Page 108 - Basic Structured Grid Generation

Structured grid generation – algebraic methods 97

values η j , ς k with
j − 1 k − 1
0 η j = 1, 0 ς k = 1, j = 1, 2,. .., ˜, k = 1, 2,..., k ˜
˜  − 1 k − 1
˜
for some ˜, k. We could then use P ξ , through eqn (4.78), to interpolate a grid between
˜
i−1
ı
these faces, taking discrete values of ξ also, with 0 ξ i = 1, i = 1, 2,..., ˜.
˜ ı−1
The bilinear ‘tensor product’ P ξ P η may be expressed in full as
P ξ P η (ξ, η, ς) = (1 − ξ)(1 − η)r(0, 0,ς) + (1 − ξ)ηr(0, 1,ς)
+ξ(1 − η)r(1, 0,ς) + ξηr(1, 1,ς). (4.81)

The effect of this transformation on the unit cube is to map all four straight edges
parallel to the ς direction onto the corresponding four curved edges r(0, 0,ς),etc.,
of R. Between these curved edges we then have linear interpolation in both ξ and η
directions. This map could be used for linear interpolation if we started with just those
four edges of R. The other bilinear products have similar properties, and are given by
P η P ς (ξ, η, ς) = (1 − η)(1 − ς)r(ξ, 0, 0)

+(1 − η)ςr(ξ, 0, 1) + η(1 − ς)r(ξ, 1, 0) + ηςr(ξ, 1, 1), (4.82)
P ξ P ς (ξ, η, ς) = (1 − ξ)(1 − ς)r(0,η, 0)
+(1 − ξ)ςr(0,η, 1) + ξ(1 − ς)r(1,η, 0) + ξςr(1,η, 1). (4.83)

Clearly these products all have the property of commutativity.
We can also formulate the ‘trilinear’ transformation P ξ P η P ς , which may be ex-
pressed in full as

P ξ P η P ς (ξ, η, ς) = (1 − ξ)(1 − η)(1 − ς)r(0, 0, 0)
+ξ(1 − η)(1 − ς)r(1, 0, 0) + (1 − ξ)η(1 − ς)r(0, 1, 0)
+(1 − ξ)(1 − η)ςr(0, 0, 1) + ξη(1 − ς)r(1, 1, 0)
+ξ(1 − η)ςr(1, 0, 1) + (1 − ξ)ηςr(0, 1, 1) + ξηςr(1, 1, 1).
(4.84)
This trilinear interpolant maps the unit cube onto a region of physical space with
the same vertices as R but with straight lines connecting the vertices.
The Boolean sum P ξ ⊕ P η ⊕ P ς may be formulated in terms of the above mappings
by successively applying the definition (4.72). We have

P ξ ⊕(P η ⊕P ς ) = P ξ ⊕(P η +P ς −P η P ς ) = P ξ +P η +P ς −P η P ς −P ξ P η −P ξ P ς +P ξ P η P ς .
(4.85)
It is straightforward to show that the same result emerges from evaluating (P ξ ⊕
P η ) ⊕ P ς , which means that Boolean summation is associative. Putting ξ = 0in the
expressions (4.78), (4.79), (4.80), (4.81), (4.82), (4.83), and (4.84), and combining the
results according to the vector sums in (4.85) shows that the face ξ = 0 of the unit
cube maps onto the curved face r(0,η, ς) of R under the Boolean sum (4.85). In fact
each face of the cube maps onto a face of R.