Test page with index 1
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Kratos
' T u t o r i a l s
Test Page 1 Title
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
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