Mapping Operator



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Mapping Operator

 

The operator represents the mapping of a specific point of the computational domain to a point in the physical domain (3.3-2). It is worth noting that in general is a nonlinear mapping of Banach spaces and differentiability is expressed as Fréchet-differentiability [Dir86]. The transformation is accomplished by the set of functions and (3.3-3).

 

 

In the further text coordinates appearing as subscripts indicate partial differentiation, for instance . In two dimensions the index 1 is associated with and . The index 2 is associated with and . The Jacobian of the transformation is given by (3.3-4), 's Fréchet-derivative .

 

gives the differential coordinate transformation (3.3-5), and consequently the shape of the distorted cell in the physical space that corresponds to an infinitesimal square cell in computational space.

 

The Jacobian's determinant measures the area of the distorted cell in the physical space.

 

For non-vanishing determinants the inverse of exists and a one-to-one mapping can be established. These mappings are called regular. We prefer a sloppy but more practical and comprehensible interpretation of regularity [Son89]:

A grid is called regular if all grid lines lie in the closures of the physical domain and if there is no intersection of grid lines of the same family and if two lines of different families intersect only once.



Martin Stiftinger
Wed Oct 19 13:03:34 MET 1994