Page 285 - Computational Fluid Dynamics for Engineers

9.4 Algebraic Methods 275

sinh<5
X = T tanh 1 (9.4.23)
8 cosh(5-l + i

The arc length distribution is then given by:
s i n h [ < » ]
s p 1+ (9A24)
^= i ^kfyj

Multidirectional and Transfinite Interpolation
Let Pi(r) be a unidirectional interpolation function in the i direction which
matches r on the N lines i = i n (n = 1, , . . . N). Let Pj(r) be a unidirectional
2
interpolation function in the j direction which matches r on the M lines j = j m
(ra = 1,2...M) (Fig. 9.10).

Pj(r)

Fig. 9.10. Multidirectional interpolation in i and j directions.

These interpolations are performed by projectors Pi and Pj, which may be
simple linear operators. The product projector Pi[Pj(r)] matches the function
Pj(r), instead of r, on the N lines i = i n. Since PJ(T) matches r on the M
lines j = j m , it follows that the product projector matches r at the N x M
points (imjm)- The same conclusion can be reached with the product projector
Pj[Pi(r)] indicating that the projectors Pi and Pj commute.
The sum projector Pi(r) -\-Pj(r) matches r + Pj(r) on the TV lines i = i n and
matches r + Pi(v) o n the M lines j = j m - It follows that the projector Pj(r) +
Pj(r) — Pi[Pj(r)] will match r on the N lines i = i n since Pi[Pj(r)] matches Pj(r)
on these lines. In the same manner, the projector Pj(r) + PJ(T) ~ Pj[Pi{v)} will
match r on the M lines j = j m . Therefore, since PiPj = PjPi, the Boolean sum
projector
Pi © Pj = Pi + Pj - PiPj (9.4.25)
will match r on the N+M lines I = i n and j = jm including the entire boundary
of the region.
In summary, the individual projectors Pi and Pj interpolate unidirectionally
between two opposing boundaries, the product projector PiPj interpolates in
two directions from the four corners and the Boolean sum projector Pi 0 Pj
interpolates from the entire boundary (Fig. 9.11).