Page 104 - Process Modelling and Simulation With Finite Element Methods

Partial Differential Equations and the Finite Element Method 91

Dimensionality Shape

1 -D

2-D

3-D
Regular Irregular
Tetrahedral Hexahedral Hexahedral

Table 2.1 Basic elements.
1 Constant
Linear
Quadratic
Cubic
Quartic
Quintic
Table 2.2 Pascal Triangle for 2-D elements.

Linear Quadratic Cubic

FEMLAB provides you with two major types of elements irrespective to the
geometry, namely Lagrange and Hermite elements. These elements are so named
because of the type of the interpolation polynomials used in them. As the name
suggests, the Lagrange polynomials are used as the basis functions in Lagrange
elements. Suppose a field variable u(x) is expressed using Lagrange polynomials
L, over a 1-D element. Then,
u(x) =L&)ul +L,(x)u, +*.-+L,(x)u, (2.58)