# Isoparametric formulation

## Parametric interpolation

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

$\hat \varphi (\xi) = N_1^{(e)}(\xi) \varphi_1^{(e)} + N_2^{(e)}(\xi) \varphi_2^{(e)}$

$g = \frac {d \hat \varphi}{dx} = \frac {d N_1^{(e)}(\xi)}{dx} \varphi_1^{(e)} + \frac {d N_2^{(e)}(\xi)}{dx} \varphi_2^{(e)}$

developing the above expressions (see Lagrangian Elements):

$\frac {d N_1^{(e)}(\xi)}{dx} = \frac {d N_1^{(e)}(\xi)}{d \xi} \frac {d \xi}{dx} = \frac {d}{d \xi} \left( \frac {1 - \xi}{2} \right) \frac {d \xi}{dx} = - \frac{1}{2} \frac {d \xi}{dx}$

$\frac {d N_2^{(e)}(\xi)}{dx} = \frac {d N_2^{(e)}(\xi)}{d \xi} \frac {d \xi}{dx} = \frac {d}{d \xi} \left( \frac {1 + \xi}{2} \right) \frac {d \xi}{dx} = \frac{1}{2} \frac {d \xi}{dx}$

and therefore:

$g = - \frac{1}{2} \left( \frac{d \xi}{dx} \right) \varphi_1^{(e)} + \frac{1}{2} \left( \frac{d \xi}{dx} \right) \varphi_2^{(e)}$

to compute $\frac{d \xi}{dx}$ 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 $x_1, x_2,... , x_m \,$ coordinates of m points of the element, any x value can be computed, as follows:

$x = \hat N_1^{(e)}(\xi) x_1 + \hat N_2^{(e)}(\xi) x_2 + ... + \hat N_m^{(e)} (\xi) x_m$

with $N_i^{(e)}$ 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, $N_i \,$ and used to interpolate the values of the unkown;
• m geometrical points, which define the geometrical interpolation functions, $\hat N_i \,$ 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 $N_i \equiv \hat N_i \,$, and the formulation is called isoparametric;