Page 64 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

50 3 Algorithm for Simulating Coupled Problems in Hydrothermal Systems

Otherwise, Eq. (3.79) can be rewritten as

ξ A = α − βη A (3.82)

where
c − c 2 ∗
∗
1
α = , (3.83)
∗
a − b ∗
2 2
a − b ∗
∗
β = 3 3 . (3.84)
∗
a − b ∗
2 2
Therefore, the corresponding solutions can be expressed for the following sub-
cases.
(1) For β = 0
ξ A = α, (3.85)

∗ ∗
c − a ξ A
2
1
η A = . (3.86)
∗
a + ξ A
3
(2) For β = 0

∗ ∗ ∗ ∗ 2 ∗ ∗
−(a β − a − α) ± (a β − a − α) − 4β(c − a α)
3
1
2
2
3
2
η A = , (3.87)
2β
ξ A = α − βη A . (3.88)
3.2.3 Consistent Interpolation Step
Based on the concept of isoparametric elements (Zienkiewicz 1977), any nodal solu-
tion at any point (i.e., point A) in mesh 1 can be consistently interpolated using the
following equation:

4

S A = N i (ξ A ,η A )S i , (3.89)
i=1

where S i is the appropriate numerical solution at a nodal point in mesh 1.
Note that since the global coordinate system used in mesh 1 is exactly the same as
in mesh 2, the nodal value of any concerned solution in mesh 2 can be interpolated
through mesh 1 and then directly transferred to mesh 2.