The triangular element and the corresponding notations used in this work are
shown in Fig. <4.1>. The well known linear approximation of an unknown
function
within a triangle is given by
The linear element form functions will be obtained from this approximation.
Assume the values of
on the nodes of the triangle are known. The nodes are
numbered by
,
and
as in Fig. <4.1>, and the corresponding field values
are
,
and
. The coefficients
,
and
are determined by
the system
where
and
are the coordinates of the i-th node in the element.
Solving (4.26) for
,
, and
leads to
is the so called Jacobi determinant
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(4.28) |
where
is the area of the triangular element and
with
. With back substituting of
,
, and
from (4.27) into (4.25)
is written in the form
which gives the element shape functions
.