Optimization and Computational Fluid Dynamics - Department of ...

84 Kyriakos C. Giannakoglou and Dimitrios I. Papadimitriou
constrained by the state (flow) equations Rm(U)=0,m=1, ..., M, which
must be satisfied over the M grid nodes. The gradient of these constraints is
expressed as
dRm
dbi
= ∂Rm
∂bi
+ ∂Rm
∂Uk
dUk
dbi
=0. (4.2)
The combination of Eqs. (4.1) and (4.2) allows the elimination of dUk
dbi and
produces the final expression for dF . By introducing the adjoint or costate
dbi
variables Ψm,m=1, ..., M, this expression becomes
dF
dbi
= ∂F ∂Rm
+ Ψm
∂bi ∂bi
(4.3)
where the adjoint variables result from the solution of the adjoint equations
R Ψ k
∂F ∂Rm
= + Ψm =0. (4.4)
∂Uk ∂Uk
Since such a development relies on the discrete state equations, Eqs. (4.2),
it is usually referred to as the discrete adjoint method to distinguish it from
the continuous one (see below). The advantage of the adjoint formulation is
that only one adjoint equation has to be solved irrespective of the value of
N.
The adjoint approach, which is associated with Eqs. (4.1) to (4.4) for the
computation of the gradient of F, can also be applied to the computation
of any quantity F = γ T U,whereU satisfies the linear system AU = φ;
γ and φ are known vectors and A is a known square matrix. Here also,
the direct computation of F (F = γ T A −1 φ) can be replaced by an adjoint
approach. The adjoint equation A T Ψ = γ is first solved and then, F results
from F = Ψ T φ,[55].
4.2.2 The Continuous Adjoint Approach
Section 4.2.1 summarizes the discrete adjoint approach as a means to compute
the gradient of an objective function F, in discrete form, constrained by the
discretized state PDE’s. Alternatively, the same problem may be handled by
the continuous adjoint approach, where the adjoint PDE’s are first produced
and then, discretized and solved.
Assume that the sensitivity derivatives of F with respect to bi can be
expressed as
δF
δbi
�
=
Ω
γ δU
�
dΩ +
δbi S
ζB1( δU
)dS +
δbi
δFg
δbi
(4.5)