FEM3D_SAMPLE
Evaluate a Finite Element Function of a 3D Argument

FEM3D_SAMPLE is a C++ program which can evaluate a finite element function of a 3D argument.

The current version of the program can only handle tetrahedral finite element meshes using linear basis functions.

The current version of the program uses a NAIVE search algorithm to locate points within the mesh. It is intended that this search algorithm be replaced by a Delaunay search.

Usage:

fem3d_sample fem_prefix sample_prefix

• fem_prefix_nodes.txt, the node coordinates.
• fem_prefix_elements.txt, the nodes that make up each element;
• fem_prefix_values.txt, the values defined at each node.

• sample_prefix_nodes.txt, the node coordinates where samples are desired.
• sample_prefix_values.txt, the values computed at each sample node.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

FEM3D_SAMPLE is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

FEM_IO, a C++ library which reads or writes node, element and data files defining a finite element model.

FEM_TO_TEC, a MATLAB program which converts FEM models into TEC graphics files.

FEM1D_SAMPLE, 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.

FEM2D_SAMPLE, a C++ program which samples a scalar or vector finite element function of 2 variables defined by FEM files, returning interpolated values at the sample points.

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_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.

HISTOGRAM_DATA_2D_SAMPLE, a C++ program which demonstrates how to construct a Probability Density Function (PDF) from a frequency table over a 2D domain, and then to use that PDF to create new samples.

Reference:

1. Hans Rudolf Schwarz,
   Finite Element Methods,
   Academic Press, 1988,
   ISBN: 0126330107,
   LC: TA347.F5.S3313.
2. Gilbert Strang, George Fix,
   An Analysis of the Finite Element Method,
   Cambridge, 1973,
   ISBN: 096140888X,
   LC: TA335.S77.
3. Olgierd Zienkiewicz,
   The Finite Element Method,
   Sixth Edition,
   Butterworth-Heinemann, 2005,
   ISBN: 0750663200,
   LC: TA640.2.Z54.

Source Code:

• fem3d_sample.cpp, the source code.

Examples and Tests:

SQ is FEM data for the function f(x,y,z)=x^2+2*y^2-z^2, on a 5x5x5 grid of nodes in the cube [0,4]x[0,4]x[0,4], organized into linear linear tetrahedrons. The sample data seeks the values of this function on a 4x4x4 evenly spaced grid.

• fem_sq_nodes.txt, defines the FEM nodes.
• fem_sq_elements.txt, defines the FEM elements.
• fem_sq_values.txt, defines the FEM nodal values.
• sample_sq_nodes.txt, defines the sample nodes.
• sample_sq_values.txt, defines the sample nodal values.
• sample_sq_values_exact.txt, the exact values of the function at the sample nodes. This should be compared with the values determined from the FEM function.

List of Routines:

• MAIN is the main program for FEM3D_SAMPLE.
• BASIS_MN_TET4: all bases at N points for a T4 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.
• FEM3D_EVALUATE samples an FEM function on a T4 tet mesh.
• 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.
• I4_MAX returns the maximum of two I4's.
• I4_MIN returns the minimum of two I4's.
• I4_UNIFORM returns a scaled pseudorandom I4.
• I4COL_COMPARE compares columns I and J of an I4COL.
• I4COL_SORT_A ascending sorts an I4COL.
• I4COL_SWAP swaps two columns of an I4COL.
• I4I4I4_SORT_A ascending sorts a triple of I4's.
• I4MAT_DATA_READ reads data from an I4MAT file.
• I4MAT_HEADER_READ reads the header from an I4MAT file.
• R4_ABS returns the absolute value of an R4.
• R4_NINT returns the nearest integer to an R4.
• R4_UNIFORM_01 returns a unit pseudorandom R4.
• R8_HUGE returns a "huge" R8.
• R8_MAX returns the maximum of two R8's.
• R8_MIN returns the minimum of two R8's.
• R8_NINT returns the nearest integer to an R8.
• R8MAT_DATA_READ reads the data from an R8MAT file.
• 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_WRITE0 writes an R8MAT file with no header.
• R8VEC_IS_NONNEGATIVE is true if all entries in an R8VEC are nonnegative.
• 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.
• TET_MESH_NEIGHBOR_TETS determines tetrahedron neighbors.
• TET_MESH_SEARCH_NAIVE naively searches a tet mesh.
• TETRAHEDRON_BARYCENTRIC returns the barycentric coordinates of a point.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the C++ source codes.