Page 60 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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In order to determine the location of P on ij, we have the following linear relation
between the distances and temperature values, namely,
THE FINITE ELEMENT METHOD
2
60.0 − 50.0 (x P − x i ) + (y P − y i ) 2
(3.48)
=
70.0 − 50.0 2 2
(x j − x i ) + (y j − y i )
From the data given, it is clear that the y coordinate on the ij side are equal to zero
and thus the above equation is simplified to
10.0 (x P − x i )
= (3.49)
20.0 (x j − x i )
which results in x P = 2.0 cm. The location of Q along ik can be determined in a similar
fashion as
60.0 − 50.0 y Q − y i
= (3.50)
100.0 − 50.0 y k − y i
which gives y Q = 0.5 cm. The x coordinate of this point is zero.
◦
The line joining P and Q will be the 60 C isothermal (Figure 3.8). It should be noted
that the same principle can be used for arbitrary triangles.

3.2.4 Area coordinates

An area, or natural, coordinate system will now be introduced for triangular elements in
order to simplify the solution process. Let us consider a point P within a triangle at any
location as shown in Figure 3.9. The local coordinates L i , L j and L k of this point can be
established by calculating appropriate non-dimensional distances or areas. For example, L i
is defined as the ratio of the perpendicular distance from point P to the side ‘jk’(OP)to
the perpendicular distance of point ‘i’ from the side ‘jk’(QR). Thus,
OP
L i = (3.51)
QR
Similarly, L j and L k are also defined. The value of L i is also equal to the ratio of the
area A i (opposite to node ‘i’) to the total area of the triangle, that is,
A i 0.5(OP)(jk) OP
L i = = = (3.52)
A 0.5(QR)(jk) QR

y k

A i O
R
A j
P
A k
i j
Q
x
Figure 3.9 Area coordinates of a triangular element