A finite element stores a finite set of unknown degrees of freedom at its nodes. These nodal values are used to approximate a field throughout the element.

For a structural element, degrees of freedom may include translations and rotations. For a thermal element, a node may carry only temperature.

Nodal vectors

An element's nodal unknowns can be collected in a vector such as

$$\mathbf u=(u_1,v_1,\theta_1,u_2,v_2,\theta_2)^T$$

for a two-node planar beam element.

Shape functions

Shape functions interpolate the field inside the element from its nodal values. In one dimension,

$$u(x)\approx\sum_i N_i(x)u_i,$$

where $N_i(x)$ are the shape functions and $u_i$ are nodal values.

Linear elements use first-order interpolation; higher-order elements add nodes or polynomial terms so that more complex variation can be represented.

Approximation lives inside the element

The nodal values are unknowns solved globally, but the shape functions determine how those values imply displacement, strain or another field between nodes.

This is the key step that turns a continuous field problem into a finite algebraic one.