Optimization and Computational Fluid Dynamics - Department of ...

8 Multi-objective Optimization in Convective Heat Transfer 233
Table 8.1 Variables defining the linear piecewise channel
Variable Symbols Range
module length L [0.8;2.0]
corrugation height h [0.0;0.5]
forward edge angle ϕin [10 ◦ ;60 ◦ ]
backward edge angle ϕout [10 ◦ ;60 ◦ ]
translation of upper wall transl [−0.5L;0.5L]
Table 8.2 Control points for the NURBS-profile and parameters required
Point x y DOFs
1 0 0 0
2 (L − Lwave)/2 0 0
3 x2 + ∆x 23 0 1
4 x5 + ρ 54 cos ϑ54 y5 + ρ 54 sinϑ54 2
5 x1 + ∆x5 y1 + ∆y5 2
6 x5 + ρ 56 cos ϑ56 y5 + ρ 56 sinϑ56 2
7 x8 − ∆x 87 0 1
8 x2 + Lwave 0 1
9 L 0 1
8.5.1.2 2D NURBS Parametrization
In order to generate the NURBS channel during the optimization process,
let’s start by defining first the wall profile. As a good compromise between the
number of DOFs and the geometrical complexity, a 9 control-point periodic
cubic NURBS has been chosen, to ensure the periodicity of the channel itself.
A large number of DOFs allows to describe minutely the profile, but if this
number is excessive, it would make the optimization process quite difficult
and expensive. For this reason, it has been also decided to fix to a unitary
value the curve’s weights. This, together with the uniform knots distribution
we adopted, makes our NURBS curve practically equivalent to a B-Spline
curve [18, 38].
As depicted in Fig. 8.5(d), both the first and the last three aligned control
points are needed to maintain the entrance and the exit of the profile parallel
to the x direction. The remaining ones give freedom to the wavy section. The
parameters required for the definition of the lower profile are explained in
Table 8.2 and numbered as in Fig. 8.5(d). The symbol ∆ means the difference
between the coordinates of two points while ρ ij is the module and ϑij the
phase of a polar-coordinate system centered on the point i. The phase is
positive counterclockwise with respect to the positive direction of the x axis.
The profile is again constructed, for convenience, in order to have the wavy