jburkardt

FEM2D_PACK\ Finite Element Routines {#fem2d_pack-finite-element-routines align=”center”} =======================

FEM2D_PACK is a C++ library which implements the finite element method.

The emphasis is on simplicity and clarity. Only the 2D case is handled, with a choice of low order triangular and quadrilateral elements.

A few routines are included for computing a “sphere grid”, that is, a finite element mesh on the surface of a sphere.

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

FEM1D_PACK, a C++ library which contains utilities for 1D finite element calculations.

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 time dependent heat equation on the unit square.

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

FEM2D_POISSON_RECTANGLE_LINEAR, a C++ program which solves the 2D Poisson equation on a rectangle, using the finite element method, and piecewise linear triangular elements.

FEM2D_SAMPLE, a C++ library which evaluates a finite element function defined on an order 3 or order 6 triangulation.

Reference: {#reference align=”center”}

Milton Abramowitz, Irene Stegun,\ Handbook of Mathematical Functions,\ National Bureau of Standards, 1964,\ ISBN: 0-486-61272-4,\ LC: QA47.A34.
Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart,\ LINPACK User’s Guide,\ SIAM, 1979,\ ISBN13: 978-0-898711-72-1.
Vladimir Krylov,\ Approximate Calculation of Integrals,\ Dover, 2006,\ ISBN: 0486445798.
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.
Arthur Stroud,\ Approximate Calculation of Multiple Integrals,\ Prentice Hall, 1971,\ ISBN: 0130438936,\ LC: QA311.S85.
Arthur Stroud, Don Secrest,\ Gaussian Quadrature Formulas,\ Prentice Hall, 1966,\ LC: QA299.4G3S7.
Olgierd Zienkiewicz,\ The Finite Element Method,\ Sixth Edition,\ Butterworth-Heinemann, 2005,\ ISBN: 0750663200.
Daniel Zwillinger, editor,\ CRC Standard Mathematical Tables and Formulae,\ 30th Edition,\ CRC Press, 1996,\ ISBN: 0-8493-2479-3.

Source Code: {#source-code align=”center”}

fem2d_pack.cpp, the source code;
fem2d_pack.hpp, the include file;

Examples and Tests: {#examples-and-tests align=”center”}

fem2d_pack_prb.cpp, the calling program;
fem2d_pack_prb_output.txt, the sample output.

A number of files are created by the program.

fem2d_pack_prb_q4.png, a PNG image of a Q4 mesh.
fem2d_pack_prb_t3.png, a PNG image of a T3 mesh.
fem2d_pack_prb_t6.png, a PNG image of a T6 mesh.
sphere_q4_nodes.txt, a node file for a Q4 mesh, which is also a polygonal surface, that approximates a sphere.
sphere_q4_elements.txt, an element file for a Q4 mesh, which is also a polygonal surface, that approximates a sphere.
sphere_q9_nodes.txt, a node file for a Q9 mesh that approximates a sphere.
sphere_q9_elements.txt, an element file for a Q9 mesh that approximates a sphere.
sphere_q16_nodes.txt, a node file for a Q16 mesh that approximates a sphere.
sphere_q16_elements.txt, an element file for a Q16 mesh that approximates a sphere.
sphere_t3_nodes.txt, a node file for a T3 mesh, which is also a polygonal surface, that approximates a sphere.
sphere_t3_elements.txt, an element file for a T3 mesh, which is also a polygonal surface, that approximates a sphere.
sphere_t6_nodes.txt, a node file for a T6 mesh that approximates a sphere.
sphere_t6_elements.txt, an element file for a T6 mesh that approximates a sphere.

List of Routines: {#list-of-routines align=”center”}

BANDWIDTH_MESH determines the bandwidth of the coefficient matrix.
BANDWIDTH_VAR determines the bandwidth for finite element variables.
BASIS_11_T3: one basis at one point for a T3 element.
BASIS_11_T3_TEST verifies BASIS_11_T3.
BASIS_MN_T4: one basis at one point for a T4 element.
BASIS_11_T4_TEST verifies BASIS_11_T4.
BASIS_11_T6: one basis at one point for the T6 element.
BASIS_11_T6_TEST verifies BASIS_11_T6.
BASIS_MN_Q4: all bases at N points for a Q4 element.
BASIS_MN_Q4_TEST verifies BASIS_MN_Q4.
BASIS_MN_T3: all bases at N points for a T3 element.
BASIS_MN_T3_TEST verifies BASIS_MN_T3.
BASIS_MN_T4: all bases at N points for a T4 element.
BASIS_MN_T4_TEST verifies BASIS_MN_T4.
BASIS_MN_T6: all bases at N points for a T6 element.
BASIS_MN_T6_TEST verifies BASIS_MN_T6.
CH_CAP capitalizes a single character.
DEGREES_TO_RADIANS converts an angle from degrees to radians.
DERIVATIVE_AVERAGE_T3 averages derivatives at the nodes of a T3 mesh.
DIV_Q4 estimates the divergence and vorticity of a discrete field.
ELEMENT_CODE returns the code for each element.
ELEMENTS_EPS creates an EPS file image of the elements of a grid.
GRID_ELEMENT returns the element grid associated with any available element.
GRID_ELEMENT_NUM returns the number of elements in a grid.
GRID_NODE_NUM returns the number of nodes in a grid.
GRID_NODES_01 returns an equally spaced rectangular grid of nodes in the unit square.
GRID_PRINT prints the elements that form a grid.
GRID_Q4_ELEMENT produces a grid of 4 node quadrilaterals.
GRID_Q4_ELEMENT_NUM counts the elements in a grid of 4 node quadrilaterals.
GRID_Q4_NODE_NUM counts the nodes in a grid of 4 node quadrilaterals.
GRID_Q8_ELEMENT produces a grid of 8 node quadrilaterals.
GRID_Q8_ELEMENT_NUM counts the elements in a grid of 8 node quadrilaterals.
GRID_Q8_NODE_NUM counts the nodes in a grid of 8 node quadrilaterals.
GRID_Q9_ELEMENT produces a grid of 9 node quadrilaterals.
GRID_Q9_ELEMENT_NUM counts the elements in a grid of 9 node quadrilaterals.
GRID_Q9_NODE_NUM counts the nodes in a grid of 9 node quadrilaterals.
GRID_Q12_ELEMENT produces a grid of 12 node quadrilaterals.
GRID_Q12_ELEMENT_NUM counts the elements in a grid of 12 node quadrilaterals.
GRID_Q12_NODE_NUM counts the nodes in a grid of 12 node quadrilaterals.
GRID_Q16_ELEMENT produces a grid of 16 node quadrilaterals.
GRID_Q16_ELEMENT_NUM counts the elements in a grid of 16 node quadrilaterals.
GRID_Q16_NODE_NUM counts the nodes in a grid of 16 node quadrilaterals.
GRID_QL_ELEMENT produces a grid of 6 node quadratics/linears.
GRID_QL_ELEMENT_NUM counts the elements in a grid of quadratic/linear quadrilaterals.
GRID_QL_NODE_NUM counts the nodes in a grid of quadratic/linear quadrilaterals.
GRID_SHAPE_2D guesses the shape N1 by N2 of a vector of data.
GRID_T3_ELEMENT produces a grid of pairs of 3 node triangles.
GRID_T3_ELEMENT_NUM counts the elements in a grid of 3 node triangles.
GRID_T3_NODE_NUM counts the nodes in a grid of 3 node triangles.
GRID_T4_ELEMENT produces a grid of pairs of 4 node triangles.
GRID_T4_ELEMENT_NUM counts the elements in a grid of 4 node triangles.
GRID_T4_NODE_NUM counts the nodes in a grid of 4 node triangles.
GRID_T6_ELEMENT produces a grid of pairs of 6 node triangles.
GRID_T6_ELEMENT_NUM counts the elements in a grid of 6 node triangles.
GRID_T6_NODE_NUM counts the nodes in a grid of 6 node triangles.
GRID_T10_ELEMENT produces a grid of pairs of 10 node triangles.
GRID_T10_ELEMENT_NUM counts the elements in a grid of 10 node triangles.
GRID_T10_NODE_NUM counts the nodes in a grid of 10 node triangles.
GRID_TEST tests the grid routines.
GRID_WIDTH computes the width of a given grid.
I4_MAX returns the maximum of two I4’s.
I4_MIN returns the smaller of two I4’s.
I4_MODP returns the nonnegative remainder of I4 division.
I4_WRAP forces an I4 to lie between given limits by wrapping.
I4MAT_TRANSPOSE_PRINT prints an I4MAT, transposed.
I4MAT_TRANSPOSE_PRINT_SOME prints some of an I4MAT, transposed.
I4MAT_WRITE writes an I4MAT file with no header.
I4VEC_PRINT prints an I4VEC.
INTERP interpolates a quantity in an element from basis node values.
INTERP_TEST tests the interpolation property of an element.
LEGENDRE_COM computes abscissas and weights for Gauss-Legendre quadrature.
LEGENDRE_SET sets abscissas and weights for Gauss-Legendre quadrature.
MAP returns the interpolation matrix for any available element.
MAP_TEST tests the map routines.
MASS_MATRIX_T6 computes the mass matrix, using 6-node triangles.
NEXT_BOUNDARY_NODE returns the next boundary node in any element.
NEXT_BOUNDARY_NODE_Q4 returns the next boundary node in a Q4 element.
NEXT_BOUNDARY_NODE_Q8 returns the next boundary node in a Q8 element.
NEXT_BOUNDARY_NODE_Q9 returns the next boundary node in a Q9 element.
NEXT_BOUNDARY_NODE_Q12 returns the next boundary node in a Q12 element.
NEXT_BOUNDARY_NODE_Q16 returns the next boundary node in a Q16 element.
NEXT_BOUNDARY_NODE_QL returns the next boundary node in a QL element.
NEXT_BOUNDARY_NODE_T3 returns the next boundary node in a T3 element.
NEXT_BOUNDARY_NODE_T4 returns the next boundary node in a T4 element.
NEXT_BOUNDARY_NODE_T6 returns the next boundary node in a T6 element.
NEXT_BOUNDARY_NODE_T10 returns the next boundary node in a T10 element.
NODE_REFERENCE returns the basis nodes for any available element.
NODE_REFERENCE_Q4 returns the basis nodes for a 4 node quadrilateral.
NODE_REFERENCE_Q8 returns the basis nodes for an 8 node quadrilateral.
NODE_REFERENCE_Q9 returns the basis nodes for a 9 node quadrilateral.
NODE_REFERENCE_Q12 returns the basis nodes for a 12 node quadrilateral.
NODE_REFERENCE_Q16 returns the basis nodes for a 16 node quadrilateral.
NODE_REFERENCE_QL returns the basis nodes for a quadratic/linear.
NODE_REFERENCE_T3 returns the basis nodes for the 3 node triangle.
NODE_REFERENCE_T4 returns the basis nodes for the 4 node triangle.
NODE_REFERENCE_T6 returns the basis nodes for a 6 node triangle.
NODE_REFERENCE_T10 returns the basis nodes for a 10 node triangle.
NS_T6_VAR_COUNT counts the Navier Stokes variables on a T6 grid.
NS_T6_VAR_SET sets the Navier Stokes variables on a T6 grid.
ORDER_CODE returns the order for each element.
PHYSICAL_TO_REFERENCE_T3 maps physical points to reference points.
POINTS_PLOT plots a pointset.
POLY returns the polynomial terms associated with any available element.
POLY_Q4 returns the monomials associated with a 4 node quadrilateral.
POLY_Q8 returns the monomials associated with an 8 node quadrilateral.
POLY_Q9 returns the monomials associated with a 9 node quadrilateral.
POLY_Q12 returns the monomials associated with a 12 node quadrilateral.
POLY_Q16 returns the monomials associated with a 16 node quadrilateral.
POLY_QL returns the monomials for a quadratic/linear quadrilateral.
POLY_T3 returns the monomials associated with a 3 node triangle.
POLY_T6 returns the monomials associated with a 6 node triangle.
POLY_T10 returns the monomials associated with a 10 node triangle.
R8_ABS returns the absolute value of an R8.
R8_EPSILON returns the roundoff unit for R8’s.
R8_HUGE returns a “huge” double precision value.
R8_NINT returns the nearest integer to an R8.
R8_POWER computes the P-th power of R.
R8_UNIFORM_01 returns a unit pseudorandom R8.
R8GE_FA performs a LINPACK-style PLU factorization of a R8GE matrix.
R8GE_INVERSE computes the inverse of a R8GE matrix factored by R8GE_FA.
R8MAT_MM multiplies two matrices.
R8MAT_PRINT prints an R8MAT, with an optional title.
R8MAT_PRINT_SOME prints some of an R8MAT.
R8MAT_WRITE writes an R8MAT file with no header.
REFERENCE_SAMPLE samples a reference element.
REFERENCE_TO_PHYSICAL_Q4 maps Q4 reference points to physical points.
REFERENCE_TO_PHYSICAL_T3 maps T3 reference points to physical points.
REFERENCE_TO_PHYSICAL_T6 maps T6 reference points to physical points.
S_EQI reports whether two strings are equal, ignoring case.
S_LEN_TRIM returns the length of a string to the last nonblank.
SERENE interpolates data using a Q8 element.
SHAPE evaluates shape functions for any available element.
SHAPE_Q4 evaluates shape functions for a 4 node quadrilateral.
SHAPE_Q8 evaluates shape functions for an 8 node quadrilateral.
SHAPE_Q9 evaluates shape functions for a 9 node quadrilateral.
SHAPE_Q12 evaluates shape functions for a 12 node quadrilateral.
SHAPE_Q16 evaluates shape functions for a 16 node quadrilateral.
SHAPE_QL evaluates shape functions for a 6 node quadratic/linear.
SHAPE_T3 evaluates shape functions for a 3 node triangle.
SHAPE_T4 evaluates shape functions for a T4 triangle.
SHAPE_T6 evaluates shape functions for a 6 node triangle.
SHAPE_T10 evaluates shape functions for a 10 node triangle.
SHAPE_TEST verifies the shape function values at the basis nodes.
SPHERE_GRID_ELEMENT_NUM returns the number of elements in a sphere grid.
SPHERE_GRID_NODE_NUM returns the number of nodes in a sphere grid.
SPHERE_GRID_Q4_ELEMENT produces a Q4 sphere grid.
SPHERE_GRID_Q4_ELEMENT_NUM counts the elements in a Q4 sphere grid.
SPHERE_GRID_Q4_NODE_NUM counts nodes in a Q4 sphere grid.
SPHERE_GRID_Q4_NODE_XYZ produces node coordinates for a Q4 sphere grid.
SPHERE_GRID_Q9_ELEMENT produces a Q9 sphere grid.
SPHERE_GRID_Q9_ELEMENT_NUM counts the elements in a Q9 sphere grid.
SPHERE_GRID_Q9_NODE_NUM counts nodes in a Q9 sphere grid.
SPHERE_GRID_Q9_NODE_XYZ produces node coordinates for a Q9 sphere grid.
SPHERE_GRID_Q16_ELEMENT produces a Q16 sphere grid.
SPHERE_GRID_Q16_ELEMENT_NUM counts the elements in a Q16 sphere grid.
SPHERE_GRID_Q16_NODE_NUM counts nodes in a Q16 sphere grid.
SPHERE_GRID_Q16_NODE_XYZ produces node coordinates for a Q16 sphere grid.
SPHERE_GRID_T3_ELEMENT produces a T3 sphere grid.
SPHERE_GRID_T3_ELEMENT_NUM counts the elements in a T3 sphere grid.
SPHERE_GRID_T3_NODE_NUM counts nodes in a T3 sphere grid.
SPHERE_GRID_T3_NODE_XYZ produces node coordinates for a T3 sphere grid.
SPHERE_GRID_T6_ELEMENT produces a T6 sphere grid.
SPHERE_GRID_T6_ELEMENT_NUM counts the elements in a T6 sphere grid.
SPHERE_GRID_T6_NODE_NUM counts nodes in a T6 sphere grid.
SPHERE_GRID_T6_NODE_XYZ produces node coordinates for a T6 sphere grid.
TIMESTAMP prints the current YMDHMS date as a time stamp.
TRIANGLE_UNIT_SET sets a quadrature rule in a unit triangle.
TRIANGLE_UNIT_SIZE returns the “size” of a unit triangle quadrature rule.

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

Last revised on 16 August 2009.