FEM2D_SAMPLE
Evaluate a Finite Element Function of a 2D Argument

FEM2D_SAMPLE is a C++ program which can evaluate a finite element function of a 2D argument.

The current version of the program can only handle finite element meshes which are made of piecewise linear triangles of order 3 or 6.

Usage:

fem2d_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.

Related Data and Programs:

DISCRETE_PDF_SAMPLE a MATLAB program which demonstrates how to construct a Probability Density Function (PDF) from a table of sample data, and then to use that PDF to create new samples.

FEKETE is a C++ library which defines a Fekete rule for quadrature or interpolation over a triangle.

FEM is a data directory which contains a description of the data files that can be used to describe a finite element model.

FEM_50 is a MATLAB program which defines a finite element method in just 50 lines of code.

FEM_50_HEAT is a MATLAB program which is a modified version of FEM_50 suitable for solving the heat equation.

FEM_BASIS_T3_DISPLAY is a MATLAB program which displays a basis function associated with a linear triangle ("T3") mesh.

FEM_BASIS_T6_DISPLAY is a MATLAB program which reads a quadratic triangle mesh and displays any associated basis function.

FEM_IO is a C++ library which reads or writes the node, element and data files that define a finite element model.

FEM_TO_TEC is a MATLAB program which can convert an FEM model into a TEC graphics file.

FEM1D is a C++ program which applies the finite element method, with piecewise linear basis functions, to a linear two point boundary value problem;

FEM1D_ADAPTIVE is a C++ program which applies the finite element method to a linear two point boundary value problem in a 1D region, using adaptive refinement to improve the solution.

FEM1D_NONLINEAR is a C++ program which applies the finite element method to a nonlinear two point boundary value problem in a 1D region.

FEM1D_PMETHOD is a C++ program which applies the p-method version of the finite element method to a linear two point boundary value problem in a 1D region.

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_HEAT is a C++ program which solves the time dependent heat equation in the unit square.

FEM2D_PACK is a C++ library which includes utitlies for 2D finite element calculations.

FEM2D_POISSON is a C++ program which solves Poisson's equation on a square, using the finite element method.

FEM2D_POISSON_LINEAR is a C++ program which solves the 2D Poisson equation on a rectangle, using the finite element method, and piecewise linear triangular elements.

FEM2D_PROJECT is a C++ program which projects a function F(X,Y) into a given finite element space of piecewise linear triangular elements.

FEM2D_SAMPLE is available in a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

FREE_FEM_HEAT is a C++ program which uses the finite element method and the backward Euler method to solve the time-dependent heat equation on an arbitrary triangulated region in 2D. (The user supplies some information in some problem-dependent subroutines.)

FREE_FEM_POISSON is a C++ program which solves Poisson's equation on a triangulated region, using the finite element method.

FREE_FEM_STOKES is a C++ program which solves the Stokes equations on a triangulated region, using the finite element method.

HOT_PIPE is a MATLAB program which can be run with FEM_50_HEAT.

HOT_POINT is a MATLAB program which can be run with FEM_50_HEAT.

MESH_BANDWIDTH is a C++ program which returns the geometric bandwidth associated with a mesh of elements of any order and in a space of arbitrary dimension.

TABLE is a data format which is used for the input and output files used by the program.

TABLE_IO is a C++ library which reads and writes TABLE files.

TRIANGULATION_ORDER3 is a dataset directory which contains a description and examples of how an order 3 (linear) triangulation is defined by a node and element file.

TRIANGULATION_ORDER6 is a dataset directory which contains a description and examples of how an order 6 (quadratic) triangulation is defined by a node and element file.

TRIANGULATION_ORDER6_CONTOUR is a MATLAB program which can make contours of a scalar quantity defined on the nodes of an order 6 triangulation.

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:

• fem2d_sample.C, the source code.
• fem2d_sample.csh, commands to compile the source code.

Examples and Tests:

SQ is FEM data for the function f(x,y)=x^2+y^2, on a 5x5 grid of nodes, organized into linear triangles. The sample data seeks the values of this function on a 4x4 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.

List of Routines:

• MAIN is the main program for FEM2D_SAMPLE.
• 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.
• GRID_SAMPLE samples a (scalar) FE function on a T3 or T6 triangulation.
• 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_SEED returns a random seed for the random number generator.
• I4_MAX returns the maximum of two integers.
• I4_MIN returns the smaller of two integers.
• 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.
• 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_WRITE0 writes an R8MAT file with no header.
• 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_ORDER6_NEIGHBOR_TRIANGLES determines triangle neighbors.
• TRIANGULATION_SEARCH_DELAUNAY searches a triangulation for a point.

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