Page 15 - Basic Structured Grid Generation

4 Basic Structured Grid Generation

of values. (In expressions involving general curvilinear co-ordinates the summation
convention applies only when one of the repeated indices appears as a subscript and
the other as a superscript.) The comparison shows that
∂x i ∂x i ∂x i i i
i 1 + i 2 + i 3 =∇x = g , (1.11)
∂y 1 ∂y 2 ∂y 3
where the gradient operator ∇, or grad, is defined in cartesians by

∂ ∂ ∂ ∂
∇= i 1 + i 2 + i 3 = i k . (1.12)
∂y 1 ∂y 2 ∂y 3 ∂y k
For a general scalar field ϕ we have
∂ϕ ∂ϕ ∂x j ∂x j ∂ϕ j ∂ϕ
∇ϕ = i k = i k = i k = g , (1.13)
j j j
∂y k ∂x ∂y k ∂y k ∂x ∂x
making use of a chain rule again and eqn (1.11); this gives the representation of the
gradient operator in general curvilinear co-ordinates.

1.3 Metric tensors

i
Given a set of curvilinear co-ordinates {x } with covariant base vectors g i and con-
i
travariant base vectors g , we can define the covariant and contravariant metric tensors
respectively as the scalar products
g ij = g i · g j (1.14)
i
ij
j
g = g · g , (1.15)
where i and j can take any values from 1 to 3. From eqns (1.5), (1.10), for the back-
i
ground cartesian components of g i and g , it follows that
∂y k ∂y k
g ij = (1.16)
i
∂x ∂x j
and
i
∂x ∂x j
ij
g = . (1.17)
∂y k ∂y k
If we write (x, y, z) for cartesians and (ξ, η, ς) for curvilinear co-ordinates, we have
the formulas
2
2
g 11 = x + y + z 2 ξ
ξ
ξ
2
2
g 22 = x + y + z 2 η
η
η
2
2
g 33 = x + y + z 2 (1.18)
ς ς ς
g 12 = g 21 = x ξ x η + y ξ y η + z ξ z η