Local and global coordinates in FEM

Question

Am I right that in FEM we can associate a local coordinate system with every node, not with every element?

Answer 1

Let me choose an example problem, in which we have an unknown vector field, otherwise the question makes no sense.

Considering $3$-dimensional linear elasticity, nodal unknowns are displacement components $u_i$, $i=1,\dots, 3$ which can be expressed either in the global or in a nodal-local system of reference. Local systems of reference are handy when e.g. Dirichlet boundary conditions are given only on the normal component of displacement (roller support) and the domain has a general shape.

Element quantities, evaluated at Gauss points, are e.g. stress/strain tensor components $\sigma_{ij}$, $\epsilon_{ij}$, $i,j=1,\dots, 3$, which can be computed in an element-local or global system of reference.

Both the nodal-local and element-local reference frames are usually user defined.

Nodal coordinates $x_i$ are typically expressed in the global reference frame.

For the iso-parametric formulation, stiffness matrices are integrated in an element parametric space, that depends on the nodal coordinates and cannot be chosen by the user.

Answer 2

I can't think of a use case where a node would need a local coordinate system as it represents a physical location in space and doesn't have any length, area or volume associated with it and thus any coordinate system seems a moot point as there is no distance or direction to measure. The element stiffness matrices are usually computed in a local or parametric coordinate system though and can be very useful.