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Generalized Laplacian systems can be solved by many different ways;
however, numerical solution often is facilitated through intermediate
mappings to meshes exhibiting many attributes of the final grid. This
consists of a composite of a transformation   from a
Cartesian to an intermediate basis , and a
transformation  x^{k}(s^{j})  from to the final
coordinate mesh . Notationally, the composite mapping
transforms for the ND problem are,
 

 (2) 
Note that coordinate system may be of a greater dimension
than , which allows for composite mapping operations of
that
project a 2D surface into 3D space (see figure for
an example).
Example
Figure 1 Meshing example for mapping a 2D
Cartesian domain to a surface in a 3D volume. Top panel: Regular
Cartesian mesh ; Middle panel: Intermediate
transformation domain ; and Bottom panel: Surface in
middle panel projected onto 3D surface where
increasing grey scale intensity represents increasing height.
Coordinate system transformations  and x^{k}(s^{j})  are
described in differential geometry through metric tensor, g_{ij},
which relates the geometry of a coordinate system to that of
Guggenheimer (1977). The metric tensor is symmetric
(i.e. g_{ij}=g_{ji}) and has elements given by,
 
(3) 
where the metric tensor superscript specifies the coordinate system in
which the operator is defined. (Note that summation notation 
g_{ii} = g_{11}+g_{22}+g_{33}  is implicit for any repeated
indicies found in the paper.) The associated metric tensor g^{ij}
is related to the metric tensor through g^{ij} = g_{ij}/g^{s} where
g^{s} is the metric tensor determinant. Through use of this
differential geometric framework, the governing set of differential
gridding equations Liseikin (2004) become,
 
(4) 
 (5) 
Equations 4 represent the N generalized Laplace's
equations acting on coordinate fields , and
equations 5 map the boundary values of each coordinate
field to the boundary of domain . As
posed, equations 4 and 5 provide no
guarantee that generated grids will exhibit appropriate
characteristics because no mesh regularization has yet been enforced.
Next: Regularization through Monitor Functions
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Previous: Theoretical Overview
Stanford Exploration Project
4/5/2006