Page 152 - Basic Structured Grid Generation

Differential models for grid generation 141

5.9 Surface-grid generation model

A natural extension of the differential model given by eqns (5.6) to grid generation on
a surface in three dimensions is obtained by replacing the Laplacian operator by the
Beltrami operator, as defined in Section 3.9. Thus if the surface is to be covered by a
2
α
1
curvilinear co-ordinate system of ξ curves, we could force ξ(= ξ ) and η(= ξ ) to
satisfy the equations
B ξ = P, B η = Q, (5.104)
where P , Q are control functions. Note that the surface may already be effectively
α
covered by parametric curves u , α = 1, 2, but these may not give rise to a satisfactory
grid. We assume here that the surface has four edges (four known space-curve seg-
α
ments) which can be mapped onto the edges of a unit square in the ξ computational
plane.
α
Substituting into eqn (3.165) with the ξ α system instead of u immediately gives
the ‘inverse’ equation
2
∂ r ∂r ∂r
αβ
a + P + Q = 2κ m N,
α
∂ξ ∂ξ β ∂ξ ∂η
which after multiplying through by a may be written as

∂r ∂r
Dr + a P + Q = RN, (5.105)
∂ξ ∂η
where the operator D is given by

∂ 2 ∂ 2 ∂ 2
D = a 22 2 − 2a 12 + a 11 2 , (5.106)
∂ξ ∂ξ∂η ∂η
and
αβ
R = 2aκ m = aa b αβ = a 22 b 11 − 2a 12 b 12 + a 11 b 22 . (5.107)
Equation (5.105) comprises three scalar equations for the cartesian components
(x, y, z) of grid-points in physical space. These can be solved numerically when the
equation of the surface is given, either explicitly, implicitly, or in terms of parameters.
Specification of the two constraining eqns (5.104) provides the core of the method.
Here we shall just consider the case where P = Q = 0, and take the surface to be
defined by the explicit equation z = f (x, y). Then eqn (5.105) reduces to the three
equations

2
2
2
∂ x ∂ x ∂ x
a 22 2 − 2a 12 + a 11 2 = RN x
∂ξ ∂ξ∂η ∂η
2
2
2
∂ y ∂ y ∂ y
a 22 2 − 2a 12 + a 11 2 = RN y (5.108)
∂ξ ∂ξ∂η ∂η
2
2
2
∂ z ∂ z ∂ z
a 22 2 − 2a 12 + a 11 2 = RN z ,
∂ξ ∂ξ∂η ∂η