Page 120 - Basic Structured Grid Generation

Structured grid generation – algebraic methods 109

4.6.1 Subdirectory: Book/univariate.gds

1. File: linear.f
This program simply generates a one-dimensional grid on a straight line based on
the linear interpolation

x = (1 − ξ)x 0 + ξx 1 , 0 ξ 1.
2. File: Eriksson.f
This generates a one-dimensional grid on a straight line using the Eriksson stretching
function
β
x = (e βξ − 1)/(e − 1), 0 ξ 1.
Here the constant β controls the degree of clustering near x = 0.
3. File: Compress1.f
This generates another grid on a straight line using the stretching function
kb 1/b k
x = (p + cξ) − p , 0 ξ 1, (4.123)
k
b
kb
where p, k,and b are constants, with c = (p + 1) − p .
4. File: Compress2.f
This generates a grid on a straight line with the stretching function

tanh[Q(1 − ξ)]
x = Pξ + (1 − P) 1 − , 0 ξ 1, (4.124)
tanh Q
mentioned above. For example, when the parameters P and Q take the values
P = 1.8, Q = 2.0, an equally spaced set of points in the ξ computational domain
maps into a set of points in the physical x domain with clustering near x = 1.
When P = 0.9, Q = 2.0, a fairly equally spaced set of points in physical space is
obtained, while the choice P = 0.1, Q = 2.0 gives clustering near x = 0.

4.6.2 Subdirectory: Book/Algebra

1. File: Algebraic1.f
This generates a two-dimensional planar boundary-conforming grid using the co-
ordinate transformation given by eqn (4.98), where the functions h 1 (x) and h 2 (x)
are specified by the user. The program calculates the so-called metrics of the trans-
formation, given by the partial derivatives ∂ξ/∂x, ∂ξ/∂y, ∂η/∂x, ∂η/∂y, which are
required in transforming the hosted partial differential equations to be solved.
In this example we can show that

∂ξ ∂ξ ∂η

= 1, = 0, =[h (x)h 1 (x) − h (x)h 2 (x) − yh (x)]
1
2
2
∂x ∂y ∂x
−2
×[h 2 (x) − h 1 (x)] ,
∂η −1
=[h 2 (x) − h 1 (x)] .
∂y