Page 157 - Basic Structured Grid Generation

146 Basic Structured Grid Generation

giving finally
∂ϕ ∂ ∂ϕ ∂v
i 2 i ij i
[v + ν(∇ x )] + g ν + ϕg · + S = 0, (5.121)
∂x i ∂x i ∂x j ∂x i
i
i
i
where the g s are given by eqns (1.8) or (1.161), v = g ·v,and i, j are summed from
1 to 2 or from 1 to 3, depending on whether the problem is in one or two dimensions.
Exercise 6. Making use of eqns (1.120), (1.118), and (1.135), show that another (con-
servative) form of the equation is
∂ √ i ij ∂ϕ √
g v ϕ + νg + gS = 0. (5.122)
∂x i ∂x j

Now, following the generation of a structured grid for the physical space, with the
i
x co-ordinate curves coinciding by definition with grid lines, the hosted equation may
i
be discretized and solved on the square or rectangular domain in x computational
space, subject to the given boundary conditions.
Typical boundary conditions
∂ϕ
c 1 ϕ + c 2 = c 3 (5.123)
∂n
in physical space transform, according to eqn (1.193), into the condition
1 ij ∂ϕ
g = c 3 , (5.124)
c 1 ϕ + c 2 ) j
g ii ∂x
i
with summation over j but not i, on a boundary x = const. in computational space.

5.12 Multiblock grid generation

When the geometry of the solution domain is complex, as is generally the case in
engineering problems, one approach is to divide it into subdomains (blocks) of simpler
geometries. Structured grid generation can be carried out in each of the sub-domains
using algebraic methods or differential models, and the resulting grids can then be
patched together at the common boundaries. A major drawback of this ‘multiblocking’,
or ‘block-structuring’, technique has been the difficulty of automating the process of
domain decomposition. Dividing the domain into subdomains while keeping track of
data relating to the boundaries between the blocks and to the ‘connectivity’ of the
blocks is not difficult when the number of subdomains is relatively small, so that
the process can be performed manually. However, the generation of this initial data-
structure can be unacceptably time-consuming when the number of subdomains is large
(say, into the hundreds).
Approaches to automating this procedure were suggested by Allwright (1988) and
Eiseman, Cheng, and Hauser (1994), and there are currently commercially available
Computer Aided Design (CAD) codes which can be used. These perform block geom-
etry generation automatically, together with curve-fitting, typically using cubic splines,
along the common boundaries of the sub-domains. When the overall solution domain