Page 116 - Basic Structured Grid Generation

Structured grid generation – algebraic methods 105

η which can be used to give an interpolation formula r(ξ, η) between inner and outer
curves, with
r(ξ, η i ) = r i (ξ), i = 1, 2, 3,...,n. (4.106)

Piecewise-linear curves from inner curve r 1 (ξ) to outer curve r n (ξ) may be con-
structed by linking together points on all n curves corresponding to the same value
of ξ. We assume that these piecewise-linear curves do not intersect each other. A
segment of such a curve between the curves r i (ξ) and r i+1 (ξ) must be the vector
r i+1 (ξ) − r i (ξ). The multisurface method produces smooth co-ordinate curves r(ξ, η)
with ξ fixed and η varying from η 1 to η n by matching tangent vectors ∂r/∂η at each sur-
face r i (ξ), i = 1, 2, 3,. ..,n−1, to the directions of the line segment r i+1 (ξ)−r i (ξ).
Thus we take
n−1
∂r
= ψ i (η)T i {r i+1 (ξ) − r i (ξ)}, (4.107)
∂η
i=1
where the T i s are positive scalars associated with the surfaces r 1 , r 2 ,.. . , r n−1 ,and the
ψ i s satisfy
ψ i (η k ) = δ ik .

These could be Lagrange polynomials, given by eqn (4.13).
Integration then yields

n−1 ! η

r(ξ, η) = r 1 (ξ) + ψ i (η ) dη T i {r i+1 (ξ) − r i (ξ)}. (4.108)
i=1 η 1
Note that if we put
−1
!
η n

T i = ψ i (η ) dη , (4.109)
η 1
we obtain an equation which is consistent at η = η n , since the right-hand side of
eqn (4.108) reduces to
n−1

r 1 (ξ) + {r i+1 (ξ) − r i (ξ)},
i=1
which clearly telescopes down to the left-hand side r(ξ, η n ) = r n (ξ). Equation (4.108)
is, of course, also consistent at η = η 1 .
Thus the multisurface equation is taken to be
! η

ψ i (η ) dη
n−1

η 1
r(ξ, η) = r 1 (ξ) + {r i+1 (ξ) − r i (ξ)}. (4.110)
!
η n

i=1 ψ i (η ) dη
η 1
In the case n = 3, where there is one intermediate surface r 2 (ξ), the approach we
take here is to choose parameter values η 1 = 0, η 2 = 1, and η 3 initially unspecified,
with value greater than one. We can then use the linear Lagrange polynomials
ψ 1 = 1 − η, ψ 2 = η.