FEM2D_PROJECT
Project Data onto a 2D Finite Element Mesh

FEM2D_PROJECT is a MATLAB program which projects a finite element function.

Let us suppose we have a region R and a triangulation of R, that is, a set of nodes N1 and triangles T1 whose union is R. Let P1(I)(X,Y) be the finite element basis function associated with node N1(I). Now let us suppose that we have a finite element function V1, that is a scalar- or vector-valued function V1(X,Y) defined over R, with the formula

V1(X,Y) = sum ( I ) V1(I) * P1(I)(X,Y)

Now suppose we have a second triangulation of R comprising a set of nodes N2 and triangles T2. Can we determine an appropriate set of finite element coefficients V2(I) which best approximate V1 in the finite element space defined by N2 and T2? The finite element coefficient vector V2 is defined by the following relationship:

Integral Sum ( I ) V2(I) P2(I)(X,Y) P2(J)(X,Y) dx dy = Integral V1(X,Y) P2(J)(X,Y) dx dy

This procedure can be used to determine the least squares approximant to data (actually, to the piecewise linear interpolant of that data) or to determine the finite element coefficients appropriate when recomputing a finite element solution from a fine mesh to a coarse mesh.

The sample data is given as three tables, each stored in a file:

• the SAMPLE_NODES file contains the 2D coordinates of sample points. Every sample point is presumed to lie within the area covered by the finite element mesh.
• the SAMPLE_ELEMENTS file contains the indices of nodes that form the elements. The elements are presumed to be 3-node triangles. This is simple a Delaunay triangulation of the sample nodes.
• the SAMPLE_VALUES file contains the value of some vector quantity V at each sample point. The dimensionality of the V data is arbitrary.

The finite element mesh is given as two tables, each stored in a file:

• the FEM_NODES file contains the 2D coordinates of nodes.
• the FEM_ELEMENTS file contains the indices of nodes that form the elements. The elements are presumed to be 3-node triangles.

The program produces a new table FEM_VALUES, of the same dimensionality as SAMPLE_VALUES. The vector FEM_VALUES can be used in conjunction with the finite element mesh data to produce a finite element function that is an approximant to the SAMPLE_VALUES data.

Usage:

fem2d_project ( 'sample_prefix', 'fem_prefix' )

• sample_prefix_nodes.txt, the node coordinates (input);
• sample_prefix_elements.txt, the 3 nodes that make up each element (input);
• sample_prefix_values.txt, the data values (input);

• fem_prefix_nodes.txt, the node coordinates (input);
• fem_prefix_elements.txt, the 3 nodes that make up each element (input);
• fem_prefix_values.txt, the data values (output).

The file fem_prefix_values.txt is created by the program, and contains the projections of the sample data values onto the finite element space, that is, these may be regarded as coefficients of finite element functions representing the projections of the sample data. Note that we may also regard this operation as the refinement or coarsening of a finite element function, in that we are transferring information from the ``sample'' space to the ``fem'' space.

Licensing:

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

Languages:

FEM2D_PROJECT is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

FEM1D_PROJECT, a MATLAB program which projects data into a finite element space, including the least squares approximation of data, or the projection of a finite element solution from one mesh to another.

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

FEM2D_PROJECT_FUNCTION, a MATLAB program which projects a function F(X,Y), given as a formula, into a given finite element space of piecewise linear triangular elements.

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

FEM3D_PROJECT, a MATLAB program which projects a function F(X,Y,Z), given as a data, into a given finite element space of piecewise linear tetrahedral elements.

Reference:

1. Hans Rudolf Schwarz,
   Finite Element Methods,
   Academic Press, 1988,
   ISBN: 0126330107,
   LC: TA347.F5.S3313.
2. Gilbert Strang, George Fix,
   An Analysis of the Finite Element Method,
   Cambridge, 1973,
   ISBN: 096140888X,
   LC: TA335.S77.
3. Olgierd Zienkiewicz,
   The Finite Element Method,
   Sixth Edition,
   Butterworth-Heinemann, 2005,
   ISBN: 0750663200,
   LC: TA640.2.Z54.

Source Code:

• fem2d_project.m, the source code.

Examples and Tests:

LINEAR starts with sample data for the vector function f(x)=[ 1, 2x, 3y ], on an 9 x 9 grid of equally spaced nodes from [0.0,8.0]x[0.0,8.0], and projects this onto a piecewise linear finite element meshes defined on equally spaced grids of dimension 4x4, 2x2 and 1x1.

• r8x8_t3_nodes.txt, the sample nodes, on an 8x8 grid.
• r8x8_t3_elements.txt, elements that can be used to form an 8x8 finite element mesh associated with the sample data. This is provide only so that a finite element function can be formed with the original sample data.
• r8x8_t3_values.txt, the sample nodal values.
• r4x4_t3_nodes.txt, the FEM nodes for a 4x4 grid.
• r4x4_t3_elements.txt, the FEM elements for a 4x4 grid.
• r4x4_t3_values.txt, the nodal values as projected from the 8x8 grid.
• r2x2_t3_nodes.txt, the FEM nodes for a 2x2 grid.
• r2x2_t3_elements.txt, the FEM elements for a 2x2 grid.
• r2x2_t3_values.txt, the nodal values as projected from the 8x8 grid.
• r1x1_t3_nodes.txt, the FEM nodes for a 1x1 grid.
• r1x1_t3_elements.txt, the FEM elements for a 1x1 grid.
• r1x1_t3_values.txt, the nodal values as projected from the 8x8 grid.

CIRCLE is a problem defined on a circular region with a small central hole. The SAMPLE data uses 49,408 nodes and 98,304 elements. This data is to be projected onto a finite element mesh (which is a simple coarsening of the mesh on which the sample data was generated) that uses 12,416 nodes and 24,576 elements.

• circle_sample_nodes.txt, the sample nodes.
• circle_sample_elements.txt, the sample elements.
• circle_sample_values.txt, the sample values.
• circle_sample.png, a plot of the sample data, created by TWOD_PLOTC.
• circle_fem_nodes.txt, the FEM nodes for a 4x4 grid.
• circle_fem_elements.txt, the FEM elements for a 4x4 grid.
• circle_fem_values.txt, the FEM nodal values as projected by FEM2D_PROJECT.
• circle_fem.png, a plot of the FEM data, created by TWOD_PLOTC.
• circle_output.txt, the output file.

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