Optimization and Computational Fluid Dynamics - Department of ...

4 Adjoint Methods for Shape Optimization 99
Gradient Value
-0.0008
-0.001
-0.0012
-0.0014
-0.0016
-0.0018
-0.002
finite differences
direct sensitivity
reduced adjoint
conventional adjoint
-0.0022
1 2 3 4 5
Design Variable
(a)
Hessian Value
0.012
0.01
0.008
0.006
0.004
0.002
0
finite differences
direct-adjoint approach
-0.002
0 5 10 15 20 25
Design Variable
Fig. 4.2 Inverse design of a 2D duct (a) objective function gradient values computed
using the metrics-free and the conventional adjoint approach, the direct approach and
finite differences (b) Hessian matrix values using the direct-adjoint approach and finite
d
differences. The first five values correspond to the first row of
2 F
and so forth (5
dbidbj columns × 5rows=25values)
solutions which are a direct measure of CPU cost. It is clear that the exact
Newton approach converges faster to the optimal solution, at least with this
particular number of design variables.
The change in the gradient vector values is shown in Fig. 4.4(a), in semilog
scale. Gradient values are zeroed upon convergence. In Fig. 4.4(b), it is
shown that the Hessian matrix values change slightly from cycle to cycle and
the greater change occurs during the first optimization cycles.
In Fig. 4.5(a), the initial, reference and optimal bump shapes are compared.
The shape computed via the Newton method is shown, although there
are practically no apparent differences between optimal solutions obtained by
any of the two methods. The corresponding pressure distributions are also
compared in the same figure, right. The slow convergence of steepest descent
and conjugate gradient algorithms results (after approximately 200 cycles)
into computations which were not run to convergence; so there are some discrepancies
between the optimal and target distributions (not evident, in the
scale of the figures shown herein).
4.7.2 Losses Minimization of a 2D Compressor
Cascade
The next application example is concerned with the minimization of viscous
losses of the flow in a 2D compressor cascade through the redesign of
the airfoil shape. The Navier Stokes equations are solved together with the
Spalart-Allmaras turbulence model for the computation of the total pressure
(b)