Optimization and Computational Fluid Dynamics - Department of ...

4 Adjoint Methods for Shape Optimization 85
where γ and ζ are functions of geometrical quantities and state variables, B1
is a differential operator and δFg is the gradient of a function depending on
δbi
the contour and/or grid sensitivity derivatives. Equation (4.5) contains both
field (over the flow domain Ω) and boundary (along its boundary S = ∂Ω)
integrals. Expressions for δU can be obtained using the derivatives of the
δbi
state equations and their boundary conditions that may be written as
L( δU
)=φ,over Ω
δbi
B2( δU
)=ǫ, along S (4.6)
δbi
where φ, ǫ are known functions and L, B2 are known operators. Using the
continuous adjoint approach, δF
δbi canbeexpressedas
δF
δbi
�
=
Ω
�
ΨφdΩ +
S
(B ∗ 1Ψ)ǫdS + δFg
δbi
(4.7)
where the adjoint field Ψ is computed by discretizing and solving the adjoint
PDE’s with appropriate boundary conditions, namely
L ∗ Ψ=γ ,over Ω
where L ∗ is the adjoint operator to L and B ∗ 1 , B∗ 2
B ∗ 2Ψ=ζ ,along S (4.8)
are boundary operators
satisfying the following equation [55],
�
ΨL(
Ω
δU
�
)dΩ− (L
δbi Ω
∗ Ψ) δU
�
dΩ = (B
δbi S
∗ 2Ψ)B1( δU
)dS
δbi
�
− (B ∗ δU
1Ψ)B2( )dS . (4.9)
δbi
Advantages and disadvantages of the discrete and continuous approaches
are discussed in Sect. 4.2.3. Nevertheless, regardless of the form (discrete
or continuous), the adjoint approach, as a means to compute the objective
function gradient in aerodynamic optimization, is much more inexpensive
than the direct approach.
4.2.3 Differences Between Discrete and
Continuous Adjoint
A marked difference that distinguishes the discrete from the continuous adjoint
approach is the way the discrete adjoint equations are derived, starting
from the state PDE’s. The two alternative ways to produce the discrete ad-
S