FEM2D_PACK
Finite Element Routines

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:

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 C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

FEM_BASIS, a C library which can define basis functions for the finite element method (FEM) for any degree in an M-dimensional simplex (1D interval, 2D triangle, 3D tetrahedron, and higher dimensional generalizations.)

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:

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.
9. Daniel Zwillinger, editor,
   CRC Standard Mathematical Tables and Formulae,
   30th Edition,
   CRC Press, 1996,
   ISBN: 0-8493-2479-3.

Source Code:

• fem2d_pack.c, the source code;
• fem2d_pack.h, the include file;

Examples and Tests:

• fem2d_pack_test.c, the calling program;
• fem2d_pack_test.txt, the sample output.

List of Routines:

• 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.
• 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.
• I4VEC_PRINT prints an I4VEC.
• MASS_MATRIX_T3 computes the mass matrix, using 3-node triangles.
• 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.
• ORDER_CODE returns the order for each element.
• PHYSICAL_TO_REFERENCE_T3 maps physical points to reference points.
• R8_ABS returns the absolute value of an R8.
• R8_EPSILON returns the R8 round off unit.
• R8_HUGE returns a "huge" R8.
• R8_NINT returns the nearest integer to an R8.
• R8_POWER computes an integer power of an R8.
• R8_UNIFORM_01 returns a pseudorandom R8 scaled to [0,1].
• 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.
• R8MAT_PRINT_SOME prints some of an R8MAT.
• R8MAT_WRITE writes an R8MAT file.
• 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.
• 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.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TRIANGLE_UNIT_SET sets a quadrature rule in the unit triangle.
• TRIANGLE_UNIT_SIZE returns the "size" of a unit triangle quadrature rule.

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