From KratosWiki
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General
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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
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