From KratosWiki

General

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 m1 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


width="25%" style="background:#EAF5FB; border:1px solid #eeeed1; fontsize:100%; mozborderradiustopright:0px; mozborderradiusbottomright:0px; padding:7px 7px 7px 7px;" valign="top"