- 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;