Page 58 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

44 3 Algorithm for Simulating Coupled Problems in Hydrothermal Systems

3

4

A
y

1 2

0 x
Fig. 3.2 Point A in a fournode quadrilateral element

→ → → → → → →
where 12, 23, 34 and 41 are vectors of four sides of the element; 1A, 2A, 3A
→ →
and 4A are vectors of each node of the element to point A respectively; k is a
normal vector of the plane where the element is located; λ i (i = 1, 2, 3, 4) are four
different constants. Note that the left-hand sides of Eqs. (3.22), (3.23), (3.24) and
(3.25) represent the cross products of two vectors.
By using the global coordinates of point A and four nodal points of the element,
Eqs (3.22), (3.23), (3.24), (3.25) and (3.26) can be expressed in the following form:

(x 2 − x 1 )(y A − y 1 ) − (x A − x 1 )(y 2 − y 1 ) = λ 1 ≥ 0, (3.27)
(x 3 − x 2 )(y A − y 2 ) − (x A − x 2 )(y 3 − y 2 ) = λ 2 ≥ 0, (3.28)

(x 4 − x 3 )(y A − y 3 ) − (x A − x 3 )(y 4 − y 3 ) = λ 3 ≥ 0, (3.29)

(x 1 − x 4 )(y A − y 4 ) − (x A − x 4 )(y 1 − y 4 ) = λ 4 ≥ 0, (3.30)

where x i and y i are the global coordinates of nodal points of the element; x A and y A
are the global coordinates of point A.
Note that when point A is located on the side of the element, two vectors related
to a node of the element are coincident. Consequently, the corresponding λ i to this
node must be equal to zero. Since Eqs. (3.27), (3.28), (3.29) and (3.30) are only
dependent on the global coordinates of five known points (i.e., point A and four
nodes of the element), they can be straightforwardly used to predict the element, in
which point A is located.
It must be pointed out that, using the point-searching strategy expressed by
Eqs. (3.27), (3.28), (3.29) and (3.30), it is very easy and accurate to relate a point
in mesh 2 to an element (containing this point) in mesh 1, since Eqs. (3.27), (3.28),
(3.29) and (3.30) only involve certain simple algebraic calculations which can be
carried out by computers at a very fast speed. Thus, to translate/transfer the solution