Optimization and Computational Fluid Dynamics - Department of ...

100 Kyriakos C. Giannakoglou and Dimitrios I. Papadimitriou
Functional Value
1
1e-005
1e-010
1e-015
1e-020
1e-025
Quasi Newton
Conjugate Gradient
Steepest Descent
1e-030
0 50 100 150 200 250
(a)
Functional Value
Cycles
1
1e-005
1e-010
1e-015
1e-020
1e-025
Functional Value
1
1e-005
1e-010
1e-015
1e-020
1e-025
1e-030
0 20 40 60 80 100 120 140
Newton
Quasi Newton
1e-030
0 10 20 30 40 50 60 70
Equivalent Flow Solutions
(c)
Newton
Quasi Newton
(b)
Cycles
Fig. 4.3 Inverse design of a 2D duct (a) convergence rates of the three gradient-based
algorithms (steepest descent, conjugate gradient and quasi-Newton) (b) convergence of
Newton and quasi-Newton algorithms in terms of optimization cycles c convergence of
Newton and quasi-Newton algorithms in terms of required equivalent flow solutions
losses used as objective function and the corresponding adjoint equations are
solved for the computation of the gradient. The sensitivity of the turbulent
viscosity is not taken into account in the adjoint formulation. Geometrical
constraints are imposed in order to prevent the blade shape from becoming
too thin. The gradient of the constraint is added to the gradient of F,using
the augmented Lagrange multipliers method [10].
Each blade side is parameterized with 13 control points using Bézier-
Bernstein polynomials [19]. Only the blade-to-blade coordinates of the control
points are allowed to vary. The first three control points (determining
the curvature of the leading edge) and the last one are kept constant on both
sides. The H-type structured grid, chosen after a grid-independency study,
has 400×200=80, 000 nodes. The flow conditions are: α1 =44 ◦ , M2,is =0.45
and the chord-based Reynolds number is Rec =8×10 5 . The grid together
with the Mach number distribution over the optimal cascade airfoil is shown
in Fig. 4.6.
The convergence history of F (total pressure losses) is shown in Fig. 4.7(a).
In Fig. 4.7(b), the convergence history of the sum of violated constraints is