Page 183 - Basic Structured Grid Generation

172 Basic Structured Grid Generation

As an example, we present an orthogonality-functional
1 2
I = (a 12 ) dξ dη. (6.70)
2
The two corresponding Euler-Lagrange equations are:
∂ 2 ∂ ∂ 2 ∂ ∂ 2
(a 12 ) − (a 12 ) − (a 12 ) = 0,
∂u ∂ξ ∂u ξ ∂η ∂u η

∂ 2 ∂ ∂ 2 ∂ ∂ 2
(a 12 ) − (a 12 ) − (a 12 ) = 0. (6.71)
∂v ∂ξ ∂v ξ ∂η ∂v η
If we put

A = u η ˜a 11 + v η ˜a 12 , B = u η ˜a 12 + v η ˜a 22 ,
C = u ξ ˜a 11 + v ξ ˜a 12 , D = u ξ ˜a 12 + v ξ ˜a 22 , (6.72)
we get
a 12 = Au ξ + Bv ξ = Cu η + Dv η , (6.73)
and eqns (6.71) may be expressed in the form:

∂ 2 ∂ 2 ∂ ∂ ∂a 12
(A u ξ ) + (C u η ) + (ABv ξ ) + (CDv η ) = a 12 ,
∂ξ ∂η ∂ξ ∂η ∂u
∂ ∂ ∂ 2 ∂ 2 ∂a 12
(ABu ξ ) + (CDu η ) + (B v ξ ) + (D v η ) = a 12 . (6.74)
∂ξ ∂η ∂ξ ∂η ∂v
Note that by eqn (6.69)

∂a 12 ∂ ˜a 11 ∂ ˜a 12 ∂ ˜a 22
= u ξ u η + (u η v ξ + u ξ v η ) + v ξ v η ,
∂u ∂u ∂u ∂u
and similarly for ∂a 12 /∂v.

6.5 Harmonic maps

The Winslow method, with variational formulation based on the functionals (6.43) and
(6.44), together with its extension to surface grid generation discussed in Section 5.9,
can be regarded as an application of harmonic maps. These may be defined in the gen-
eral context of differentiable mappings from an n-dimensional ‘manifold’ M with co-
i
ij
ordinates x and contravariant metric tensor γ , which we may regard here simply as
the n-dimensional physical domain, to an n-dimensional manifold N with co-ordinates
i
ξ and covariant metric tensor G ij , which will be the computational domain. Thus
3
2
for practical purposes here N is a square or rectangle in E or a cube in E ,and M
2
is a two-dimensional region in E or a two-dimensional surface or three-dimensional
3
region in E .
Associated with any mapping is a so-called energy density
k
1 ∂ξ ∂ξ l
i ij i
e(ξ (x)) = γ (x)G kl (ξ (x)) , (6.75)
i
2 ∂x ∂x j