# Local and global coordinates in FEM

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

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Can you precise what you actually mean? –  shuhalo Jan 28 '13 at 8:52
Can you give us an example to elaborate what you asked? –  Shuhao Cao Jan 28 '13 at 15:52
Welcome to SciComp! I think there's a good question here (especially after seeing @StefanoM's answer), but the current formulation is not really helpful for other people. It would be good if you could expand your question by describing explicitly what you mean by "local coordinate system with every node" and why you are interested in doing this. –  Christian Clason Jan 29 '13 at 13:05

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.

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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.

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