FEM3D_PROJECT\
Project Data onto a 3D Finite Element Mesh {#fem3d_project-project-data-onto-a-3d-finite-element-mesh align=”center”}
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FEM3D_PROJECT is a C++ program which projects a finite element
function.
Let us suppose we have a region R and a “tet mesh” (tetrahedral mesh) of
R, that is, a set of nodes N1 and tetrahedrons T1 whose union is R. Let
P1(I)(X,Y,Z) 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,Z) defined over R,
with the formula
V1(X,Y,Z) = sum ( 1 <= I <= NODE_NUM1 ) V1(I) * P1(I)(X,Y,Z)
Now suppose we have a second tet mesh of R comprising a set of nodes N2
and tetrahedrons 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 ( 1 <= I <= NODE_NUM2 ) V2(I) P2(I)(X,Y,Z)
P2(J)(X,Y,Z) dx dy dz = Integral V1(X,Y,Z) P2(J)(X,Y,Z) dx dy dz
Thus, in particular, the function V1(X,Y,Z), 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 3D 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 4-node
tetrahedrons that form a Delaunay tetrahedralization 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 3D coordinates of nodes.
- the FEM_ELEMENTS file contains the indices of nodes that form
the elements. The elements are presumed to be 4-node tetrahedrons.
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”}
fem3d_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 4 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 4 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”}
FEM3D_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_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 triangular elements.
FEM3D, a data directory which contains
examples of 3D FEM files, three text files that describe a 3D finite
element geometry;
FEM3D_PACK, a C++ library
which contains utilities for 3D finite element calculations.
FEM3D_SAMPLE, a C++
program which evaluates a finite element function defined on 3D
tetrahedral mesh.
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, 4z ], on an 8x8x8 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
4x4x4, 2x2x2 and 1x1x1.
- r8x8x8_t3_nodes.txt, the sample nodes, on
an 8x8x8 grid.
- r8x8_t4_elements.txt, elements that can
be used to form an 8x8x8 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_t4_values.txt, the sample nodal
values.
- r4x4x4_t4_nodes.txt, the FEM nodes for a
4x4x4 grid.
- r4x4x4_t4_elements.txt, the FEM elements
for a 4x4x4 grid.
- r4x4x4_t4_values.txt, the nodal values as
projected from the 8x8x8 grid.
- r2x2x2_t4_nodes.txt, the FEM nodes for a
2x2x2 grid.
- r2x2x2_t4_elements.txt, the FEM elements
for a 2x2x2 grid.
- r2x2x2_t4_values.txt, the nodal values as
projected from the 8x8x8 grid.
- r1x1x1_t4_nodes.txt, the FEM nodes for a
1x1x1 grid.
- r1x1x1_t4_elements.txt, the FEM elements
for a 1x1x1 grid.
- r1x1x1_t4_values.txt, the nodal values as
projected from the 8x8x8 grid.
List of Routines: {#list-of-routines align=”center”}
- MAIN is the main program for FEM3D_PROJECT.
- BASIS_MN_TET4: all bases at N points for a TET4 element.
- CH_CAP capitalizes a single character.
- CH_EQI is a case insensitive comparison of two characters for
equality.
- CH_TO_DIGIT returns the integer value of a base 10 digit.
- FEM3D_TRANSFER “transfers” from one finite element mesh to
another.
- FILE_COLUMN_COUNT counts the number of columns in the first
line of a file.
- FILE_ROW_COUNT counts the number of row records in a file.
- GET_UNIT returns a free FORTRAN unit number.
- I4I4I4_SORT_A ascending sorts a triple of I4’s.
- I4COL_COMPARE compares columns I and J of an I4COL.
- I4COL_SORT_A ascending sorts an I4COL.
- I4COL_SWAP swaps columns J1 and J2 of an I4COL.
- I4MAT_DATA_READ reads data from an I4MAT file.
- I4MAT_HEADER_READ reads the header from an I4MAT.
- I4MAT_WRITE writes an I4MAT file.
- PROJECTION evaluates an FEM function on a TET4 mesh.
- R8GE_FSS factors and solves multiple R8GE systems.
- R8MAT_DATA_READ reads data from an R8MAT file.
- R8MAT_DET_4D computes the determinant of a 4 by 4 R8MAT.
- R8MAT_HEADER_READ reads the header from an R8MAT file.
- R8MAT_SOLVE uses Gauss-Jordan elimination to solve an N by N
linear system.
- R8MAT_WRITE writes an R8MAT file.
- 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.
- TET_MESH_NEIGHBOR_TETS determines tetrahedron neighbors.
- TET_MESH_SEARCH_DELAUNAY searches a Delaunay tet mesh for a
point.
- TET_MESH_SEARCH_NAIVE naively searches a tet mesh.
- TETRAHEDRON_BARYCENTRIC: barycentric coordinates of a point.
- TETRAHEDRON_UNIT_QUAD04: 4 point quadrature rule for the unit
tetrahedron.
- TETRAHEDRON_VOLUME computes the volume of a tetrahedron in 3D.
- TIMESTAMP prints the current YMDHMS date as a time stamp.
You can go up one level to the C++ source codes.
Last revised on 25 August 2009.