Optimization and Computational Fluid Dynamics - Department of ...

104 Kyriakos C. Giannakoglou and Dimitrios I. Papadimitriou
Cp
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-0.2 0 0.2 0.4 0.6 0.8 1
(a)
x
initial
optimal
Cp
0.6
0.55
0.5
0.45
0.4
0.35
0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1
(b)
x
initial
optimal
Fig. 4.10 Total pressure losses minimization in a 2D compressor cascade (a) pressure
coefficient distribution for the initial and optimal cascade airfoil and (b) focusonthe
separation region
Cf
0.006
0.005
0.004
0.003
0.002
0.001
0
-0.001
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
(a)
x
initial
optimal
Cf
0.006
0.005
0.004
0.003
0.002
0.001
0
-0.001
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9
(b)
x
initial
optimal
Fig. 4.11 Total pressure losses minimization in a 2D compressor cascade (a) friction
coefficient distribution for the initial and optimal cascade airfoil and (b) focusonthe
separation region
4.8 Conclusions
Discrete and continuous adjoint approaches for use in aerodynamic shape
optimization problems were presented. These can be used to compute the
gradient of objective functions in inviscid or viscous flows. Different objective
functions (the standard one, used in inverse shape design problems, or
others, which are appropriate when the target is the minimization of entropy
generation or total pressure losses in internal flows), were handled. An improved
formulation of the adjoint problem avoids the computation of field
integrals containing metrics and other geometrical sensitivities and reduces
the overall computational cost. The extension of the adjoint formulation for
the computation of the Hessian matrix of a functional was then presented in
both discrete and continuous forms. The application of the Newton method