Optimization and Computational Fluid Dynamics - Department of ...

114 Nicolas R. Gauger
Fig. 5.1 Cost function computation
design vector + initial grids
⇓
surface deformation
⇓
computational grid deformation
⇓
flow computation
⇓
cost function value
initial shape or surface mesh and its dependent computational grid based on
the parameterization and afterwards evaluating the cost function. A schema
of the procedure is illustrated in Fig. 5.1.
5.2.1 Surface Deformation
The basic idea for deforming the surface of an airfoil is to compute functions
and then add their values to the upper and lower side of the surface.
Therefore, every design parameter is used to scale a specific function which
is afterwards added to the shape.
Several kinds of functions are considered for the deformation. The first are
the Hicks-Henne functions which are defined as
� � log 0,5 b
ha,b :[0, 1] → [0, 1] : ha,b(x)= sin(πx log a ) .
These functions are positive, defined and mapped in the interval [0, 1] where
their peak is at position a. Furthermore, they are analytically smooth at zero
and one.
The used parameterization operates with Hicks-Henne functions with a
fixed b of 3.0andavaries from 3 n+3
n+5 to n+5 where n is the number of design
parameters.
The second kind of functions considered is transformed cosine functions.
These cosine functions are defined for q ∈ [0, 1] as cq(x):[0, 1] → [0, 1] where
and
cq(x)=
�
cq(x)=
� 1
2
�
π)) for x ≤ 2q
0 for x>2q
(1 − cos(x
q
1
2 (1 − cos(x−2q+1
1−q
0 for
�
x 1
2 .