Page 59 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

3.2 Mathematical Formulation of the Algorithm 45

of any point from mesh 1 to mesh 2, only one element in mesh 1 needs to be
inversely mapped. This certainly reduces the number of elements, to which the
inverse mapping is applied, to the minimum. From the computational point of view,
the present algorithm is both effective and efficient.

3.2.2 Inverse Mapping Step

This section deals with the second key issue related to the development of the con-
sistent point-searching interpolation algorithm. In order to interpolate the solution
at point A consistently in the finite element sense, it is necessary to find out the
local coordinates of this point by an inverse mapping. For a four-node quadrilateral
isoparametric element shown in Fig. 3.3, it is possible to implement the inverse
mapping analytically. The following gives the related mathematical equations.

η

3

4
ξ
y
0

1 2

x
0
(Physical element)
η

4 3

ξ
0

1 2
(Parent element)
Fig. 3.3 A four-node quadrilateral isoparametric element (Zienkiewicz 1977; Zhao et al. 1999f)