FEM_BASIS_T4_DISPLAY
Display a T4 Finite Element Basis Function

FEM_BASIS_T4_DISPLAY is a MATLAB program which reads data defining a T4 mesh of 4-node triangles, and displays the basis function associated with any node in the mesh.

The entire mesh is shown (which could be a problem for large meshes!). However, a finite element basis function is nonzero over only a relatively small number of elements. Over these elements, the basis function is cubic, and the curved nature of this part of the surface is suggested by taking many plot points in this region.

The best part of the presentation is that MATLAB allows the user to turn the plot or zoom in and out, which makes it much easier to understand the information that is being presented.

The triangular mesh is defined by a node file containing the coordinates of nodes, and a triangle file containing sets of the four nodes that make up each triangle.

Note that the four nodes making up each triangle are to be listed in a particular order, in which the vertices come first, followed by the centroid, as suggested by the following diagram:

        3
        |\
        | \
        |  \
        |   \
        | 4  \
        |     \
        |      \
        1-------2
      

Usage:

fem_basis_t4_display ( 'node_file', 'element_file' )

• 'node_file' lists the node coordinates
• 'element_file' lists the node indices that form elements

Licensing:

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

Related Programs:

BEZIER_SURFACE_DISPLAY is a MATLAB program which displays a Bezier surface.

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

FEM_50_HEAT is a MATLAB program which is a modified version of FEM_50 suitable for solving the heat equation.

FEM_BASIS_T3_DISPLAY is a MATLAB program which displays a basis function associated with a 3-node T3 mesh.

FEM_BASIS_T6_DISPLAY is a MATLAB program which displays a basis function associated with a 6-node T6 mesh.

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

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

FEM2D_PACK is a MATLAB library which is useful for 2D finite element calculations.

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

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

FREE_FEM_HEAT is a MATLAB program which solves the time dependent heat equation in an arbitrary triangulated 2D region.

FREE_FEM_NAVIER_STOKES is a MATLAB program which solves the steady incompressible Navier Stokes equations on an arbitrary triangulated region, using the finite element method.

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

FREE_FEM_STOKES is a MATLAB program which solves the Stokes equations on a triangulated region, using the finite element method.

POLYGONAL_SURFACE_DISPLAY is a MATLAB program which reads two files defining a polygonal surface and displays it within MATLAB.

QUAD_SURFACE_DISPLAY is a MATLAB program which reads files defining a 3D quadrilateral mesh surface and displays it within MATLAB.

TET_MESH_DISPLAY is a MATLAB program which can read in the node and tetra files defining a tet mesh and display a wireframe image.

TRI_SURFACE_DISPLAY is a MATLAB program which reads data defining a triangular mesh of a 3D surface and displays it.

TRIANGLE is a C program which computes a triangulation of a geometric region.

TRIANGULATION_DISPLAY_OPEN_GL is a C++ program which reads files defining a triangulation and displays an image using Open GL.

TRIANGULATION_ORDER6 is a directory which contains a description and examples of order 6 triangulations.

TRIANGULATION_ORDER6_CONTOUR is a MATLAB program which can make contours of a scalar quantity defined on the nodes of an order 6 triangulation.

TRIANGULATION_PLOT is a MATLAB program which creates an EPS image of a triangulation.

Reference:

1. Klaus-Juergen Bathe,
   Finite Element Procedures,
   Prentice Hall, 1996.
2. Eric Becker, Graham Carey, John Oden,
   Finite Elements, An Introduction, Volume I,
   Prentice-Hall, 1981.
3. Herbert Edelsbrunner,
   Geometry and Topology for Mesh Generation,
   Cambridge, 2001,
   LC: QA377.E36,
   ISBN: 0-521-79309-2.
4. Max Gunzburger,
   Finite Element Methods for Viscous Incompressible Flows,
   A Guide to Theory, Practice, and Algorithms,
   Academic Press, 1989,
   LC: TA357.G86.
5. Gilbert Strang, George Fix,
   An Analysis of the Finite Element Method,
   Prentice Hall, 1973,
   LC: TA335.S77.

Source code:

• basis_11_t4.m evaluates one of the basis functions associated with a T4 triangular element.
• fem_basis_t4_display.m gets information about a T4 mesh, and the node for which the corresponding T4 basis function is desired, and displays a plot.
• i4_modp.m performs a modulus function, always with a positive result.
• i4_wrap.m uses "wrapping" to force a value to lie within a given range.
• s_escape_tex.m de-escapes TeX escape strings.
• timestamp.m prints the current YMDHMS date as a time stamp.

Tests and Examples

RECTANGLE is a simple 5 by 4 mesh of 44 nodes and 24 elements. Files you may copy include:

• rectangle_nodes_t4.txt, the node file.
• rectangle_nodes_t4.png, a PNG image of the nodes.
• rectangle_triangles_t4.txt, the element file.
• rectangle_triangles_t4.png, a PNG image of the elements.
• basis12.png, a PNG image of the basis function associated with node 12.
• basis35.png, a PNG image of the basis function associated with node 35.

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