Optimization and Computational Fluid Dynamics - Department of ...

3 Mathematical Aspects of CFD-based Optimization 67
B ≈Lpp −LpyA −1 cp − (LpyA −1 cp) ⊤ + c ⊤ p A−⊤ LyyA −1 cp
is recommended. Let us compare this with the SQP-formulation of equation
(3.12) in the separability framework
⎡ ⎤⎛
⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
⎣
Lyy Lyp c ⊤ x
Lpy Lpp c ⊤ p
cx cp 0
⎦⎝
∆y
∆p⎠
= ⎝
∆λ
−L ⊤ y
−L ⊤ p
−c
⎠ ,
⎝ yk+1
p k+1
λ k+1
⎠ =
⎝ yk
p k
λ k
⎠+τ· ⎝ ∆y
∆p⎠
. (3.23)
∆λ
Because of the triangular structure of the system matrix in Eqs. (3.21) or
(3.22), the reduced SQP formulation is much more modular than the full SQP
formulation (3.23). This is the reason why the matrices in Eqs. (3.21) or (3.22)
are used as pre-conditioner in large scale linear-quadratic optimal control
problems [2] or as pre-conditioners in Lagrange-Newton-Krylov methods as
discussed in [3, 4]. Indeed, one observes for the (iteration) matrix
⎡
M = I − ⎣
0 0 c⊤ x
0 Bc⊤ p
cx cp 0
⎤
⎦
−1⎡
Lyy Lyp c⊤ x
⎣
Lpy Lpp c ⊤ p
cx cp 0
the fact that M �= 0 in general, but M 3 = 0, which is the basis of the
convergence considerations in [23].
Now, let us discuss CFD-optimization in the context of one-shot aerodynamic
shape optimization as in [20, 21]. The typical problem formulation
there is
min f(y,p) (drag) (3.24)
y,p
s.t. c(y,p) = 0 (CFD-model) (3.25)
h(y,p) ≥ 0 (lift constraint) (3.26)
where y collects the state variables of an Euler or Navier-Stokes flow model
and p is a finite-dimensional vector parameterizing the shape of a part of an
aircraft by means of a suitable spline family. The lift constraint h is a scalar
valued function. Additionally, one often also has to treat a pitching moment
constraint, which is also a scalar valued function. Engineering knowledge tells
us that the lift constraint will be active at the solution. Therefore, we can
formulate the constraint right from the beginning in the form of an equality
constraint. In this context, a full SQP-approach as in equation (3.23) reads
⎡
⎤⎛
⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
⎢
⎣
Lyy Lyp h ⊤ x c⊤ x
Lpy Lpp h⊤ p c⊤ p
hx hp 0 0
cx cp 0 0
∆y
⎥⎜
⎥⎜∆p
⎟
⎦⎝∆μ⎠
∆λ
=
⎜
⎝
−L⊤ y
−L⊤ p
−h
−c
⎟
⎠ ,
⎤
⎦
y
⎜
⎝
k+1
pk+1 μk+1 λk+1 ⎟
⎠ =
y
⎜
⎝
k
pk μk λk ∆y
⎟ ⎜
⎟
⎠ + τ · ⎜∆p
⎟
⎝∆μ⎠
∆λ
.
(3.27)