Page 60 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

46 3 Algorithm for Simulating Coupled Problems in Hydrothermal Systems

The forward mapping of the physical element in the global system to the parent
element in the local system reads:

4

x = N i x i , (3.31)
i=1
4

y = N i y i , (3.32)
i=1
where
1
N 1 = (1 − ξ)(1 − η), (3.33)
4
1
N 2 = (1 + ξ)(1 − η), (3.34)
4
1
N 3 = (1 + ξ)(1 + η), (3.35)
4
1
N 4 = (1 − ξ)(1 + η), (3.36)
4
where N 1 , N 2 , N 3 and N 4 are shape functions of the element.
For a given point (i.e., point A), the corresponding local coordinates are ξ A and
η A . Substituting these local coordinates into Eqs. (3.31), (3.32), (3.33), (3.34), (3.35)
and (3.36) yields the following inverse mapping:

a 2 ξ A + a 3 η A + a 4 ξ A η A = 4x A − a 1 , (3.37)
b 2 ξ A + b 3 η A + b 4 ξ A η A = 4y A − b 1 , (3.38)

where

a 1 = x 1 + x 2 + x 3 + x 4 , (3.39)

a 2 =−x 1 + x 2 + x 3 − x 4 , (3.40)
a 3 =−x 1 − x 2 + x 3 + x 4 , (3.41)

a 4 = x 1 − x 2 + x 3 − x 4 , (3.42)
b 1 = y 1 + y 2 + y 3 + y 4 , (3.43)

b 2 =−y 1 + y 2 + y 3 − y 4 , (3.44)

b 3 =−y 1 − y 2 + y 3 + y 4 , (3.45)
b 4 = y 1 − y 2 + y 3 − y 4 . (3.46)