Basis Functions and Shape Functions

• While the weak form is essentially what you need for adding physics to MOOSE, in traditional finite element software more work is necessary.

• We need to discretize our weak form and select a set of simple "basis functions" amenable for manipulation by a computer.

Example of linear Lagrange shape function associated with single node on triangular mesh

1D linear Lagrange shape functions

Shape Functions

• Our discretized expansion of u takes on the following form:

• The here are called "basis functions"

• These form the basis for the "trial function",

• Analogous to the we used earlier

• The gradient of can be expanded similarly:

• In the Galerkin finite element method, the same basis functions are used for both the trial and test functions:

• Substituting these expansions back into our weak form, we get:

• The left-hand side of the equation above is what we generally refer to as the component of our "Residual Vector" and write as .

• Shape Functions are the functions that get multiplied by coefficients and summed to form the solution.

• Individual shape functions are restrictions of the global basis functions to individual elements.

• They are analogous to the functions from polynomial fitting (in fact, you can use those as shape functions).

• Typical shape function families: Lagrange, Hermite, Hierarchic, Monomial, Clough-Toucher - MOOSE has support for all of these.

• Lagrange shape functions are the most common. - They are interpolary at the nodes, i.e., the coefficients correspond to the values of the functions at the nodes.

Linear Lagrange

Cubic Lagrange

Cubic Hermite