FEM_BASIS
Finite Element Basis Functions
on an M-dimensional Simplex

FEM_BASIS is a MATLAB library which can evaluate basis functions associated with an M-dimensional simplex (a 1D interval, a 2D triangle, a 3D tetrahedron, and the higher-dimensional generalizations).

A complete polynomial space is generated, up to the user-specified degree. Thus, for instance, in 2D, simply by specifying the degree, the user can request a space of:

        degree 0,  1 function,  1;
        degree 1,  3 functions, 1, x, y;
        degree 2,  6 functions, 1, x, y, x^2, xy, y^2;
        degree 3, 10 functions, 1, x, y, x^2, xy, y^2, x^3, x^2y, xy^2, y^3
      

The feature of this library is to generate a Lagrange form of the basis set of the given degree, and to evaluate those basis functions at any point.

The Triangular Case:

We will discuss the triangular case in detail; it is general enough to show all the details of the procedure, while still being simple enough to describe in pictures and tables.

Given the maximum degree D for the polynomial basis defined on a reference triangle, we have ( ( D + 1 ) * ( D + 2 ) ) / 2 monomials of degree at most D. Normally, the reference triangle is supplied with barycentric coordinates (xsi1,xsi2,xsi3) which are nonnegative and add to 1. For our purposes, we will use barycentric coordinates that have been multiplied by D. This means that all the coordinates we are interested in will be integers, which we will now identify by (I,J,K) = D * (xsi1,xsi2,xsi3). We can then define a grid of integer coordinates so that 0 <= I, J, K <= D and I+J+K = D, with (I,J,K) corresponding to

• the basis point (X,Y)(I,J,K) = ( I/D, J/D );
• the basis monomial P(I,J,K)(X,Y) = X^I Y^J.

For example, with D = 2, we have simply:

        F
        |\
        C-E
        |\|\
        A-B-D
      

         I J K    X    Y    P(I,J,K)(X,Y)

      A (0 0 2) (0.0, 0.0)  1
      B (1 0 1) (0.5, 0.0)  x
      C (0 1 1) (0.0, 0.5)  y
      D (2 0 0) (1.0, 0.0)  x^2
      E (1 1 0) (0.5, 0.5)  x y
      F (0 2 0) (0.0, 1.0)  y^2
      

Instead of the monomials P(I,J,K)(X,Y), we want a set of polynomials L(I,J,K)(X,Y) which span the same space, but have the Lagrange property, namely L(I,J,K)(X,Y) is 1 if (X,Y) is equal to (X,Y)(I,J,K), and 0 if (X,Y) is equal to any other of the basis points. One effect of this change is to tie each basis point (X,Y) to a particular basis polynomial L(I,J,K)(X,Y), namely, the polynomial which is 1 there.

This is easily arranged. Given an index (I,J,K), we compute a polynomial that has:

1. I factors of the form (X-0) * (X-1/D) * ... * (X-(I-1)/D);
2. J factors of the form (Y-0) * (Y-1/D) * ... * (Y-(J-1)/D);
3. K factors of the form (X+Y-(D-0)/D) * (X+Y-(D-1)/D) * ... * (X+Y-(D-(K-1))/D).

This polynomial is the product of I+J+K linear factors, in other words, it is a polynomial of degree D. This polynomial is 0 at all basis points except (X,Y)(I,J,K). If we divide this polynomial by its value at the basis point, we arrive at the desired Lagrange polynomial L(I,J,K)(X,Y).

This method is applicable for a basis set of any polynomial degree; moreover, it is a straightforward matter to extend it to higher dimensions.

The Triangular Prism:

A (right) triangular prism is a 3D shape which can be formed by drawing a triangle in the XY plane and then "lifting" the triangle along a line in the Z direction a fixed distance.

A finite element basis family can be defined for this shape, using the product of basis functions in the XY triangle and basis functions for the Z line. The basis functions are then defined by scaled barycentric coordinates I(1), I(2), I(3) for the triangle, and an independent set of scaled barycentric coordinates J(1) and J(2) for the line. A function is provided to evaluate basis functions for such an element.

Licensing:

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

Languages:

FEM_BASIS is available in a C++ version and a FORTRAN90 version and a MATLAB version and

Related Data and Programs:

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

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

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

FEM3D_PACK is a MATLAB library which contains utilities for 3D finite element calculations.

Reference:

1. John Burkardt,
   The Finite Element Basis for Simplices in Arbitrary Dimensions,
   http://people.sc.fsu.edu/~jburkardt/presentations/fem_basis.pdf.
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,
   TA640.2.Z54.

Source Code:

• fem_basis_1d.m, evaluates an arbitrary 1D basis function.
• fem_basis_2d.m, evaluates an arbitrary 2D basis function.
• fem_basis_3d.m, evaluates an arbitrary 3D basis function.
• fem_basis_md.m, evaluates an arbitrary M-dimensional basis function.
• fem_basis_prism_triangle.m, evaluates an arbitrary basis function in a 3D prism formed with a triangular base in XY.
• timestamp.m, prints the current YMDHMS date as a timestamp.

Examples and Tests:

• fem_basis_test.m, a sample calling program.
• fem_basis_test01.m, tests FEM_BASIS_1D.
• fem_basis_test02.m, tests FEM_BASIS_2D.
• fem_basis_test03.m, tests FEM_BASIS_3D.
• fem_basis_test04.m, repeats TEST01 using FEM_BASIS_MD.
• fem_basis_test05.m, repeats TEST02 using FEM_BASIS_MD.
• fem_basis_test06.m, repeats TEST03 using FEM_BASIS_MD.
• fem_basis_test07.m, tests FEM_BASIS_PRISM_TRIANGLE using order 2 in XY and in Z.
• fem_basis_test08.m, tests FEM_BASIS_PRISM_TRIANGLE using order 3 in XY and order 1 in Z.
• fem_basis_test_output.txt, the output file.

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