Boundary fitted coordinates

The basic idea of a boundary-conforming curvilinear coordinate system is to have some coordinate line (in 2D, surface in 3D) coincident with each boundary segment, analogous to the way in which lines of constant radial coordinate coincide with circles in the cylindrical coordinate system. The other curvilinear coordinate, analogous to the angular coordinate in the cylindrical system, will vary along the boundary segment and clearly must do so monotonically, else the same pair of values of the curvilinear coordinates will occur at two different physical points. (It should be clear that the curvilinear coordinate that varies along a boundary segment must have the same direction and range of variation over some opposing segment, e.g., as the angular variable varies from 0 to 2$\pi$ over both of two concentric circles in cylindrical coordinates).

With the values of the curvilinear coordinates thus specified on the boundary, it then remains to generate values of these coordinates in the field from these boundary values. There must, or course, be a unique correspondence between the Cartesian (or other basis system) and the curvilinear coordinates, i.e., the mapping of the physical region onto the transformed region must be one-to-one, so that every point in the physical field corresponds to one, and only one, point in the transformed field, and vice versa. Coordinate lines of the same family must not cross, and lines of different families must not cross more than once.

In this chapter a two-dimensional region will be considered in most of the discussions in the interest of economy of presentation. Generalization to three dimensions will be evident in most cases and will be mentioned specifically only when necessary. As noted above, the curvilinear coordinates may be normalized to any intervals, just as the radial and angular coordinates of the cylindrical coordinate system can be expressed in many different units. Since the interest of the present discussion is numerical application, it will be generally convenient to define the increments of all the curvilinear coordinates to be uniformly unity, and then to normalize these coordinates to the interval $[1,N^i]$, where $N^i$ is the total number of grid points to be used in the i direction. (The three curvilinear coordinates will be indicated as $\xi^i, i = 1,2,3$, in general. In two dimensions, however, the notation $(\xi,\eta)$ will often be used for the two coordinates $\xi^1$ and $\xi^2$ .) The computational field, i.e., the field in the transformed space, thus will have rectangular boundaries and will be covered by a square grid. (It will become clear later that the actual values of the increments in the curvilinear coordinates are immaterial since they do not appear in the final numer1cal expressions. Therefore no generality is lost in making the grid square and of unit increment in the transformed field.)