Page 168 - Basic Structured Grid Generation

Variational methods and adaptive grid generation 157

6.3 One-dimensional grid generation

6.3.1 Variational approach

According to the analysis of Section 5.7, the transformation x = x(ξ) between the one-
dimensional computational domain 0 ξ 1 and the physical domain a x b,
where the grid-point density is governed by a weight function ϕ(ξ), must satisfy
eqn (5.83). This equation is actually the Euler-Lagrange equation of the variational
problem δI = 0, where the functional I is given by
1 1 1 dx 2
I = dξ. (6.13)
2 0 ϕ(ξ) dξ
In fact, because the integrand is reasonably simple, we can reach without difficulty
the stronger conclusion that the solution of (5.83), subject to the end conditions x(0) =
a, x(1) = b, minimizes I. For suppose that x =ˆx(ξ) satisfies eqn (5.83) and the end
conditions. If we consider a varied function x =ˆx(ξ) + v(ξ), also satisfying the end
conditions, so that v(0) = v(1) = 0, we obtain, substituting into eqn (6.13),
1 b 1 dˆx dv 2 1 1 dˆx dv 1 1 1 dv 2
ˆ
I = + dξ = I + dξ + dξ,
2 a ϕ(ξ) dξ dξ 0 ϕ(ξ) dξ dξ 2 0 ϕ(ξ) dξ
ˆ
where I is the value of I when x =ˆx. The middle term vanishes since, on integration
by parts, we get
1 1
1 dˆx d 1 dˆx
v − v dξ,
ϕ(ξ) dξ 0 0 dξ ϕ(ξ) dξ
where the integrated part is zero because of the end conditions satisfied by v(ξ) and
the integral is zero because ˆx(ξ) satisfies eqn (5.83).
Hence we have the exact equation

1 1 1 dv 2
ˆ
ˆ
I = I + dξ I,
2 0 ϕ(ξ) dξ
since ϕ(ξ) takes positive values. So the minimizing property of ˆx(ξ) is established.
On the other hand, if we consider the functional
1 1 1 dx 2
˜
I = dξ, (6.14)
2 0 [ϕ(x)] 2 dξ
the extremal which makes I stationary must satisfy the Euler-Lagrange equation

2
d ∂I ˜ ∂I ˜ d x ξ 1 dϕ dx
− = +
3
dξ ∂x ξ ∂x dξ [ϕ(x)] 2 [ϕ(x)] dx dξ
2
1 d x 2 dx 2 dϕ 1 dϕ dx 2
= − +
2
3
[ϕ(x)] dξ 2 [ϕ(x)] 3 dξ dx [ϕ(x)] dx dξ
2
1 d x 1 dϕ dx 2
= − = 0,
2
3
[ϕ(x)] dξ 2 [ϕ(x)] dx dξ