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<html>
<head>
<title>
FEM2D_PROJECT - Project Data onto a 2D Finite Element Mesh
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
FEM2D_PROJECT <br> Project Data onto a 2D Finite Element Mesh
</h1>
<hr>
<p>
<b>FEM2D_PROJECT</b>
is a C++ program which
projects a finite element function.
</p>
<p>
Let us suppose we have a region R and a triangulation of R, that is,
a set of nodes N1 and triangles T1 whose union is R. Let P1(I)(X,Y)
be the finite element basis function associated with node N1(I).
Now let us suppose that we have a finite element function V1, that is
a scalar- or vector-valued function V1(X,Y) defined over R,
with the formula
<blockquote>
V1(X,Y) = sum ( I ) V1(I) * P1(I)(X,Y)
</blockquote>
</p>
<p>
Now suppose we have a second triangulation of R comprising
a set of nodes N2 and triangles T2. Can we determine an appropriate
set of finite element coefficients V2(I) which best approximate V1 in
the finite element space defined by N2 and T2? The finite element
coefficient vector V2 is defined by the following relationship:
<blockquote>
Integral Sum ( I ) V2(I) P2(I)(X,Y) P2(J)(X,Y) dx dy = Integral V1(X,Y) P2(J)(X,Y) dx dy
</blockquote>
Thus, in particular, the function V1(X,Y), which is defined on the first finite
element space, must be evaluated in a computation that uses the second finite element
space.
</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 as three tables, each stored in a file:
<ul>
<li>
the <b>SAMPLE_NODES</b> file contains the 2D coordinates of sample points.
Every sample point is presumed to lie within the area covered by the finite
element mesh.
</li>
<li>
the <b>SAMPLE_ELEMENTS</b> file contains the indices of nodes that
form the elements. The elements are presumed to be 3-node triangles.
This is simple a Delaunay triangulation of the sample nodes.
</li>
<li>
the <b>SAMPLE_VALUES</b> file contains the value of some vector quantity
V at each sample point. The dimensionality of the V data is arbitrary.
</li>
</ul>
</p>
<p>
The finite element mesh is given as two tables, each stored in a file:
<ul>
<li>
the <b>FEM_NODES</b> file contains the 2D coordinates of nodes.
</li>
<li>
the <b>FEM_ELEMENTS</b> file contains the indices of nodes that
form the elements. The elements are presumed to be 3-node triangles.
</li>
</ul>
</p>
<p>
The program produces a new table <b>FEM_VALUES</b>, of the same dimensionality
as <b>SAMPLE_VALUES</b>. The vector <b>FEM_VALUES</b> can be used in conjunction with
the finite element mesh data to produce a finite element function that is
an approximant to the <b>SAMPLE_VALUES</b> data.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>fem2d_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 (input);
</li>
<li>
<i>sample_prefix</i><b>_elements.txt</b>, the 3 nodes that make up each element (input);
</li>
<li>
<i>sample_prefix</i><b>_values.txt</b>, the data values (input);
</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 (input);
</li>
<li>
<i>fem_prefix</i><b>_elements.txt</b>, the 3 nodes that make up each element (input);
</li>
<li>
<i>fem_prefix</i><b>_values.txt</b>, the data values (output).
</li>
</ul>
</p>
<p>
The file <i>fem_prefix</i>_values.txt is created by the program, and contains
the projections of the sample data values onto the finite element space, that is,
these may be regarded as coefficients of finite element functions
representing the projections of the sample data. Note that we may also regard
this operation as the refinement or coarsening of a finite element function,
in that we are transferring information from the ``sample'' space to the ``fem''
space.
</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_PROJECT</b> is available in
<a href = "../../cpp_src/fem2d_project/fem2d_project.html">a C++ version</a> and
<a href = "../../f_src/fem2d_project/fem2d_project.html">a FORTRAN90 version</a> and
<a href = "../../m_src/fem2d_project/fem2d_project.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/fem1d_project/fem1d_project.html">
FEM1D_PROJECT</a>,
a C++ program which
projects data into a finite element space, including the least squares
approximation of data, or the projection of a finite element solution
from one mesh to another.
</p>
<p>
<a href = "../../data/fem2d/fem2d.html">
FEM2D</a>,
a data directory which
contains examples of 2D FEM files,
three text files that describe a 2D finite element geometry;
</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/fem2d_pack/fem2d_pack.html">
FEM2D_PACK</a>,
a C++ library which
includes utitlies for 2D finite element calculations.
</p>
<p>
<a href = "../../cpp_src/fem2d_poisson/fem2d_poisson.html">
FEM2D_POISSON</a>,
a C++ program which
solves the 2D Poisson equation on a rectangle, using the finite element method,
and piecewise quadratic triangular elements.
</p>
<p>
<a href = "../../cpp_src/fem2d_poisson_linear/fem2d_poisson_linear.html">
FEM2D_POISSON_LINEAR</a>,
a C++ program which
solves the 2D Poisson equation on a rectangle, using the finite element method,
and 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/fem2d_sample/fem2d_sample.html">
FEM2D_SAMPLE</a>,
a C++ program which
evaluates a finite element function defined on an order 3 or order 6 triangulation.
</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 = "fem2d_project.cpp">fem2d_project.cpp</a>, the source code.
</li>
<li>
<a href = "fem2d_project.sh">fem2d_project.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<b>LINEAR</b> starts with sample data for the vector function f(x)=[ 1, 2x, 3y ],
on an 9 x 9 grid of equally spaced nodes from [0.0,8.0]x[0.0,8.0], and projects this onto
a piecewise linear finite element meshes defined on equally spaced grids of
dimension 4x4, 2x2 and 1x1.
<ul>
<li>
<a href = "r8x8_t3_nodes.txt">r8x8_t3_nodes.txt</a>,
the sample nodes, on an 8x8 grid.
</li>
<li>
<a href = "r8x8_t3_elements.txt">r8x8_t3_elements.txt</a>,
elements that can be used to form an 8x8 finite element mesh associated
with the sample data. This is provide only so that a finite element
function can be formed with the original sample data.
</li>
<li>
<a href = "r8x8_t3_values.txt">r8x8_t3_values.txt</a>,
the sample nodal values.
</li>
<li>
<a href = "r4x4_t3_nodes.txt">r4x4_t3_nodes.txt</a>,
the FEM nodes for a 4x4 grid.
</li>
<li>
<a href = "r4x4_t3_elements.txt">r4x4_t3_elements.txt</a>,
the FEM elements for a 4x4 grid.
</li>
<li>
<a href = "r4x4_t3_values.txt">r4x4_t3_values.txt</a>,
the nodal values as projected from the 8x8 grid.
</li>
<li>
<a href = "r2x2_t3_nodes.txt">r2x2_t3_nodes.txt</a>,
the FEM nodes for a 2x2 grid.
</li>
<li>
<a href = "r2x2_t3_elements.txt">r2x2_t3_elements.txt</a>,
the FEM elements for a 2x2 grid.
</li>
<li>
<a href = "r2x2_t3_values.txt">r2x2_t3_values.txt</a>,
the nodal values as projected from the 8x8 grid.
</li>
<li>
<a href = "r1x1_t3_nodes.txt">r1x1_t3_nodes.txt</a>,
the FEM nodes for a 1x1 grid.
</li>
<li>
<a href = "r1x1_t3_elements.txt">r1x1_t3_elements.txt</a>,
the FEM elements for a 1x1 grid.
</li>
<li>
<a href = "r1x1_t3_values.txt">r1x1_t3_values.txt</a>,
the nodal values as projected from the 8x8 grid.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for FEM2D_PROJECT.
</li>
<li>
<b>BASIS_MN_T3:</b> all bases at N points for a T3 element.
</li>
<li>
<b>BASIS_MN_T6:</b> all bases at N points for a 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>FEM2D_TRANSFER</b> "transfers" from one finite element mesh to another.
</li>
<li>
<b>FILE_COLUMN_COUNT</b> counts the 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>I4_MAX</b> returns the maximum of two I4's.
</li>
<li>
<b>I4_MIN</b> returns the minimum of two I4's.
</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 I4COL.
</li>
<li>
<b>I4MAT_MIN</b> returns the minimum of an I4MAT.
</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>PROJECTION</b> evaluates an FEM function on a T3 or T6 triangulation.
</li>
<li>
<b>R4_ABS</b> returns the absolute value of an R4.
</li>
<li>
<b>R4_NINT</b> returns the nearest integer to an R4.
</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>R8GE_FSS_NEW</b> factors and solves multiple R8GE systems.
</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_WRITE0</b> writes an R8MAT file with no header.
</li>
<li>
<b>R8MAT_ZERO_NEW</b> returns a new zeroed R8MAT.
</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>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>TRIANGULATION_ORDER3_NEIGHBOR_TRIANGLES</b> determines triangle neighbors.
</li>
<li>
<b>TRIANGULATION_SEARCH_DELAUNAY</b> searches a Delaunay triangulation for a point.
</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 08 August 2009.
</i>
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