Optimization and Computational Fluid Dynamics - Department of ...

4 Adjoint Methods for Shape Optimization 101
Absolute Gradient Value
0.01
0.0001
1e-006
1e-008
1e-010
1e-012
1e-014
1st cycle
10th cycle
converged
1e-016
1 2 3 4 5
Design Variable
(a)
Hessian Value
0.018
0.016
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
-0.002
1st cycle
10th cycle
converged
0 5 10 15 20 25
Design Variable
Fig. 4.4 Inverse design of a 2D duct (a) change in the objective function gradient values
(absolute values) during the optimization (b) corresponding change in Hessian matrix
values
y
0.1
0.08
0.06
0.04
0.02
0
0 0.5 1 1.5 2
(a)
x
reference
initial
optimal
Cp
0.94
0.93
0.92
0.91
0.9
0.89
0.88
0.87
0.86
0.85
0.84
(b)
0 0.5 1 1.5 2
(b)
x
reference
initial
optimal
Fig. 4.5 Inverse design of a 2D duct (a) comparison of the initial, optimal and reference
airfoil contours (not in scale) (b) comparison of the initial, optimal and target pressure
distributions
also shown. This curve quantifies the difference (absolute value) in the actual
blade airfoil thickness, at specific chord-wise positions, from the corresponding
minimum allowed ones. The latter are defined to be equal to the 90% of
an existing (reference) airfoil. If any of these local constraints are violated,
a penalty value is added to the total penalty value. So, zero penalty values
correspond to a non-violated thickness constraint. During the first five cycles,
losses are reduced almost by 1.5%, while none of the thickness constraints are
violated. The reduction rate in total pressure losses decreased during the subsequent
cycles and oscillations became apparent due to occasional constraint
violations. Upon convergence (for which around 40 cycles are needed), the
thickness constraints are not violated any more or are just slightly violated.
The reduction rate of the gradient values for all design variables are shown
in Fig. 4.8. The objective function gradient with respect to the suction side