FEM2D_PACK
2D Finite Element Routines

FEM2D_PACK is a MATLAB library which contains utilities for implementing 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:

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

Languages:

FEM2D_PACK is available in a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

FEM_BASIS, a MATLAB library which can define and evaluate basis functions for any degree in an M-dimensional simplex (1D interval, 2D triangle, 3D tetrahedron, and higher dimensional generalizations.)

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

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

FEM2D, a data directory which contains examples of 2D FEM files, text files that describe a 2D finite element geometry and associated nodal values;

FEM2D_HEAT, a MATLAB program which solves the time dependent heat equation in the unit square.

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

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

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

FEM3D_PACK, a MATLAB library which contains utilities for 3D finite element calculations.

ONED, a MATLAB library which contains functions useful for 1D finite element calculations.

THREED, a MATLAB library which contains functions useful for 3D finite element calculations.

TWOD, a MATLAB library which contains functions useful for 2D finite element calculations.

Reference:

1. Milton Abramowitz, Irene Stegun,
   Handbook of Mathematical Functions,
   National Bureau of Standards, 1964,
   ISBN: 0-486-61272-4,
   LC: QA47.A34.
2. Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart,
   LINPACK User's Guide,
   SIAM, 1979,
   ISBN13: 978-0-898711-72-1.
3. Vladimir Krylov,
   Approximate Calculation of Integrals,
   Dover, 2006,
   ISBN: 0486445798.
4. Hans Rudolf Schwarz,
   Finite Element Methods,
   Academic Press, 1988,
   ISBN: 0126330107,
   LC: TA347.F5.S3313.
5. Gilbert Strang, George Fix,
   An Analysis of the Finite Element Method,
   Cambridge, 1973,
   ISBN: 096140888X,
   LC: TA335.S77.
6. Arthur Stroud,
   Approximate Calculation of Multiple Integrals,
   Prentice Hall, 1971,
   ISBN: 0130438936,
   LC: QA311.S85.
7. Arthur Stroud, Don Secrest,
   Gaussian Quadrature Formulas,
   Prentice Hall, 1966,
   LC: QA299.4G3S7.
8. Olgierd Zienkiewicz,
   The Finite Element Method,
   Sixth Edition,
   Butterworth-Heinemann, 2005,
   ISBN: 0750663200,
   LC: TA640.2.Z54
9. Daniel Zwillinger, editor,
   CRC Standard Mathematical Tables and Formulae,
   30th Edition,
   CRC Press, 1996,
   ISBN: 0-8493-2479-3.

Source Code:

• bandwidth_mesh.m computes the bandwidth associated with the finite element mesh;
• bandwidth_var.m computes the bandwidth associated with an arbitrary assignment of variables to the nodes of the finite element mesh;
• basis_11_t3.m one basis at one point for the T3 element;
• basis_11_t3_test.m verifies BASIS_11_T3;
• basis_11_t4.m one basis at one point for the T4 element;
• basis_11_t4_test.m verifies BASIS_11_T4;
• basis_11_t6.m one basis at one point for the T6 element;
• basis_11_t6_test.m verifies BASIS_11_T6;
• basis_mn_q4.m all basis functions for N points in a Q4 element;
• basis_mn_q4_test.m verifies BASIS_MN_Q4.
• basis_mn_t3.m all basis functions for N points in a T3 element.
• basis_mn_t3_test.m verifies BASIS_MN_T3.
• basis_mn_t4.m all basis functions for N points in a T4 element.
• basis_mn_t4_test.m verifies BASIS_MN_T4.
• basis_mn_t6.m evaluates all basis functions for a T6 element.
• basis_mn_t6_test.m verifies the basis values at the basis nodes for a T6 element.
• ch_cap.m capitalizes a single character.
• degrees_to_radians.m converts an angle from degrees to radians.
• derivative_average_t3.m averages derivatives at the nodes of a T3 mesh.
• div_q4.m estimates the divergence and vorticity of a discrete field.
• element_code.m returns the code for each element.
• elements_eps.m creates an EPS file image of the elements of a grid.
• grid_element.m returns the grid associated with any available element.
• grid_element_num.m counts the elements in a grid given the element code.
• grid_node_num.m counts the nodes in a grid given the element code.
• grid_nodes.m defines a rectangular grid of equally spaced nodes in the unit square.
• grid_print.m prints the elements that form a grid.
• grid_q4_element.m produces a grid of Q4 elements.
• grid_q4_element_num.m counts the elements in a grid of Q4 elements.
• grid_q4_node_num.m counts the nodes in a grid of Q4 elements.
• grid_q8_element.m produces a grid of Q8 elements.
• grid_q8_element_num.m counts the elements in a grid of Q8 elements.
• grid_q8_node_num.m counts the nodes in a grid of Q8 elements.
• grid_q9_element.m produces a grid of Q9 elements.
• grid_q9_element_num.m counts the elements in a grid of Q9 elements.
• grid_q9_node_num.m counts the nodes in a grid of Q9 elements.
• grid_q12_element.m produces a grid of Q12 elements.
• grid_q12_element_num.m counts the elements in a grid of Q12 elements.
• grid_q12_node_num.m counts the nodes in a grid of Q12 elements.
• grid_q16_element.m produces a grid of Q16 elements.
• grid_q16_element_num.m counts the elements in a grid of Q16 elements.
• grid_q16_node_num.m counts the nodes in a grid of Q16 elements.
• grid_ql_element.m produces a grid of QL elements.
• grid_ql_element_num.m counts the elements in a grid of QL elements.
• grid_ql_node_num.m counts the nodes in a grid of QL elements.
• grid_shape_2d.m guesses the shape N1 by N2 of a vector of data.
• grid_t3_element.m produces a grid of pairs of T3 elements.
• grid_t3_element_num.m counts the elements in a grid of T3 elements.
• grid_t3_node_num.m counts the nodes in a grid of T3 elements.
• grid_t4_element.m produces a grid of pairs of T4 elements.
• grid_t4_element_num.m counts the elements in a grid of T4 elements.
• grid_t4_node_num.m counts the nodes in a grid of T4 elements.
• grid_t6_element.m produces a grid of pairs of T6 elements.
• grid_t6_element_num.m counts the elements in a grid of T6 elements.
• grid_t6_node_num.m counts the nodes in a grid of T6 elements.
• grid_t10_element.m produces a grid of pairs of T10 elements.
• grid_t10_element_num.m counts the elements in a grid of T10 elements.
• grid_t10_node_num.m counts the nodes in a grid of T10 elements.
• grid_test.m tests the grid routines.
• grid_width.m computes the width of a given grid.
• i4_modp.m returns the nonnegative remainder of integer division.
• i4_sign.m returns the sign of an integer.
• i4_wrap.m forces an integer to lie between given limits by wrapping.
• i4mat_write.m, writes an I4MAT file.
• i4vec_print.m prints an I4VEC;
• interp.m interpolates a quantity in an element from basis node values.
• interp_test.m tests the interpolation power of the basis functions for a given element;
• legendre_com.m computes abscissas and weights for Gauss-Legendre quadrature.
• legendre_set.m sets abscissas and weights for Gauss-Legendre quadrature.
• map.m returns the interpolation matrix for any available element.
• map_test.m tests the map routines.
• mass_matrix_t6.m computes the mass matrix for T6 elements.
• next_boundary_node.m returns the next boundary node in any element.
• next_boundary_node_q4.m returns the next boundary node in a Q4 element.
• next_boundary_node_q8.m returns the next boundary node in a Q8 element.
• next_boundary_node_q9.m returns the next boundary node in a Q9 element.
• next_boundary_node_q12.m returns the next boundary node in a Q12 element.
• next_boundary_node_q16.m returns the next boundary node in a Q16 element.
• next_boundary_node_ql.m returns the next boundary node in a QL element.
• next_boundary_node_t3.m returns the next boundary node in a T3 element.
• next_boundary_node_t4.m returns the next boundary node in a T4 element.
• next_boundary_node_t6.m returns the next boundary node in a T6 element.
• next_boundary_node_t10.m returns the next boundary node in a T10 element.
• node_reference.m returns the basis nodes for any available element.
• node_reference_q4.m returns the basis nodes for a Q4 element.
• node_reference_q8.m returns the basis nodes for a Q8 element.
• node_reference_q9.m returns the basis nodes for a Q9 element.
• node_reference_q12.m returns the basis nodes for a Q12 element.
• node_reference_q16.m returns the basis nodes for a Q16 element.
• node_reference_ql.m returns the basis nodes for a QL element.
• node_reference_t3.m returns the basis nodes for a T3 element.
• node_reference_t4.m returns the basis nodes for a T4 element.
• node_reference_t6.m returns the basis nodes for a T6 element.
• node_reference_t10.m returns the basis nodes for a T10 element.
• ns_t6_var_count.m counts the Navier Stokes variables on a T6 grid.
• ns_t6_var_set.m sets the Navier Stokes variables on a T6 grid.
• order_code.m returns the order for each element.
• physical_to_reference_t3.m maps points in physical space into a T3 reference element.
• points_plot.m creates an EPS file containing an image of the nodes.
• poly.m returns the polynomial terms associated with any available element.
• poly_q4.m returns the monomials associated with a Q4 element.
• poly_q8.m returns the monomials associated with an Q8 element.
• poly_q9.m returns the monomials associated with a Q9 element.
• poly_q12.m returns the monomials associated with a Q12 element.
• poly_q16.m returns the monomials associated with a Q16 element.
• poly_ql.m returns the monomials for a QL element.
• poly_t3.m returns the monomials associated with a T3 element.
• poly_t6.m returns the monomials associated with a T6 element.
• poly_t10.m returns the monomials associated with a T10 element.
• r8_epsilon.m returns the machine epsilon.
• r8_power.m returns a power of a real number.
• r8_uniform_01.m returns a pseudorandom value in [0,1].
• r8ge_fa.m performs a LINPACK style PLU factorization of a DGE matrix.
• r8ge_inverse.m computes the inverse of a matrix factored by DGE_FA.
• r8mat_print.m prints an R8MAT, with an optional title.
• r8mat_print_some.m prints some of an R8MAT, with an optional title.
• r8mat_write.m, writes an R8MAT file.
• reference_sample.m samples a reference element.
• reference_to_physical_q4.m maps points in a Q4 reference triangle into physical space.
• reference_to_physical_t3.m maps points in a T3 reference triangle into physical space.
• reference_to_physical_t6.m maps points in a T6 reference triangle into physical space.
• s_eqi.m is a case insensitive comparison of two strings for equality.
• s_l2norm.m computes the "big" L2 norm of a scalar function over a region.
• s_len_trim.m returns the length of a string to the last nonblank.
• serene.m interpolates data using a Q8 element.
• shape.m evaluates shape functions for any available element.
• shape_test.m verifies the shape function values at the basis nodes.
• shape_q4.m evaluates shape functions for a Q4 element.
• shape_q8.m evaluates shape functions for a Q8 element.
• shape_q9.m evaluates shape functions for a Q9 element.
• shape_q12.m evaluates shape functions for a Q12 element.
• shape_q16.m evaluates shape functions for a Q16 element.
• shape_ql.m evaluates shape functions for a QL element.
• shape_t3.m evaluates shape functions for a T3 element.
• shape_t4.m evaluates shape functions for a T4 element.
• shape_t6.m evaluates shape functions for a T6 element.
• shape_t10.m evaluates shape functions for a T10 element.
• sphere_grid_element_num.m returns the number of elements in a sphere grid.
• sphere_grid_node_num.m returns the number of nodes in a sphere grid.
• sphere_grid_q4_element.m produces a Q4 sphere grid.
• sphere_grid_q4_element_num.m counts the elements in a Q4 sphere grid.
• sphere_grid_q4_node_num.m counts the nodes in a Q4 sphere grid.
• sphere_grid_q4_node_xyz.m sets the nodes in a Q4 sphere grid.
• sphere_grid_q9_element.m produces a Q9 sphere grid.
• sphere_grid_q9_element_num.m counts the elements in a Q9 sphere grid.
• sphere_grid_q9_node_num.m counts the nodes in a Q9 sphere grid.
• sphere_grid_q9_node_xyz.m sets the nodes in a Q9 sphere grid.
• sphere_grid_q16_element.m produces a Q16 sphere grid.
• sphere_grid_q16_element_num.m counts the elements in a Q16 sphere grid.
• sphere_grid_q16_node_num.m counts the nodes in a Q16 sphere grid.
• sphere_grid_q16_node_xyz.m sets the nodes in a Q16 sphere grid.
• sphere_grid_t3_element.m produces a T3 sphere grid.
• sphere_grid_t3_element_num.m counts the elements in a T3 sphere grid.
• sphere_grid_t3_node_num.m counts the nodes in a T3 sphere grid.
• sphere_grid_t3_node_xyz.m sets the nodes in a T3 sphere grid.
• sphere_grid_t6_element.m produces a T6 sphere grid.
• sphere_grid_t6_element_num.m counts the elements in a T6 sphere grid.
• sphere_grid_t6_node_num.m counts the nodes in a T6 sphere grid.
• sphere_grid_t6_node_xyz.m sets the nodes in a T6 sphere grid.
• timestamp.m prints the current YMDHMS date as a time stamp.
• triangle_unit_set.m sets a quadrature rule in a unit triangle.
• triangle_unit_size.m returns the order of a quadrature rule in a unit triangle.

Examples and Tests:

• fem2d_pack_test.m, runs all the tests;
• fem2d_pack_test01.m, tests BANDWIDTH_MESH;
• fem2d_pack_test02.m, tests BANDWIDTH_VAR;
• fem2d_pack_test03.m, tests BASIS_11_**_TEST;
• fem2d_pack_test04.m, tests BASIS_MN_**_TEST;
• fem2d_pack_test05.m, tests DERIVATIVE_AVERAGE_T3;
• fem2d_pack_test06.m, tests DIV_Q4;
• fem2d_pack_test07m, tests ELEMENTS_EPS with Q4 elements;
• fem2d_pack_test08.m, tests ELEMENTS_EPS with T3 elements;
• fem2d_pack_test09.m, tests ELEMENTS_EPS with T4 elements;
• fem2d_pack_test10.m, tests ELEMENTS_EPS with T6 elements;
• fem2d_pack_test105.m, tests GRID_NODES_01;
• fem2d_pack_test11.m, tests GRID_TEST;
• fem2d_pack_test12.m, tests INTERP_TEST;
• fem2d_pack_test13.m, tests MAP_TEST;
• fem2d_pack_test14.m, tests MASS_MATRIX_T6;
• fem2d_pack_test15.m, tests PHYSICAL_TO_REFERENCE_T3 and REFERENCE_TO_PHYSICAL_T3.
• fem2d_pack_test16.m, tests PHYSICAL_TO_REFERENCE_T6.
• fem2d_pack_test17.m, tests S_L2NORM;
• fem2d_pack_test18.m, tests SHAPE_TEST;
• fem2d_pack_test19.m, tests SPHERE_GRID_Q4;
• fem2d_pack_test20.m, tests SPHERE_GRID_Q9;
• fem2d_pack_test21.m, tests SPHERE_GRID_Q16;
• fem2d_pack_test22.m, tests SPHERE_GRID_T3;
• fem2d_pack_test23.m, tests SPHERE_GRID_T6;
• fem2d_pack_test24.m, tests TRIANGLE_UNIT_SET.
• fem2d_pack_test.out, the sample output.
• fem2d_pack_test_q4.png, a PNG image of a Q4 mesh.
• fem2d_pack_test_t3.png, a PNG image of a T3 mesh.
• fem2d_pack_test_t4.png, a PNG image of a T4 mesh.
• fem2d_pack_test_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.

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