
<a href = "../../m_src/fem1d_project/fem1d_project.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<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 1D linear two point boundary value problem.
</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 1D linear two point boundary value problem
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, with piecewise linear elements,
to a two point boundary value problem in one spatial dimension.
</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/fem1d_nonlinear/fem1d_nonlinear.html">
FEM1D_NONLINEAR</a>,
a C++ program which
applies the finite element method to a 1D nonlinear two point boundary value problem.
</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 1D linear
two point boundary value problem.
</p>
<p>
<a href = "../../cpp_src/fem1d_sample/fem1d_sample.html">
FEM1D_SAMPLE</a>,
a C++ program which
samples a scalar or vector finite element function of one variable,
defined by FEM files,
returning interpolated values at the sample points.
</p>
<p>
<a href = "../../cpp_src/fem2d_project/fem2d_project.html">
FEM2D_PROJECT</a>,
a C++ program which
projects a function F(X,Y), given as a data, into a given finite element space
of piecewise linear triangular elements.
</p>
<p>
<a href = "../../cpp_src/fem2d_project_function/fem2d_project_function.html">
FEM2D_PROJECT_FUNCTION</a>,
a C++ program which
projects a function F(X,Y), given as a formula, into a given finite element space
of piecewise linear triangular elements.
</p>
<p>
<a href = "../../cpp_src/fem3d_project/fem3d_project.html">
FEM3D_PROJECT</a>,
a C++ program which
projects a function F(X,Y,Z), given as a data, into a given finite element space
of piecewise linear tetrahedral elements.
</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 = "fem1d_project.cpp">fem1d_project.cpp</a>, the source code.
</li>
<li>
<a href = "fem1d_project.sh">fem1d_project.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>POWERS</b> starts with sample data for the vector function f(x)=[ 1, x, x^2], on a grid
of 11 equally spaced nodes from 0 to 10, and projects this onto a piecewise linear
grid of 6 equally spaced nodes from 0 to 10.
<ul>
<li>
<a href = "sample_powers_nodes.txt">sample_powers_nodes.txt</a>,
the sample nodes.
</li>
<li>
<a href = "sample_powers_values.txt">sample_powers_values.txt</a>,
the sample nodal values.
</li>
<li>
<a href = "fem_powers_nodes.txt">fem_powers_nodes.txt</a>,
the FEM nodes.
</li>
<li>
<a href = "fem_powers_elements.txt">fem_powers_elements.txt</a>,
the FEM elements.
</li>
<li>
<a href = "fem_powers_values.txt">fem_powers_values.txt</a>,
the projected FEM nodal values.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for FEM1D_PROJECT.
</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>FEM1D_APPROXIMATE</b> approximates data at sample points with an FEM function.
</li>
<li>
<b>FILE_COLUMN_COUNT</b> counts the number of columns in the first line of a file.
</li>
<li>
<b>FILE_ROW_COUNT</b> counts the number of row records in a file.
</li>
<li>
<b>I4MAT_DATA_READ</b> reads data from an I4MAT file.
</li>
<li>
<b>I4MAT_HEADER_READ</b> reads the header from an I4MAT file.
</li>
<li>
<b>PIECEWISE_LINEAR_PRODUCT_QUAD:</b> piecewise linear product integral.
</li>
<li>
<b>R8_ABS</b> returns the absolute value of an R8.
</li>
<li>
<b>R8_EPSILON</b> returns the R8 roundoff unit.
</li>
<li>
<b>R8_MIN</b> returns the minimum of two R8's.
</li>
<li>
<b>R83_NP_FA</b> factors a R83 system without pivoting.
</li>
<li>
<b>R83_NP_SL</b> solves a R83 system factored by R83_NP_FA.
</li>
<li>
<b>R8MAT_DATA_READ</b> reads the data from an R8MAT file.
</li>
<li>
<b>R8MAT_HEADER_READ</b> reads the header from an R8MAT file.
</li>
<li>
<b>R8MAT_WRITE</b> writes an R8MAT file.
</li>
<li>
<b>R8MAT_ZERO_NEW</b> returns a new zeroed R8MAT.
</li>
<li>
<b>R8VEC_BRACKET3</b> finds the interval containing or nearest a given value.
</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>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 05 July 2013.
</i>
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