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<html>
<head>
<title>
FEM2D_PACK - 2D Finite Element Routines
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
FEM2D_PACK <br> 2D Finite Element Routines
</h1>
<hr>
<p>
<b>FEM2D_PACK</b>
is a MATLAB library which
contains utilities for implementing the finite element method.
</p>
<p>
The emphasis is on simplicity and clarity.
Only the 2D case is handled, with a choice
of low order triangular and quadrilateral elements.
</p>
<p>
A few routines are included for computing a "sphere grid",
that is, a finite element mesh on the surface of a sphere.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>FEM2D_PACK</b> is available in
<a href = "../../c_src/fem2d_pack/fem2d_pack.html">a C version</a> and
<a href = "../../cpp_src/fem2d_pack/fem2d_pack.html">a C++ version</a> and
<a href = "../../f77_src/fem2d_pack/fem2d_pack.html">a FORTRAN77 version</a> and
<a href = "../../f_src/fem2d_pack/fem2d_pack.html">a FORTRAN90 version</a> and
<a href = "../../m_src/fem2d_pack/fem2d_pack.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/fem_basis/fem_basis.html">
FEM_BASIS</a>,
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.)
</p>
<p>
<a href = "../../m_src/fem_io/fem_io.html">
FEM_IO</a>,
a MATLAB library which
reads or writes
the node, element and data files that define a finite element model.
</p>
<p>
<a href = "../../m_src/fem1d_pack/fem1d_pack.html">
FEM1D_PACK</a>,
a MATLAB library which
contains utilities for 1D finite element calculations.
</p>
<p>
<a href = "../../data/fem2d/fem2d.html">
FEM2D</a>,
a data directory which
contains examples of 2D FEM files,
text files that describe a 2D finite element geometry
and associated nodal values;
</p>
<p>
<a href = "../../m_src/fem2d_heat/fem2d_heat.html">
FEM2D_HEAT</a>,
a MATLAB program which
solves the time dependent heat equation in the unit square.
</p>
<p>
<a href = "../../m_src/fem2d_poisson/fem2d_poisson.html">
FEM2D_POISSON</a>,
a MATLAB program which
solves Poisson's equation
on a square, using the finite element method.
</p>
<p>
<a href = "../../m_src/fem2d_poisson_rectangle_linear/fem2d_poisson_rectangle_linear.html">
FEM2D_POISSON_RECTANGLE_LINEAR</a>,
a MATLAB program which
solves the 2D Poisson equation on a rectangle, using the finite element method,
and piecewise linear triangular elements.
</p>
<p>
<a href = "../../m_src/fem2d_sample/fem2d_sample.html">
FEM2D_SAMPLE</a>,
a MATLAB library which
evaluates a finite element function defined on an order 3 or order 6 triangulation.
</p>
<p>
<a href = "../../m_src/fem3d_pack/fem3d_pack.html">
FEM3D_PACK</a>,
a MATLAB library which
contains utilities for 3D finite element calculations.
</p>
<p>
<a href = "../../m_src/oned/oned.html">
ONED</a>,
a MATLAB library which
contains functions useful for 1D finite element calculations.
</p>
<p>
<a href = "../../m_src/threed/threed.html">
THREED</a>,
a MATLAB library which
contains functions useful for 3D finite element calculations.
</p>
<p>
<a href = "../../m_src/twod/twod.html">
TWOD</a>,
a MATLAB library which
contains functions useful for 2D finite element calculations.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Milton Abramowitz, Irene Stegun,<br>
Handbook of Mathematical Functions,<br>
National Bureau of Standards, 1964,<br>
ISBN: 0-486-61272-4,<br>
LC: QA47.A34.
</li>
<li>
Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart,<br>
LINPACK User's Guide,<br>
SIAM, 1979,<br>
ISBN13: 978-0-898711-72-1.
</li>
<li>
Vladimir Krylov,<br>
Approximate Calculation of Integrals,<br>
Dover, 2006,<br>
ISBN: 0486445798.
</li>
<li>
Hans Rudolf Schwarz,<br>
Finite Element Methods,<br>
Academic Press, 1988,<br>
ISBN: 0126330107,<br>
LC: TA347.F5.S3313.
</li>
<li>
Gilbert Strang, George Fix,<br>
An Analysis of the Finite Element Method,<br>
Cambridge, 1973,<br>
ISBN: 096140888X,<br>
LC: TA335.S77.
</li>
<li>
Arthur Stroud,<br>
Approximate Calculation of Multiple Integrals,<br>
Prentice Hall, 1971,<br>
ISBN: 0130438936,<br>
LC: QA311.S85.
</li>
<li>
Arthur Stroud, Don Secrest,<br>
Gaussian Quadrature Formulas,<br>
Prentice Hall, 1966,<br>
LC: QA299.4G3S7.
</li>
<li>
Olgierd Zienkiewicz,<br>
The Finite Element Method,<br>
Sixth Edition,<br>
Butterworth-Heinemann, 2005,<br>
ISBN: 0750663200,<br>
LC: TA640.2.Z54
</li>
<li>
Daniel Zwillinger, editor,<br>
CRC Standard Mathematical Tables and Formulae,<br>
30th Edition,<br>
CRC Press, 1996,<br>
ISBN: 0-8493-2479-3.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "bandwidth_mesh.m">bandwidth_mesh.m</a>
computes the bandwidth associated with the finite element mesh;
</li>
<li>
<a href = "bandwidth_var.m">bandwidth_var.m</a>
computes the bandwidth associated with an arbitrary assignment
of variables to the nodes of the finite element mesh;
</li>
<li>
<a href = "basis_11_t3.m">basis_11_t3.m</a>
one basis at one point for the T3 element;
</li>
<li>
<a href = "basis_11_t3_test.m">basis_11_t3_test.m</a>
verifies BASIS_11_T3;
</li>
<li>
<a href = "basis_11_t4.m">basis_11_t4.m</a>
one basis at one point for the T4 element;
</li>
<li>
<a href = "basis_11_t4_test.m">basis_11_t4_test.m</a>
verifies BASIS_11_T4;
</li>
<li>
<a href = "basis_11_t6.m">basis_11_t6.m</a>
one basis at one point for the T6 element;
</li>
<li>
<a href = "basis_11_t6_test.m">basis_11_t6_test.m</a>
verifies BASIS_11_T6;
</li>
<li>
<a href = "basis_mn_q4.m">basis_mn_q4.m</a>
all basis functions for N points in a Q4 element;
</li>
<li>
<a href = "basis_mn_q4_test.m">basis_mn_q4_test.m</a>
verifies BASIS_MN_Q4.
</li>
<li>
<a href = "basis_mn_t3.m">basis_mn_t3.m</a>
all basis functions for N points in a T3 element.
</li>
<li>
<a href = "basis_mn_t3_test.m">basis_mn_t3_test.m</a>
verifies BASIS_MN_T3.
</li>
<li>
<a href = "basis_mn_t4.m">basis_mn_t4.m</a>
all basis functions for N points in a T4 element.
</li>
<li>
<a href = "basis_mn_t4_test.m">basis_mn_t4_test.m</a>
verifies BASIS_MN_T4.
</li>
<li>
<a href = "basis_mn_t6.m">basis_mn_t6.m</a>
evaluates all basis functions for a T6 element.
</li>
<li>
<a href = "basis_mn_t6_test.m">basis_mn_t6_test.m</a>
verifies the basis values at the basis nodes for a T6 element.
</li>
<li>
<a href = "ch_cap.m">ch_cap.m</a>
capitalizes a single character.
</li>
<li>
<a href = "degrees_to_radians.m">degrees_to_radians.m</a>
converts an angle from degrees to radians.
</li>
<li>
<a href = "derivative_average_t3.m">derivative_average_t3.m</a>
averages derivatives at the nodes of a T3 mesh.
</li>
<li>
<a href = "div_q4.m">div_q4.m</a>
estimates the divergence and vorticity of a discrete field.
</li>
<li>
<a href = "element_code.m">element_code.m</a>
returns the code for each element.
</li>
<li>
<a href = "elements_eps.m">elements_eps.m</a>
creates an EPS file image of the elements of a grid.
</li>
<li>
<a href = "grid_element.m">grid_element.m</a>
returns the grid associated with any available element.
</li>
<li>
<a href = "grid_element_num.m">grid_element_num.m</a>
counts the elements in a grid given the element code.
</li>
<li>
<a href = "grid_node_num.m">grid_node_num.m</a>
counts the nodes in a grid given the element code.
</li>
<li>
<a href = "grid_nodes_01.m">grid_nodes.m</a>
defines a rectangular grid of equally spaced nodes in the unit square.
</li>
<li>
<a href = "grid_print.m">grid_print.m</a>
prints the elements that form a grid.
</li>
<li>
<a href = "grid_q4_element.m">grid_q4_element.m</a>
produces a grid of Q4 elements.
</li>
<li>
<a href = "grid_q4_element_num.m">grid_q4_element_num.m</a>
counts the elements in a grid of Q4 elements.
</li>
<li>
<a href = "grid_q4_node_num.m">grid_q4_node_num.m</a>
counts the nodes in a grid of Q4 elements.
</li>
<li>
<a href = "grid_q8_element.m">grid_q8_element.m</a>
produces a grid of Q8 elements.
</li>
<li>
<a href = "grid_q8_element_num.m">grid_q8_element_num.m</a>
counts the elements in a grid of Q8 elements.
</li>
<li>
<a href = "grid_q8_node_num.m">grid_q8_node_num.m</a>
counts the nodes in a grid of Q8 elements.
</li>
<li>
<a href = "grid_q9_element.m">grid_q9_element.m</a>
produces a grid of Q9 elements.
</li>
<li>
<a href = "grid_q9_element_num.m">grid_q9_element_num.m</a>
counts the elements in a grid of Q9 elements.
</li>
<li>
<a href = "grid_q9_node_num.m">grid_q9_node_num.m</a>
counts the nodes in a grid of Q9 elements.
</li>
<li>
<a href = "grid_q12_element.m">grid_q12_element.m</a>
produces a grid of Q12 elements.
</li>
<li>
<a href = "grid_q12_element_num.m">grid_q12_element_num.m</a>
counts the elements in a grid of Q12 elements.
</li>
<li>
<a href = "grid_q12_node_num.m">grid_q12_node_num.m</a>
counts the nodes in a grid of Q12 elements.
</li>
<li>
<a href = "grid_q16_element.m">grid_q16_element.m</a>
produces a grid of Q16 elements.
</li>
<li>
<a href = "grid_q16_element_num.m">grid_q16_element_num.m</a>
counts the elements in a grid of Q16 elements.
</li>
<li>
<a href = "grid_q16_node_num.m">grid_q16_node_num.m</a>
counts the nodes in a grid of Q16 elements.
</li>
<li>
<a href = "grid_ql_element.m">grid_ql_element.m</a>
produces a grid of QL elements.
</li>
<li>
<a href = "grid_ql_element_num.m">grid_ql_element_num.m</a>
counts the elements in a grid of QL elements.
</li>
<li>
<a href = "grid_ql_node_num.m">grid_ql_node_num.m</a>
counts the nodes in a grid of QL elements.
</li>
<li>
<a href = "grid_shape_2d.m">grid_shape_2d.m</a>
guesses the shape N1 by N2 of a vector of data.
</li>
<li>
<a href = "grid_t3_element.m">grid_t3_element.m</a>
produces a grid of pairs of T3 elements.
</li>
<li>
<a href = "grid_t3_element_num.m">grid_t3_element_num.m</a>
counts the elements in a grid of T3 elements.
</li>
<li>
<a href = "grid_t3_node_num.m">grid_t3_node_num.m</a>
counts the nodes in a grid of T3 elements.
</li>
<li>
<a href = "grid_t4_element.m">grid_t4_element.m</a>
produces a grid of pairs of T4 elements.
</li>
<li>
<a href = "grid_t4_element_num.m">grid_t4_element_num.m</a>
counts the elements in a grid of T4 elements.
</li>
<li>
<a href = "grid_t4_node_num.m">grid_t4_node_num.m</a>
counts the nodes in a grid of T4 elements.
</li>
<li>
<a href = "grid_t6_element.m">grid_t6_element.m</a>
produces a grid of pairs of T6 elements.
</li>
<li>
<a href = "grid_t6_element_num.m">grid_t6_element_num.m</a>
counts the elements in a grid of T6 elements.
</li>
<li>
<a href = "grid_t6_node_num.m">grid_t6_node_num.m</a>
counts the nodes in a grid of T6 elements.
</li>
<li>
<a href = "grid_t10_element.m">grid_t10_element.m</a>
produces a grid of pairs of T10 elements.
</li>
<li>
<a href = "grid_t10_element_num.m">grid_t10_element_num.m</a>
counts the elements in a grid of T10 elements.
</li>
<li>
<a href = "grid_t10_node_num.m">grid_t10_node_num.m</a>
counts the nodes in a grid of T10 elements.
</li>
<li>
<a href = "grid_test.m">grid_test.m</a>
tests the grid routines.
</li>
<li>
<a href = "grid_width.m">grid_width.m</a>
computes the width of a given grid.
</li>
<li>
<a href = "i4_modp.m">i4_modp.m</a>
returns the nonnegative remainder of integer division.
</li>
<li>
<a href = "i4_sign.m">i4_sign.m</a>
returns the sign of an integer.
</li>
<li>
<a href = "i4_wrap.m">i4_wrap.m</a>
forces an integer to lie between given limits by wrapping.
</li>
<li>
<a href = "i4mat_write.m">i4mat_write.m</a>,
writes an I4MAT file.
</li>
<li>
<a href = "i4vec_print.m">i4vec_print.m</a>
prints an I4VEC;
</li>
<li>
<a href = "interp.m">interp.m</a>
interpolates a quantity in an element from basis node values.
</li>
<li>
<a href = "interp_test.m">interp_test.m</a>
tests the interpolation power of the basis functions for a given element;
</li>
<li>
<a href = "legendre_com.m">legendre_com.m</a>
computes abscissas and weights for Gauss-Legendre quadrature.
</li>
<li>
<a href = "legendre_set.m">legendre_set.m</a>
sets abscissas and weights for Gauss-Legendre quadrature.
</li>
<li>
<a href = "map.m">map.m</a>
returns the interpolation matrix for any available element.
</li>
<li>
<a href = "map_test.m">map_test.m</a>
tests the map routines.
</li>
<li>
<a href = "mass_matrix_t3.m">mass_matrix_t3.m</a>
computes the mass matrix for T3 elements.
</li>
<li>
<a href = "mass_matrix_t6.m">mass_matrix_t6.m</a>
computes the mass matrix for T6 elements.
</li>
<li>
<a href = "next_boundary_node.m">next_boundary_node.m</a>
returns the next boundary node in any element.
</li>
<li>
<a href = "next_boundary_node_q4.m">next_boundary_node_q4.m</a>
returns the next boundary node in a Q4 element.
</li>
<li>
<a href = "next_boundary_node_q8.m">next_boundary_node_q8.m</a>
returns the next boundary node in a Q8 element.
</li>
<li>
<a href = "next_boundary_node_q9.m">next_boundary_node_q9.m</a>
returns the next boundary node in a Q9 element.
</li>
<li>
<a href = "next_boundary_node_q12.m">next_boundary_node_q12.m</a>
returns the next boundary node in a Q12 element.
</li>
<li>
<a href = "next_boundary_node_q16.m">next_boundary_node_q16.m</a>
returns the next boundary node in a Q16 element.
</li>
<li>
<a href = "next_boundary_node_ql.m">next_boundary_node_ql.m</a>
returns the next boundary node in a QL element.
</li>
<li>
<a href = "next_boundary_node_t3.m">next_boundary_node_t3.m</a>
returns the next boundary node in a T3 element.
</li>
<li>
<a href = "next_boundary_node_t4.m">next_boundary_node_t4.m</a>
returns the next boundary node in a T4 element.
</li>
<li>
<a href = "next_boundary_node_t6.m">next_boundary_node_t6.m</a>
returns the next boundary node in a T6 element.
</li>
<li>
<a href = "next_boundary_node_t10.m">next_boundary_node_t10.m</a>
returns the next boundary node in a T10 element.
</li>
<li>
<a href = "node_reference.m">node_reference.m</a>
returns the basis nodes for any available element.
</li>
<li>
<a href = "node_reference_q4.m">node_reference_q4.m</a>
returns the basis nodes for a Q4 element.
</li>
<li>
<a href = "node_reference_q8.m">node_reference_q8.m</a>
returns the basis nodes for a Q8 element.
</li>
<li>
<a href = "node_reference_q9.m">node_reference_q9.m</a>
returns the basis nodes for a Q9 element.
</li>
<li>
<a href = "node_reference_q12.m">node_reference_q12.m</a>
returns the basis nodes for a Q12 element.
</li>
<li>
<a href = "node_reference_q16.m">node_reference_q16.m</a>
returns the basis nodes for a Q16 element.
</li>
<li>
<a href ="node_reference_ql.m">node_reference_ql.m</a>
returns the basis nodes for a QL element.
</li>
<li>
<a href = "node_reference_t3.m">node_reference_t3.m</a>
returns the basis nodes for a T3 element.
</li>
<li>
<a href = "node_reference_t4.m">node_reference_t4.m</a>
returns the basis nodes for a T4 element.
</li>
<li>
<a href = "node_reference_t6.m">node_reference_t6.m</a>
returns the basis nodes for a T6 element.
</li>
<li>
<a href = "node_reference_t10.m">node_reference_t10.m</a>
returns the basis nodes for a T10 element.
</li>
<li>
<a href = "ns_t6_var_count.m">ns_t6_var_count.m</a>
counts the Navier Stokes variables on a T6 grid.
</li>
<li>
<a href = "ns_t6_var_set.m">ns_t6_var_set.m</a>
sets the Navier Stokes variables on a T6 grid.
</li>
<li>
<a href = "order_code.m">order_code.m</a>
returns the order for each element.
</li>
<li>
<a href = "physical_to_reference_t3.m">physical_to_reference_t3.m</a>
maps points in physical space into a T3 reference element.
</li>
<li>
<a href = "points_plot.m">points_plot.m</a>
creates an EPS file containing an image of the nodes.
</li>
<li>
<a href = "poly.m">poly.m</a>
returns the polynomial terms associated with any available element.
</li>
<li>
<a href = "poly_q4.m">poly_q4.m</a>
returns the monomials associated with a Q4 element.
</li>
<li>
<a href = "poly_q8.m">poly_q8.m</a>
returns the monomials associated with an Q8 element.
</li>
<li>
<a href = "poly_q9.m">poly_q9.m</a>
returns the monomials associated with a Q9 element.
</li>
<li>
<a href = "poly_q12.m">poly_q12.m</a>
returns the monomials associated with a Q12 element.
</li>
<li>
<a href = "poly_q16.m">poly_q16.m</a>
returns the monomials associated with a Q16 element.
</li>
<li>
<a href = "poly_ql.m">poly_ql.m</a>
returns the monomials for a QL element.
</li>
<li>
<a href = "poly_t3.m">poly_t3.m</a>
returns the monomials associated with a T3 element.
</li>
<li>
<a href = "poly_t6.m">poly_t6.m</a>
returns the monomials associated with a T6 element.
</li>
<li>
<a href = "poly_t10.m">poly_t10.m</a>
returns the monomials associated with a T10 element.
</li>
<li>
<a href = "r8_power.m">r8_power.m</a>
returns a power of a real number.
</li>
<li>
<a href = "r8_uniform_01.m">r8_uniform_01.m</a>
returns a pseudorandom value in [0,1].
</li>
<li>
<a href = "r8ge_fa.m">r8ge_fa.m</a>
performs a LINPACK style PLU factorization of a DGE matrix.
</li>
<li>
<a href = "r8ge_inverse.m">r8ge_inverse.m</a>
computes the inverse of a matrix factored by DGE_FA.
</li>
<li>
<a href = "r8mat_print.m">r8mat_print.m</a>
prints an R8MAT, with an optional title.
</li>
<li>
<a href = "r8mat_print_some.m">r8mat_print_some.m</a>
prints some of an R8MAT, with an optional title.
</li>
<li>
<a href = "r8mat_write.m">r8mat_write.m</a>,
writes an R8MAT file.
</li>
<li>
<a href = "reference_sample.m">reference_sample.m</a>
samples a reference element.
</li>
<li>
<a href = "reference_to_physical_q4.m">reference_to_physical_q4.m</a>
maps points in a Q4 reference triangle into physical space.
</li>
<li>
<a href = "reference_to_physical_t3.m">reference_to_physical_t3.m</a>
maps points in a T3 reference triangle into physical space.
</li>
<li>
<a href = "reference_to_physical_t6.m">reference_to_physical_t6.m</a>
maps points in a T6 reference triangle into physical space.
</li>
<li>
<a href = "s_eqi.m">s_eqi.m</a>
is a case insensitive comparison of two strings for equality.
</li>
<li>
<a href = "s_l2norm.m">s_l2norm.m</a>
computes the "big" L2 norm of a scalar function over a region.
</li>
<li>
<a href = "s_len_trim.m">s_len_trim.m</a>
returns the length of a string to the last nonblank.
</li>
<li>
<a href = "serene.m">serene.m</a>
interpolates data using a Q8 element.
</li>
<li>
<a href = "shape.m">shape.m</a>
evaluates shape functions for any available element.
</li>
<li>
<a href = "shape_test.m">shape_test.m</a>
verifies the shape function values at the basis nodes.
</li>
<li>
<a href = "shape_q4.m">shape_q4.m</a>
evaluates shape functions for a Q4 element.
</li>
<li>
<a href = "shape_q8.m">shape_q8.m</a>
evaluates shape functions for a Q8 element.
</li>
<li>
<a href = "shape_q9.m">shape_q9.m</a>
evaluates shape functions for a Q9 element.
</li>
<li>
<a href = "shape_q12.m">shape_q12.m</a>
evaluates shape functions for a Q12 element.
</li>
<li>
<a href = "shape_q16.m">shape_q16.m</a>
evaluates shape functions for a Q16 element.
</li>
<li>
<a href = "shape_ql.m">shape_ql.m</a>
evaluates shape functions for a QL element.
</li>
<li>
<a href = "shape_t3.m">shape_t3.m</a>
evaluates shape functions for a T3 element.
</li>
<li>
<a href = "shape_t4.m">shape_t4.m</a>
evaluates shape functions for a T4 element.
</li>
<li>
<a href = "shape_t6.m">shape_t6.m</a>
evaluates shape functions for a T6 element.
</li>
<li>
<a href = "shape_t10.m">shape_t10.m</a>
evaluates shape functions for a T10 element.
</li>
<li>
<a href = "sphere_grid_element_num.m">sphere_grid_element_num.m</a>
returns the number of elements in a sphere grid.
</li>
<li>
<a href = "sphere_grid_node_num.m">sphere_grid_node_num.m</a>
returns the number of nodes in a sphere grid.
</li>
<li>
<a href = "sphere_grid_q4_element.m">sphere_grid_q4_element.m</a>
produces a Q4 sphere grid.
</li>
<li>
<a href = "sphere_grid_q4_element_num.m">sphere_grid_q4_element_num.m</a>
counts the elements in a Q4 sphere grid.
</li>
<li>
<a href = "sphere_grid_q4_node_num.m">sphere_grid_q4_node_num.m</a>
counts the nodes in a Q4 sphere grid.
</li>
<li>
<a href = "sphere_grid_q4_node_xyz.m">sphere_grid_q4_node_xyz.m</a>
sets the nodes in a Q4 sphere grid.
</li>
<li>
<a href = "sphere_grid_q9_element.m">sphere_grid_q9_element.m</a>
produces a Q9 sphere grid.
</li>
<li>
<a href = "sphere_grid_q9_element_num.m">sphere_grid_q9_element_num.m</a>
counts the elements in a Q9 sphere grid.
</li>
<li>
<a href = "sphere_grid_q9_node_num.m">sphere_grid_q9_node_num.m</a>
counts the nodes in a Q9 sphere grid.
</li>
<li>
<a href = "sphere_grid_q9_node_xyz.m">sphere_grid_q9_node_xyz.m</a>
sets the nodes in a Q9 sphere grid.
</li>
<li>
<a href = "sphere_grid_q16_element.m">sphere_grid_q16_element.m</a>
produces a Q16 sphere grid.
</li>
<li>
<a href = "sphere_grid_q16_element_num.m">sphere_grid_q16_element_num.m</a>
counts the elements in a Q16 sphere grid.
</li>
<li>
<a href = "sphere_grid_q16_node_num.m">sphere_grid_q16_node_num.m</a>
counts the nodes in a Q16 sphere grid.
</li>
<li>
<a href = "sphere_grid_q16_node_xyz.m">sphere_grid_q16_node_xyz.m</a>
sets the nodes in a Q16 sphere grid.
</li>
<li>
<a href = "sphere_grid_t3_element.m">sphere_grid_t3_element.m</a>
produces a T3 sphere grid.
</li>
<li>
<a href = "sphere_grid_t3_element_num.m">sphere_grid_t3_element_num.m</a>
counts the elements in a T3 sphere grid.
</li>
<li>
<a href = "sphere_grid_t3_node_num.m">sphere_grid_t3_node_num.m</a>
counts the nodes in a T3 sphere grid.
</li>
<li>
<a href = "sphere_grid_t3_node_xyz.m">sphere_grid_t3_node_xyz.m</a>
sets the nodes in a T3 sphere grid.
</li>
<li>
<a href = "sphere_grid_t6_element.m">sphere_grid_t6_element.m</a>
produces a T6 sphere grid.
</li>
<li>
<a href = "sphere_grid_t6_element_num.m">sphere_grid_t6_element_num.m</a>
counts the elements in a T6 sphere grid.
</li>
<li>
<a href = "sphere_grid_t6_node_num.m">sphere_grid_t6_node_num.m</a>
counts the nodes in a T6 sphere grid.
</li>
<li>
<a href = "sphere_grid_t6_node_xyz.m">sphere_grid_t6_node_xyz.m</a>
sets the nodes in a T6 sphere grid.
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>
prints the current YMDHMS date as a time stamp.
</li>
<li>
<a href = "triangle_unit_set.m">triangle_unit_set.m</a>
sets a quadrature rule in a unit triangle.
</li>
<li>
<a href = "triangle_unit_size.m">triangle_unit_size.m</a>
returns the order of a quadrature rule in a unit triangle.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "fem2d_pack_test.m">fem2d_pack_test.m</a>,
runs all the tests;
</li>
<li>
<a href = "fem2d_pack_test01.m">fem2d_pack_test01.m</a>,
tests BANDWIDTH_MESH;
</li>
<li>
<a href = "fem2d_pack_test02.m">fem2d_pack_test02.m</a>,
tests BANDWIDTH_VAR;
</li>
<li>
<a href = "fem2d_pack_test03.m">fem2d_pack_test03.m</a>,
tests BASIS_11_**_TEST;
</li>
<li>
<a href = "fem2d_pack_test04m">fem2d_pack_test04.m</a>,
tests BASIS_MN_**_TEST;
</li>
<li>
<a href = "fem2d_pack_test05.m">fem2d_pack_test05.m</a>,
tests DERIVATIVE_AVERAGE_T3;
</li>
<li>
<a href = "fem2d_pack_test06.m">fem2d_pack_test06.m</a>,
tests DIV_Q4;
</li>
<li>
<a href = "fem2d_pack_test07.m">fem2d_pack_test07m</a>,
tests ELEMENTS_EPS with Q4 elements;
</li>
<li>
<a href = "fem2d_pack_test08.m">fem2d_pack_test08.m</a>,
tests ELEMENTS_EPS with T3 elements;
</li>
<li>
<a href = "fem2d_pack_test09.m">fem2d_pack_test09.m</a>,
tests ELEMENTS_EPS with T4 elements;
</li>
<li>
<a href = "fem2d_pack_test10.m">fem2d_pack_test10.m</a>,
tests ELEMENTS_EPS with T6 elements;
</li>
<li>
<a href = "fem2d_pack_test105.m">fem2d_pack_test105.m</a>,
tests GRID_NODES_01;
</li>
<li>
<a href = "fem2d_pack_test11.m">fem2d_pack_test11.m</a>,
tests GRID_TEST;
</li>
<li>
<a href = "fem2d_pack_test12.m">fem2d_pack_test12.m</a>,
tests INTERP_TEST;
</li>
<li>
<a href = "fem2d_pack_test13.m">fem2d_pack_test13.m</a>,
tests MAP_TEST;
</li>
<li>
<a href = "fem2d_pack_test135.m">fem2d_pack_test135.m</a>,
tests MASS_MATRIX_T3;
</li>
<li>
<a href = "fem2d_pack_test14.m">fem2d_pack_test14.m</a>,
tests MASS_MATRIX_T6;
</li>
<li>
<a href = "fem2d_pack_test15.m">fem2d_pack_test15.m</a>,
tests PHYSICAL_TO_REFERENCE_T3 and REFERENCE_TO_PHYSICAL_T3.
</li>
<li>
<a href = "fem2d_pack_test16.m">fem2d_pack_test16.m</a>,
tests PHYSICAL_TO_REFERENCE_T6.
</li>
<li>
<a href = "fem2d_pack_test17.m">fem2d_pack_test17.m</a>,
tests S_L2NORM;
</li>
<li>
<a href = "fem2d_pack_test18.m">fem2d_pack_test18.m</a>,
tests SHAPE_TEST;
</li>
<li>
<a href = "fem2d_pack_test19.m">fem2d_pack_test19.m</a>,
tests SPHERE_GRID_Q4;
</li>
<li>
<a href = "fem2d_pack_test20.m">fem2d_pack_test20.m</a>,
tests SPHERE_GRID_Q9;
</li>
<li>
<a href = "fem2d_pack_test21.m">fem2d_pack_test21.m</a>,
tests SPHERE_GRID_Q16;
</li>
<li>
<a href = "fem2d_pack_test22.m">fem2d_pack_test22.m</a>,
tests SPHERE_GRID_T3;
</li>
<li>
<a href = "fem2d_pack_test23.m">fem2d_pack_test23.m</a>,
tests SPHERE_GRID_T6;
</li>
<li>
<a href = "fem2d_pack_test24.m">fem2d_pack_test24.m</a>,
tests TRIANGLE_UNIT_SET.
</li>
<li>
<a href = "fem2d_pack_test.out">fem2d_pack_test.out</a>,
the sample output.
</li>
<li>
<a href = "fem2d_pack_test_q4.png">fem2d_pack_test_q4.png</a>,
a PNG image of