Mapping and Shape Functions


Introduction
A finite element is called 'isoparametric' if the interpolation functions describing its geometry are identical to the interpolation functions describing the distribution of displacements.
In many FEA software packages the coordinate system of an element is transformed into a coordinate system called the natural coordinate system. In order to avoid mathematical difficulties element distortion must be controlled. This approach simplifies the addition of new element types to a software package.


Natural coordinates can be used to define variation of a quantity along an element, in the above case, elastic displacement. Assuming elastic displacement, paricle displacement varies linearly between points (nodes) 1 and 2.
This linear variation is called an interpolation function.

In (b): r = 0A/02

From (a): r = [x - ( x2 + x1 )/2] / [( x2 - x1 )/2] .. eqn (a1)

It is assumed that the displacement, u, varies linearly with r:

Eqn (c): u = a1 + a2 r

At node 1: u = u1 and r = -1, so u1 = a1 - a2

At node 2: u = u2 and r = 1, so u2 = a1 + a2

Adding gives: u1 + u2 = 2 a1, or a1 = (u1 + u2)/2

Subtracting: u2 - u1 = 2 a2, or a2 = (u2 - u1)/2

Substituting in (c) for a1 and a2:




N1 has a value of 1 at node 1 and falls linearly to zero at node 2.
N2 has a value of 1 at node 2 and falls linearly to zero at node 1.

At any value of r, N1 + N = (1 - r)/2 + (1 + r)/2 = 1

The co-ordinates of a point in the element (master) can be related to co-ordinates in the real element by re-arranging eqn (a1):

r(x2 - x1) = 2x - (x2 + x1)

or: x = [(1 - r)/2]x1 + [(1 + r)/2]x2

or: x = N1x1 + N2x2 ... This is a 'mapping relationship'


Having linear variation of displacement in an element which means constant stress is very limiting and it is much more useful to allow linear variation of stress within an element which requires higher order polynomial for interpolation, i.e. a quadratic 3 node linear element for the example above (and see in the section: Example on reducing errors ..).

At r = -1, u = u1, so u1 = a1 - a2 + a3

at r = 0, u = u3, so u3 = a1

at r = 1, u = u2, so u2 = a1 + a2 + a3

Re-arranging gives:

Where N1, N2 and N3 are the shape functions.

N1 = -r(1 - r)/2

N2 = r(1 + r)/2

N3 = (1 + r)(1 - r)

These are shown plotted below:


For any value of r between -1 and 1, the sum of the shape functions, in this case: N1 + N2 + N3 = 1

This procedure means that 1 dimensional, 2D and 3D elements which in the real world may have 1 or more curved edges, can be mapped onto the straight line r; the square r, s; or the cube r, s, t.

David Grieve, 16th November 2012