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.

ddg_hat_function.png

basis_shape_functions_edited.png
Copyright Oden, Becker, Carey 1981

Shape Functions

  • Our discretized expansion of $$$u$$$ takes on the following form: $$$$u \approx u_h = \sum_{j=1}^N u_j \phi_j$$$$

    • The $$$\phi_j$$$ here are called "basis functions"
    • These $$$\phi_j$$$ form the basis for the "trial function", $$$u_h$$$
    • Analogous to the $$$x^n$$$ we used earlier
  • The gradient of $$$u$$$ can be expanded similarly: $$$$\nabla u \approx \nabla u_h = \sum_{j=1}^N u_j \nabla \phi_j$$$$

  • In the Galerkin finite element method, the same basis functions are used for both the trial and test functions: $$$$\psi = \{\phi_i\}_{i=1}^N$$$$
  • Substituting these expansions back into our weak form, we get: $$$$\left(\nabla\psi_i, k\nabla u_h \right) - \langle\psi_i, k\nabla u_h\cdot \hat{n} \rangle + \left(\psi_i, \vec{\beta} \cdot \nabla u_h\right) - \left(\psi_i, f\right) = 0, \quad i=1,\ldots,N$$$$

  • The left-hand side of the equation above is what we generally refer to as the $$$i^{th}$$$ component of our "Residual Vector" and write as $$$R_i(u_h)$$$.

  • 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 $$$x^n$$$ 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.

Example 1D Shape Functions

linear_lagrange.png
Linear Lagrange

quadratic_lagrange.png
Quadratic Lagrange

cubic_lagrange.png
Cubic Lagrange

cubic_hermite.png
Cubic Hermite

2D Lagrange Shape Functions

Example bi-quadratic basis functions defined on the Quad9 element:

  • $$$\psi_0$$$ is associated to a "corner" node, it is zero on the opposite edges.
  • $$$\psi_4$$$ is associated to a "mid-edge" node, it is zero on all other edges.
  • $$$\psi_8$$$ is associated to the "center" node, it is symmetric and $$$\geq 0$$$ on the element.

quad9_phi0.png
$$$$\psi_0$$$$

quad9_phi4.png
$$$$\psi_4$$$$

quad9_phi8.png
$$$$\psi_8$$$$