FEM2D_PACK
Finite Element Routines

FEM2D_PACK is a FORTRAN77 library which can help in implementing the finite element method.

The emphasis is on simplicity and clarity. Primarily, the 2D planar case is handled, with a choice of low order triangular ("T") and quadrilateral ("Q") 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.

Related Data and Programs:

DUNAVANT is a FORTRAN90 library which defines Dunavant rules for quadrature on a triangle.

FEKETE is a FORTRAN90 library which defines Fekete rules for interpolation or quadrature on a triangle.

FEM is a data directory which contains a description and examples of files that describe a finite element model.

FEM_50 is a MATLAB program which implements a finite element calculation in just 50 lines of code.

FEM_50_HEAT is a MATLAB program which implements a finite element calculation specifically for 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 displays a basis function associated with a quadratic triangle ("T6") mesh.

FEM_IO is a FORTRAN90 library which reads or writes node, element and data files defining a finite element model.

FEM_TO_TEC is a MATLAB program which converts FEM models into TEC graphics files.

FEM1D is a FORTRAN77 program which applies the finite element method to a 1D linear two point boundary value problem.

FEM1D_ADAPTIVE is a FORTRAN77 program which applies the finite element method to a 1D linear two point boundary value problem using adaptive refinement to improve the solution.

FEM1D_NONLINEAR is a FORTRAN77 program which applies the finite element method to a 1D nonlinear two point boundary value problem.

FEM1D_PMETHOD is a FORTRAN77 program which applies the p-method version of the finite element method to a 1D linear two point boundary value problem.

FEM2D_HEAT is a FORTRAN90 program which solves the 2D time dependent heat equation on the unit square.

FEM2D_POISSON is a FORTRAN90 program which solves the 2D Poisson equation on a square, using the finite element method.

FEM2D_SAMPLE is a FORTRAN77 library which evaluates a finite element function defined on an order 3 or order 6 triangulation.

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

FREE_FEM_HEAT is a FORTRAN90 program which uses the finite element method and the backward Euler method to solve the 2D time-dependent heat equation on an arbitrary triangulated region.

FREE_FEM_NAVIER_STOKES is a FORTRAN90 program which solves the 2D incompressible Navier Stokes equations in an arbitrary triangulated region.

FREE_FEM_POISSON is a FORTRAN90 program which solves Poisson's equation on a triangulated region, using the finite element method.

FREE_FEM_STOKES is a FORTRAN90 program which solves the 2D incompressible Stokes equations in an arbitrary triangulated region.

HOT_PIPE is a MATLAB program which uses FEM_50_HEAT to solve the heat equation for a pipe model.

HOT_POINT is a MATLAB program which uses FEM_50_HEAT to solve the heat equation for a 2D region with a hot point.

MESH_BANDWIDTH is a FORTRAN90 program which returns the geometric bandwidth associated with a mesh of elements of any order and in a space of arbitrary dimension.

POLYGONAL_SURFACE is a data directory which contains a description of the data files that can be used to describe a polygonal surface. FEM2D_PACK includes routines for creating a polygonal surface grid of a sphere.

TRIANGULATION_ORDER3 is a data directory which contains a description and examples of order 3 (linear) triangulations.

TRIANGULATION_ORDER6 is a data directory which contains a description and examples of an order 6 triangulation.

TRIANGULATION_ORDER6_CONTOUR is a MATLAB program which makes contour plots of data defined on an order 6 triangulation.

WANDZURA is a FORTRAN90 library which defines Wandzura rules for quadrature on a triangle.

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,
   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:

• fem2d_pack.f, the source code;
• fem2d_pack.csh, commands to compile the source code.

Examples and Tests:

• fem2d_pack_prb.f, the calling program;
• fem2d_pack_prb.csh, commands to compile, link and run the calling program;
• fem2d_pack_prb_output.txt, the output file.
• 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_t4.png, a PNG image of a T4 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:

• BANDWIDTH_MESH: bandwidth of finite element mesh.
• BANDWIDTH_VAR determines the bandwidth for finite element variables.
• BASIS_MN_T3: all bases at N points for a T3 element.
• BASIS_11_Q4: one basis at one point for a Q4 element.
• BASIS_11_Q4_TEST verifies BASIS_11_Q4.
• BASIS_MN_T6: all bases at N points for a T6 element.
• CH_CAP capitalizes a single character.
• DEGREES_TO_RADIANS converts an angle measure from degrees to radians.
• DTABLE_WRITE0 writes a DTABLE file with no header.
• ELEMENT_CODE returns the code for each element.
• GET_UNIT returns a free FORTRAN unit number.
• I4_MODP returns the nonnegative remainder of integer division.
• I4_WRAP forces an I4 to lie between given limits by wrapping.
• ITABLE_WRITE0 writes an ITABLE file with no header.
• R8_HUGE returns a "huge" R8.
• R8_POWER computes the P-th power of an R8.
• R8_SWAP switches two R8's.
• R8_UNIFORM_01 returns a unit pseudorandom R8.
• R8MAT_PRINT prints an R8MAT.
• R8MAT_PRINT_SOME prints some of an R8MAT.
• REFERENCE_TO_PHYSICAL_T3 maps T3 reference points to physical points.
• REFERENCE_TO_PHYSICAL_T6 maps T6 reference points to physical points.
• TIMESTAMP prints out the current YMDHMS date as a timestamp.

You can go up one level to the FORTRAN77 source codes.