Page 142 - Basic Structured Grid Generation

Differential models for grid generation 131

It follows that
T T T
p r i+1 = p r i − λ i p Ap i = 0 (5.63)
i
i
i
when λ i satisfies eqn (5.60).
Suppose that, following a search for a minimum along the direction p i from the
‘point’ u i , we have reached the point u i+1 . We can calculate r i+1 from eqn (5.62),
and now we seek the value of α i in eqn (5.61) which will yield the new direction of
search p i+1 .
Substituting u = u i+1 + λ i+1 p i+1 = u i+1 + λ i+1 (r i+1 + α i p i ) into eqn (5.55) gives,
in a similar manner to the derivation of eqn (5.57),
1 T
T
r
E = u Au i+1 − d u i+1 − λ i+1 r T i+1 i+1 − λ i+1 α i r T p
2 i+1 i+1 i
2 T
1
2
T
+ (λ i+1 ) {r T Ar i+1 + 2α i (p Ar i+1 ) + (α i ) p Ap i }
2 i+1 i i
T
2 T
= E i+1 − λ i+1 r T r 1 2 2 T i+1 Ar i+1 + 2α i (p Ar i+1 ) + (α i ) p Ap i },
i+1 i+1 + (λ i+1 ) {r
i
i
using eqn (5.63), where E i+1 is the value of E at u i+1 . To minimize E, we clearly
must choose α i so that the final term in brackets is minimized, and thus we obtain
T
p Ar i+1
i
α i =− . (5.64)
T
p Ap i
i
The iterative scheme is now complete, but we make the observation that
T
p T Ap i = r T i+1 Ap i + α i p Ap i = 0 (5.65)
i
i+1
when α i satisfies eqn (5.64). Thus the direction p i+1 is orthogonal to the direction of
the vector Ap i .Since A is symmetric, it is also the case that p i is orthogonal to the
vector Ap i+1 . The two directions p i , p i+1 are said to be conjugate with respect to the
symmetric matrix A.
To summarize, given a starting guessed solution u 0 , an initial residual r 0 = d−Au 0 ,
and an initial direction p 0 = r 0 , the conjugate gradient method is defined by the iterated
cycle of steps given by
T
T
• λ i = (r p i )/(p Ap i );
i i
• u i+1 = u i + λ i p i ;
• r i+1 = r i − λ i Ap i ;
T
T
• α i =−(p Ar i+1 )/(p Ap i );
i i
• p i+1 = r i+1 + α i p i .
5.6 Numerical solutions of Winslow equations
Obtaining a grid in two dimensions typically involves the following steps:
1. Set up a rectangular grid in the computational domain and obtain an initial guess for
the values of x and y at the grid points. This may be achieved by Transfinite Inter-
polation, or Hermite interpolation, or simply by a univariate interpolation between
two opposite boundaries.