Page 19 - Basic Structured Grid Generation

8 Basic Structured Grid Generation

1.4 Line, area, and volume elements

Lengths of general infinitesimal line-elements are given by eqn (1.21). An element of
1 2 3 2
the x co-ordinate curve on which dx = dx = 0 is therefore given by (ds) =
1 2
i
g 11 (dx ) . Thus arc-length along the x -curve is
√ i
ds = g ii dx (1.42)
(with no summation over i).
1
1
A line-element along the x -curve may be written ∂r dx 1 = g 1 dx ,and simi-
∂x 1
2
2
larly a line-element along the x -curve is g 2 dx . The infinitesimal vector area of the
parallelogram of which these two line-elements form the sides is the vector product
2
1
(g 1 dx × g 2 dx ), which has magnitude
2
1
dA 3 =|g 1 × g 2 | dx dx . (1.43)
Again by the Lagrange vector identity we have
2
|g 1 × g 2 | = (g 1 × g 2 ) · (g 1 × g 2 ) = (g 1 · g 1 )(g 2 · g 2 ) − (g 1 · g 2 )(g 1 · g 2 )
2
= g 11 g 22 − (g 12 ) .

1 2
2
Hence dA 3 = g 11 g 22 − (g 12 ) dx dx , giving the general expression

k
k
j
j
2
dA i = g jj g kk − (g jk ) dx dx = G i dx dx , (1.44)
using eqn (1.34), where i, j, k must be taken in cyclic order 1, 2, 3, and again there is
no summation over j and k.
1
3
2
The parallelepiped generated by line-elements g 1 dx , g 2 dx , g 3 dx , along the co-
ordinate curves has infinitesimal volume
3
1
2
2
3
1
dV = g 1 dx · (g 2 dx × g 3 dx ) ={g 1 · (g 2 × g 3 )}dx dx dx .
By eqn (1.31) we have
√ 1 2 3
dV = g dx dx dx . (1.45)
1.5 Generalized vectors and tensors
A vector field u (a function of position r) may be expressed at a point P in terms of
1
the covariant base vectors g 1 , g 2 , g 3 , or in terms of the contravariant base vectors g ,
2
3
g , g . Thus we have
1 2 3 i
u = u g 1 + u g 2 + u g 3 = u g i (1.46)
2
1
3
i
= u 1 g + u 2 g + u 3 g = u i g , (1.47)
i
where u and u i are called the contravariant and covariant components of u, respect-
j
ively. Taking the scalar product of both sides of eqn (1.46) with g gives
j i j i j j
u · g = u g i · g = u δ = u .
i