Page 13 - Basic Structured Grid Generation

2 Basic Structured Grid Generation

The position vector r of a point P in space with respect to some origin O may be
expressed as
r = y 1 i 1 + y 2 i 2 + y 3 i 3 , (1.1)

where {i 1 , i 2 , i 3 }, alternatively written as {i, j, k}, are unit vectors in the direction of the
rectangular cartesian axes. We assume that there is an invertible relationship between
this background set of cartesian co-ordinates and the set of curvilinear co-ordinates, i.e.
1 2 3
y i = y i (x ,x ,x ), i = 1, 2, 3, (1.2)
with the inverse relationship
i
i
x = x (y 1 ,y 2 ,y 3 ), i = 1, 2, 3. (1.3)
We also assume that these relationships are differentiable. Differentiating eqn (1.1)
i
with respect to x gives the set of covariant base vectors
∂r
g i = , i = 1, 2, 3, (1.4)
∂x i
with background cartesian components

∂y j
(g i ) j = , j = 1, 2, 3. (1.5)
∂x i
At any point P each of these vectors is tangential to a co-ordinate curve passing
i
through P, i.e. a curve on which one of the x s varies while the other two remain
constant (Fig. 1.1). In general the g i s are neither unit vectors nor orthogonal to each
3
other. But so that they may constitute a set of basis vectors for vectors in E we demand
that they are not co-planar, which is equivalent to requiring that the scalar triple product
{g 1 · (g 2 × g 3 )} = 0. Furthermore, this condition is equivalent to the requirement that
the Jacobian of the transformation (1.2), i.e. the determinant of the matrix of partial
j
derivatives (∂y i /∂x ), is non-zero; this condition guarantees the existence of the inverse
relationship (1.3).

3
x varies
g 2 2
y 3 g 3 x varies
P
g
1
1
x varies
i 3
i 2 y 2
O
i 1

y 1

Fig. 1.1 Covariant base vectors at a point P in three dimensions.