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Differential N-D Mesh Generation

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 - xk(sj) - from to the final coordinate mesh . Notationally, the composite mapping transforms for the N-D problem are,
 (2)
Note that coordinate system may be of a greater dimension than , which allows for composite mapping operations of that project a 2-D surface into 3-D space (see figure  for an example).

Example
Figure 1
Meshing example for mapping a 2-D Cartesian domain to a surface in a 3-D volume. Top panel: Regular Cartesian mesh ; Middle panel: Intermediate transformation domain ; and Bottom panel: Surface in middle panel projected onto 3-D surface where increasing grey scale intensity represents increasing height.

Coordinate system transformations - and xk(sj) - are described in differential geometry through metric tensor, gij, which relates the geometry of a coordinate system to that of Guggenheimer (1977). The metric tensor is symmetric (i.e. gij=gji) 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 - gii = g11+g22+g33 - is implicit for any repeated indicies found in the paper.) The associated metric tensor gij is related to the metric tensor through gij = gij/|gs| where gs 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 Up: Theoretical Overview Previous: Theoretical Overview
Stanford Exploration Project
4/5/2006