Page 186 - Basic Structured Grid Generation

Variational methods and adaptive grid generation 175

which follows, similarly to what was presented in Exercise 10, Chapter 1, in the deriva-
k
tion of eqn (1.113), from differentiating with respect to x the Chain Rule
i
∂ξ ∂x l l
= δ .
j
j
∂x ∂ξ i
l
i
Thus, multiplying eqn (6.85) by ∂x /∂ξ (implying summation over i)gives
i
2 l
i
∂ξ ∂ξ m ∂ x 1 ∂ξ ∂x l ∂ √ jk
jk
γ = √ ( γγ ),
m
k
k
i
j
∂x ∂x ∂ξ ∂ξ i γ ∂x ∂ξ ∂x j
which, with further use of the Chain Rule, gives
i
2 l
∂ξ ∂ξ m ∂ x 1 ∂ √ jl
jk
γ = √ ( γγ ). (6.86)
m
j
k
∂x ∂x ∂ξ ∂ξ i γ ∂x j
i
m
k
j
Note that γ jk (∂ξ /∂x )(∂ξ /∂x ) is a tensor transformation rule for contravariant
second-order tensor components applied to the contravariant metric tensor γ jk .The
im
result is a set of contravariant components, say g , representing the contravariant
i
metric tensor in the new curvilinear co-ordinate system ξ in M. Hence we can
write
2 l
∂ x 1 ∂ √ jl
im
g = √ ( γγ ). (6.87)
m
∂ξ ∂ξ i γ ∂x j
This is a quasi-linear system of elliptic partial differential equations which may be
used as the basis of an algorithm for generating structured two-dimensional adaptive
grids, grids on surfaces, and three-dimensional grids.
6.5.1 Surface grids
3
As an example, we consider M to be a two-dimensional surface in E defined in
terms of cartesian co-ordinates by z = f (x, y). In fact we take x, y to be para-
2
1
metric co-ordinates for the surface, so that x = x, x = y. The covariant metric
tensor components of the surface with these co-ordinates are given by eqn (3.23),
so that
2
2
γ 11 = 1 + (f x ) , γ 12 = (f x )(f y ), γ 22 = 1 + (f y ) , (6.88)
and
2
2
γ = det(γ ij ) = 1 + (f x ) + (f y ) , (6.89)
where subscripts denote partial derivatives.
From eqn (3.30) we immediately have the contravariant components
1 + (f y ) 2 12 (f x )(f y )
11
γ = , γ =− ,
2
2
1 + (f x ) + (f y ) 2 1 + (f x ) + (f y ) 2
1 + (f x ) 2
22
γ = . (6.90)
2
1 + (f x ) + (f y ) 2