Page 151 - Basic Structured Grid Generation

140 Basic Structured Grid Generation

with (P (ξ)) i = P(ξ i ) = P i . Hence eqn (5.80) becomes

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x i+1 − 2x i + x i−1 x i+1 − x i−1
=−P i ,
( ξ) 2 2( ξ)
which we write as
−a i x i+1 + b i x i − c i x i+1 = h i , i = 1, 2,...,(m − 1), (5.99)

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where a i = c i = 1, b i = 2, and h i = P i (x i+1 − x i−1 ) /8( ξ).
Because of the dependence of the terms h i on the solution, these equations can be
solved in an iterative manner as follows:
(a) Guess a reasonable initial set of values x i , for example by linear interpolation,
given the end-conditions x 0 = a and x m = b.
(b) Evaluate the set of values h i .
(c) Solve the set of matrix equations for a new set of values x i ; this can be done by
Gaussian elimination.
(d) Return to step (b) and continue the iteration until the difference between succes-
sive sets of values x i , as measured by max i |x new − x old |, is less than some prescribed
i i
tolerance.

5.8 Three-dimensional grid generation

Extending eqn (5.6) to three dimensions leads naturally to the set of Poisson equations
i
for the curvilinear co-ordinates x :
2 i
i
∇ x = P , i = 1, 2, 3, (5.100)
i
where the P s are three control functions. Writing eqn (1.114) in its full three-dimens-
ional vector form
2
∂ r ∂r
2 j
g ij =−(∇ x ) , (5.101)
i
∂x ∂x j ∂x j
leads immediately to the inverted form
2
∂ r j ∂r
ij
gg + gP = 0, (5.102)
i
∂x ∂x j ∂x j
1
3
2
which may be expressed, using eqn (1.34) and putting x = ξ, x = η, x = ζ,as
∂r
j
Dr + gP = 0,
∂x j
where the second-order differential operator D is given by
∂ 2 ∂ 2 ∂ 2 ∂ 2 ∂ 2 ∂ 2
D = G 1 2 + G 2 2 + G 3 2 + 2G 4 + 2G 5 + 2G 6 . (5.103)
∂ξ ∂η ∂ζ ∂ξ∂η ∂ξ∂ζ ∂η∂ζ