Optimization and Computational Fluid Dynamics - Department of ...

68 H.G. Bock and V. Schulz
This approach is not implementable in general because one usually starts
out with a flow solver for c(y,p) = 0 and seeks a modular coupling with
an optimization approach, which does not necessarily change the whole code
structure, as would be the case with formulation (3.23). A modular but nevertheless
efficient alternative is an approximate reduced SQP approach as in
Eq. (3.22), which is adapted to the case of the additional lift (or pitching)
constraint, as established in [16].
⎡
0 0 0 A
⎢
⊤
⎤⎛
⎞ ⎛
∆y −L
⎥⎜∆p⎟
⎜
⊤ ⎞ ⎛
y y
⎟ ⎜
k+1
pk+1 ⎞ ⎛
y
⎟ ⎜
k
pk ⎞ ⎛ ⎞
∆y
⎟ ⎜∆p⎟
0 Bγc ⊤ p
⎢
⎣0
γ⊤ 0 0
Acp 0 0
where
⎥⎜
⎟
⎦⎝∆μ⎠
∆λ
= ⎜
⎝
−L ⊤ p
−h
−c
⎟
⎠ ,
⎜
⎝
μ k+1
λ k+1
⎟
⎠ = ⎜
⎝
μ k
λ k
γ = h ⊤ p + c ⊤ p α, such that A ⊤ α = −h ⊤ x .
⎟
⎠ +τ · ⎜ ⎟
⎝∆μ⎠
∆λ
(3.28)
An algorithmic version of this modular formulation is given by the following
steps:
(1) generate λ k by performing N iterations of an adjoint solver with right
hand side f ⊤ y (y k ,p k )startinginλ k
(2) generate α k by performing N iterations of an adjoint solver with right
hand side h ⊤ y (y k ,p k )startinginα k
(3) compute approximate reduced gradients
g = f ⊤ p + c⊤ p λk+1 , γ = h ⊤ p + c⊤ p αk+1
(4) generate Bk+1 as an approximation of the consistently reduced Hessian
(5) solve the QP �
B γ
γ⊤ ��
∆p
0 μk+1 � � �
−g
=
−h
(6) update p k+1 = p k + ∆p
(7) compute the corresponding shape geometry and adjust the computational
mesh
(8) generate y k+1 by performing N iterations of the forward state solver
starting from an interpolation of y k at the new mesh.
This highly modular algorithmic approach is not an exact transcription of
Eq. (3.28), but is shown in [16] to be asymptotically equivalent and to converge
to the same solution. The overall algorithmic effort for this algorithm
is typically in the range of factor 7 to 10 compared to a forward stationary
simulation.