FEM2D_PROJECT\
Project Data onto a 2D Finite Element Mesh {#fem2d_project-project-data-onto-a-2d-finite-element-mesh align=”center”}
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FEM2D_PROJECT is a C++ program which projects a finite element
function.
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
V1(X,Y) = sum ( I ) V1(I) * P1(I)(X,Y)
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:
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
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.
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.
The sample data is given as three tables, each stored in a file:
- the SAMPLE_NODES file contains the 2D coordinates of sample
points. Every sample point is presumed to lie within the area
covered by the finite element mesh.
- the SAMPLE_ELEMENTS 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.
- the SAMPLE_VALUES file contains the value of some vector
quantity V at each sample point. The dimensionality of the V data is
arbitrary.
The finite element mesh is given as two tables, each stored in a file:
- the FEM_NODES file contains the 2D coordinates of nodes.
- the FEM_ELEMENTS file contains the indices of nodes that form
the elements. The elements are presumed to be 3-node triangles.
The program produces a new table FEM_VALUES, of the same
dimensionality as SAMPLE_VALUES. The vector FEM_VALUES can be
used in conjunction with the finite element mesh data to produce a
finite element function that is an approximant to the SAMPLE_VALUES
data.
Usage: {#usage align=”center”}
fem2d_project sample_prefix fem_prefix
where sample_prefix is the common prefix for the SAMPLE files:
- sample_prefix_nodes.txt, the node coordinates (input);
- sample_prefix_elements.txt, the 3 nodes that make up each
element (input);
- sample_prefix_values.txt, the data values (input);
and fem_prefix is the common prefix for the FEM files:
- fem_prefix_nodes.txt, the node coordinates (input);
- fem_prefix_elements.txt, the 3 nodes that make up each
element (input);
- fem_prefix_values.txt, the data values (output).
The file fem_prefix_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.
Licensing: {#licensing align=”center”}
The computer code and data files described and made available on this
web page are distributed under the GNU LGPL
license.
Languages: {#languages align=”center”}
FEM2D_PROJECT is available in a C++
version and a
FORTRAN90 version and a
MATLAB version.
FEM1D_PROJECT, 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.
FEM2D, a data directory which contains
examples of 2D FEM files, three text files that describe a 2D finite
element geometry;
FEM2D_HEAT, a C++ program
which solves the 2D time dependent heat equation on the unit square.
FEM2D_PACK, a C++ library
which includes utitlies for 2D finite element calculations.
FEM2D_POISSON, a C++
program which solves the 2D Poisson equation on a rectangle, using the
finite element method, and piecewise quadratic triangular elements.
FEM2D_POISSON_LINEAR,
a C++ program which solves the 2D Poisson equation on a rectangle, using
the finite element method, and piecewise linear triangular elements.
FEM2D_PROJECT_FUNCTION,
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.
FEM2D_SAMPLE, a C++
program which evaluates a finite element function defined on an order 3
or order 6 triangulation.
FEM3D_PROJECT, 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.
Reference: {#reference align=”center”}
- Hans Rudolf Schwarz,\
Finite Element Methods,\
Academic Press, 1988,\
ISBN: 0126330107,\
LC: TA347.F5.S3313.
- Gilbert Strang, George Fix,\
An Analysis of the Finite Element Method,\
Cambridge, 1973,\
ISBN: 096140888X,\
LC: TA335.S77.
- Olgierd Zienkiewicz,\
The Finite Element Method,\
Sixth Edition,\
Butterworth-Heinemann, 2005,\
ISBN: 0750663200,\
LC: TA640.2.Z54.
Source Code: {#source-code align=”center”}
Examples and Tests: {#examples-and-tests align=”center”}
LINEAR 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.
- r8x8_t3_nodes.txt, the sample nodes, on an
8x8 grid.
- r8x8_t3_elements.txt, 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.
- r8x8_t3_values.txt, the sample nodal values.
- r4x4_t3_nodes.txt, the FEM nodes for a 4x4
grid.
- r4x4_t3_elements.txt, the FEM elements for
a 4x4 grid.
- r4x4_t3_values.txt, the nodal values as
projected from the 8x8 grid.
- r2x2_t3_nodes.txt, the FEM nodes for a 2x2
grid.
- r2x2_t3_elements.txt, the FEM elements for
a 2x2 grid.
- r2x2_t3_values.txt, the nodal values as
projected from the 8x8 grid.
- r1x1_t3_nodes.txt, the FEM nodes for a 1x1
grid.
- r1x1_t3_elements.txt, the FEM elements for
a 1x1 grid.
- r1x1_t3_values.txt, the nodal values as
projected from the 8x8 grid.
List of Routines: {#list-of-routines align=”center”}
- MAIN is the main program for FEM2D_PROJECT.
- BASIS_MN_T3: all bases at N points for a T3 element.
- BASIS_MN_T6: all bases at N points for a T6 element.
- CH_CAP capitalizes a single character.
- CH_EQI is true if two characters are equal, disregarding case.
- CH_TO_DIGIT returns the integer value of a base 10 digit.
- FEM2D_TRANSFER “transfers” from one finite element mesh to
another.
- FILE_COLUMN_COUNT counts the columns in the first line of a
file.
- FILE_ROW_COUNT counts the number of row records in a file.
- I4_MAX returns the maximum of two I4’s.
- I4_MIN returns the minimum of two I4’s.
- I4COL_COMPARE compares columns I and J of an I4COL.
- I4COL_SORT_A ascending sorts the columns of an I4COL.
- I4COL_SWAP swaps two columns of an I4COL.
- I4MAT_MIN returns the minimum of an I4MAT.
- I4MAT_DATA_READ reads data from an I4MAT file.
- I4MAT_HEADER_READ reads the header from an I4MAT file.
- PROJECTION evaluates an FEM function on a T3 or T6
triangulation.
- R4_ABS returns the absolute value of an R4.
- R4_NINT returns the nearest integer to an R4.
- R8_ABS returns the absolute value of an R8.
- R8_EPSILON returns the R8 roundoff unit.
- R8_MIN returns the minimum of two R8’s.
- R8GE_FSS_NEW factors and solves multiple R8GE systems.
- R8MAT_DATA_READ reads the data from an R8MAT file.
- R8MAT_HEADER_READ reads the header from an R8MAT file.
- R8MAT_WRITE0 writes an R8MAT file with no header.
- R8MAT_ZERO_NEW returns a new zeroed R8MAT.
- S_LEN_TRIM returns the length of a string to the last
nonblank.
- S_TO_I4 reads an I4 from a string.
- S_TO_I4VEC reads an I4VEC from a string.
- S_TO_R8 reads an R8 from a string.
- S_TO_R8VEC reads an R8VEC from a string.
- S_WORD_COUNT counts the number of “words” in a string.
- SORT_HEAP_EXTERNAL externally sorts a list of items into
ascending order.
- TIMESTAMP prints the current YMDHMS date as a time stamp.
- TRIANGULATION_ORDER3_NEIGHBOR_TRIANGLES determines triangle
neighbors.
- TRIANGULATION_SEARCH_DELAUNAY searches a Delaunay
triangulation for a point.
You can go up one level to the C++ source codes.
Last revised on 08 August 2009.