Page 178 - Basic Structured Grid Generation

Variational methods and adaptive grid generation 167

and
1 1 1 1
   
ϕ ξ x ξ + ψ η x η x ξ x η
ϕ 2 ψ 2 ϕ 2 ψ 2
    ϕ ξ
S =−   =−   (6.50)
 1 1   1 1  ψ η
ϕ ξ y ξ + ψ η y η y ξ y η
ϕ 2 ψ 2 ϕ 2 ψ 2
and partial differentiation (including that of the vector r) is denoted by suffixes.
6.4.3 The weighted area-functional

The area-functional generally provides a more satisfactory grid than the length-functional.
The objective here is to generate a grid such that the area of grid-cells is proportional to
some given positive weight function ϕ(ξ, η) in computational space. Given that the area
of grid-cells in the physical domain corresponding to uniform grid-cells of fixed area in
√
the computational domain is proportional to g, a natural way to define a functional by
analogy with the length functionals above would be to put
1 g 1 J 2
I A = dξ dη = dξ dη, (6.51)
2 R ϕ(ξ, η) 2 R ϕ(ξ, η)

where the Jacobian J = (x ξ y η − x η y ξ ).
Exercise 2. Show that the Euler-Lagrange equations in this case can be written as

Jr η Jr ξ
− = 0.
ϕ ξ ϕ η
It may be seen that on further differentiation these equations reduce to

(J/ϕ) ξ r η − (J/ϕ) η r ξ = 0. (6.52)
Exercise 3. Show that this is equivalent to
A 11 r ξξ + A 12 r ξη + A 22 r ηη + S = 0

as in (6.48), with
2
(y η ) −x η y η −2y ξ y η x ξ y η + x η y ξ
A 11 = , A 12 = ,
(x η ) 2
−x η y η x ξ y η + x η y ξ −2x ξ x η
2
(y ξ ) −x ξ y ξ J y η −y ξ ϕ ξ
A 22 = 2 , S =− . (6.53)
−x ξ y ξ (x ξ ) ϕ −x η x ξ ϕ η

6.4.4 Orthogonality-functional

An orthogonality (O)-functional without weight function is
1 2
I O = (g 12 ) dξ dη, (6.54)
2 R