Optimization and Computational Fluid Dynamics - Department of ...

4 Adjoint Methods for Shape Optimization 89
An = Aknk and nk are the outward normal to the boundary, unit vector
components. It is important to note that Eq. (4.18) includes a field integral
of grid metric sensitivities δ
δbi
� ∂ξ m
∂xk
�
, the computation of which increases
the CPU cost and becomes a source of inaccuracies. The standard way to
compute grid metric sensitivities is through finite differences (after perturbating
one design variable at a time, modifying the contour and remeshing).
If Eq. (4.16a) or
δFaug
δbi
= δF
�
T ∂
+ Ψ
δbi Ω ∂xk
� ∂fk
∂bi
�
dΩ (4.22)
is used instead, the adjoint equations and boundary conditions remain exactly
the same (Eqs. (4.19), (4.20) and (4.21)) but the gradient expression becomes
δF
δbi
= 1
�
�
2 δ(dS) ∂U
(p − ptar) −
2 Sw δbi Sw ∂xk
�
+ (Ψk+1p − Ψ T f k) δ(nkdS)
δbi
Sw
T
An T Ψ δxk
dS
δbi
(4.23)
which differs from Eq. (4.18) since Eq. (4.23) is free of field integrals. Thus,
the gradient values are computed with less computational cost and greater
accuracy. In addition, Eq. (4.23) is more general and can be used with either
structured or unstructured grids.
Proof of Eq. (4.23):
The integration by parts of the second term on the right-hand side (r.h.s.) of
Eq. (4.22) gives
�
Ω
T ∂
Ψ
∂xk
� ∂fk
∂bi
� �
dΩ = −
�
+
Ω
∂U T
∂bi
Si,o,w
�
A T �
∂Ψ
k dΩ
∂xk
T ∂fk
Ψ nkdS . (4.24)
∂bn
Employing the no-penetration condition, the second integral on the r.h.s of
Eq. (4.24) is written as
�
�
�
T ∂fk
T δfk
T ∂fk δxm
Ψ nkdS = Ψ nkdS − Ψ nkdS
Sw ∂bi Sw δbi Sw ∂xm δbi
�
�
δp
= Ψk+1nk dS + (Ψk+1p − Ψ
Sw δbi Sw
T fk) δ(nkdS)
δbi
� T
∂U
− An
∂xk
T Ψ δxk
dS . (4.25)
δbi
Sw