Page 27 - Basic Structured Grid Generation

16 Basic Structured Grid Generation

k
By simply substituting the result (1.106) for ∂g ij /∂x into the right-hand side of the
following equation, it is easy to verify the important result

1 ∂g jk ∂g ik ∂g ij
[ij, k]= + − . (1.108)
2 ∂x i ∂x j ∂x k
k
Neither [ij, k] nor is a third-order tensor. In a system of cartesian co-ordinates,
ij
with constant base vectors, all components of the Christoffel symbols are zero, and
tensor components which are all zero would remain zero under a transformation to a
k
different co-ordinate system. In fact the transformation rule for under transformation
ij
i
to co-ordinates x may be derived as follows, using eqns (1.62), (1.68), and (1.96):
k ∂g i k ∂ ∂x m ∂x k n
= · g = g m · g
ij j j i n
∂x ∂x ∂x ∂x
m
2 m
∂ x ∂x k n ∂x ∂x k ∂g m ∂x l n
= g m · g + · g
j
i
i
l
∂x ∂x ∂x n ∂x ∂x n ∂x ∂x j
2 m
k
m
∂ x ∂x k n ∂x ∂x ∂x l n
= δ + ml
n
i
j
i
∂x ∂x ∂x n m ∂x ∂x ∂x j
from eqn (1.99). Thus we have
2 m
k
k ∂ x ∂x k ∂x ∂x m ∂x l n
= + . (1.109)
ml
ij
i
j
n
i
∂x ∂x ∂x m ∂x ∂x ∂x j
This equation does not follow the transformation rule for a mixed third-order tensor
because of the presence of the first term on the right side.
i
A useful special case occurs when we let the new co-ordinates x coincide with
the background rectangular cartesian co-ordinates y 1 ,y 2 ,y 3 . The components of the
Christoffel symbol associated with the new co-ordinates are then identically zero, and
eqn (1.109) becomes
m
2 m
∂ x ∂y k ∂y k ∂x ∂x l n
0 = + .
ml
n
∂y i ∂y j ∂x m ∂x ∂y i ∂y j
p
Multiplying through by ∂x /∂y k (implying summation over k), using a chain rule
again, and re-arranging, we obtain
2 p
m
∂ x ∂x ∂x l p
=− . (1.110)
ml
∂y i ∂y j ∂y i ∂y j
Contraction on i and j (putting j = i, implying summation) gives the formulas
2
m
2 p
∂ x 2 p ∂x ∂x l p ml p ml ∂ y n ∂x p
=∇ x =− ml =−g ml =−g , (1.111)
m
l
∂y i ∂y i ∂y i ∂y i ∂x ∂x ∂y n
2
where ∇ is the Laplacian operator
∂ 2 ∂ 2 ∂ 2 ∂ 2
= + + .
∂y i ∂y i ∂x 2 ∂y 2 ∂z 2