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Parametric interpolation

For a two nodes lineal element, the unknow can be written:

and the gradient:

developing the above expressions (see Lagrangian Elements):

and therefore:

to compute is necessary to know the relation between x and ξ, that can be obtained by using a parametric interpolation of the geometry.

For example, by knowing the coordinates of m points of the element, any x value can be computed, as follows:

with geometrical interpolation functions equivalent to the Shape Functions (having the 1 value for the i node and 0 for the m-1 other nodes).

As an example, check the following Parametric interpolation for a cubic function.

Therefore, for each element, it can be considered two kinds of points:

• N nodes, which define the Shape Functions, and used to interpolate the values of the unkown;
• m geometrical points, which define the geometrical interpolation functions, used to interpolate the geometry;

For complex geometries, m could be greater than N, and in this case is called superparametric formulation;

For simple geometries, m could be smaller than N, and in this case is called subparametric formulation;

If m is equal to N, then , and the formulation is called isoparametric;

Isoparametric formulation for a two nodes element

Isoparametric formulation for a three nodes quadratic element