Page 85 - Basic Structured Grid Generation

74 Basic Structured Grid Generation

In fact, as the Laplacian operator associated with a particular surface, this second-
order differential operator is known as the Beltrami operator of the surface, and
given the special notation B . Since the 2 × 2 matrix represented by C iα C iβ ,using
eqn (3.142), is

1 T 11 12
(a )
a a
T
C C = a 2 T a 1 a 2 = a 12 22 , (3.158)
(a ) a a
we have
αβ
C iα C iβ = aa . (3.159)
To make this identity appear consistent, we should have written the index α as a
superscript in the definition of C in eqn (3.141). However, a consistent form for the
Beltrami operator is now given, from eqn (3.157), by

1 ∂ √ αβ ∂ϕ
B ϕ = √ aa , (3.160)
a ∂u α ∂u β
which has precisely the same form as eqn (1.147). This can be written in terms of the
covariant metric tensor as
∂ϕ ∂ϕ ∂ϕ ∂ϕ
    
a 22 1 − a 12 2 a 11 2 − a 12 1
∂u ∂u  ∂  ∂u ∂u 
1  ∂ 
√  + √  , (3.161)
a ∂u 1  a ∂u 2  a
B ϕ = √ 
which leads to the identities
1 ∂ a 22 ∂ a 12
1 ∂ a 11 ∂ a 12
B u = √ √ − √ , B v = √ √ − √ ,
a ∂u a ∂v a a ∂v a ∂u a
α
writing u as (u, v).
Using eqns (3.56) and (3.61), we also have
∂ αβ ∂ϕ γ αβ ∂ϕ
B ϕ = a + γα a
∂u α ∂u β ∂u β
2
∂ ϕ δα β δβ α ∂ϕ γ αβ ∂ϕ
αβ
= a − (a δα + a ) + γα a
δα
α
∂u ∂u β ∂u β ∂u β
2
2
∂ ϕ δα β ∂ϕ αβ ∂ ϕ δ ∂ϕ
αβ
= a − a δα = a − αβ (3.162)
α
α
∂u ∂u β ∂u β ∂u ∂u β ∂u δ
after some manipulation of indices.
αβ
Exercise 19. Show, using eqn (3.57), that B ϕ = a ϕ ,αβ in terms of covariant
derivatives.
γ
If we consider the special case ϕ = u , we obtain
γ
β
γ
δα
δα
γ
B u =−a δ =−a , (3.163)
δα β δα
which may be compared directly with eqn (1.111). Hence we can write
2
∂ ϕ β ∂ϕ
αβ
B ϕ = a + ( B u ) . (3.164)
α
∂u ∂u β ∂u β