Optimization and Computational Fluid Dynamics - Department of ...

4 Adjoint Methods for Shape Optimization 87
where U is the vector of conservative flow variables, U = � ρ,ρv T ,E � T ,
f inv
k = � ρuk ,ρukv T + pδ T k ,uk(E + p) � T is the vector of the inviscid fluxes,
ρ is the local fluid density, uk are the velocity v components, p is pressure,
E = ρe + 1
2 ρv2 is the total energy per unit volume and δk is the vector of
Kronecker symbols. For the sake of simplicity, throughout this section which
is exclusively concerned with inviscid flows, the inviscid fluxes f inv
k will be
denoted by fk.
In inverse design problems, the goal is to compute the aerodynamic shape
that produces a given pressure (or any other) distribution over the shape
contour. The term “aerodynamic shape” is used to denote isolated or cascade
airfoils, ducts, etc. (in 2D) or wings, blades, ducts, etc. (in 3D). The objective
function is written as
F = 1
�
(p − ptar)
2
2 dS (4.11)
Sw
where ptar(S) is the target pressure distribution along the solid walls Sw.
The gradient of F with respect to the design variables controlling the shape
is expressed as
δF
δbi
= 1
2
�
Sw
�
2 δ(dS)
(p − ptar) + (p − ptar)
δbi Sw
δp
dS . (4.12)
δbi
It is straightforward to derive expressions for δU
δbi
using the so-called direct
approach. Starting from the Euler equations, we obtain
� �
δ ∂fk
=
δbi ∂xk
δ
� �
∂U
Ak = 0 . (4.13)
δbi ∂xk
Equation (4.13) can be solved for the flow sensitivities δU , provided that
� � δbi
δ ∂U
the sensitivities of spatial derivatives can be transformed to the
δbi ∂xk�
, which appears in
spatial derivatives of flow sensitivities ∂
∂xk
Eq. (4.12), is expressed in terms of δU
δbi
� δU
δbi
.Since δp
δbi
,anexpressionfor δF
δbi
can be produced.
Unfortunately, to compute the grid sensitivities that appear in any possible
form of Eq. (4.13) (see, for instance, Eq. (4.17) for grid metrics sensitivities)
through finite differences, 2N calls to the grid generation software are needed.
The detailed development and solution algorithm are omitted.
Alternatively, the partial derivatives of Eq. (4.10) with respect to bi
� �
∂ ∂fk
=
∂bi ∂xk
∂
� �
∂U
Ak = 0 (4.14)
∂xk ∂bi
can be used to produce expressions for ∂U . The assumption that the Jacobian
∂bi
matrix derivatives can be omitted is made. Once ∂U
∂bi
computed using
are known, δU
δbi
can be