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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)}

and the gradient:

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;

Isoparametric formulation for a two nodes element

Isoparametric formulation for a three nodes quadratic element

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