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# Differential 2-D Mesh Generation

Generating a 2-D coordinate system through differential methods requires solving for coordinates within domain X2. Incorporating l monitor functions for grid regularization expands the dimensionality of the mapping to,
 (9)
Coordinate system is related to an underlying Cartesian grid, which is chosen to be a unit square defined by . Transformation is assumed to be piece-wise smooth and known on the boundary of such that: . Within this framework, the 2-D gridding equations become,
 (10)
where Laplacian operator and metric tensor gij are written explicitly Liseikin (2004),
 (11) (12)

One convenient way to solve the set of elliptical equations 10 is by transforming them to a set of parabolic equations (i.e. include time-dependence) that have a common steady-state solution. Thus, equations 10 are reformulated to include time-dependence - - leading to the six governing equations,
 (13) (14) (15) (16) (17) (18)
Solutions and satisfying equations 13-18 will converge to the solutions of equations 10 as . Hence, the answer to within tolerance factor occurs at some Tn. Details of an iterative scheme and an algorithm for computation to solve equations 13 are provided in Appendix A.

Next: Numerical Examples Up: Shragge: Differential gridding methods Previous: Regularization through Monitor Functions
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4/5/2006