Optimization and Computational Fluid Dynamics - Department of ...

92 Kyriakos C. Giannakoglou and Dimitrios I. Papadimitriou
alternative formulations, a distinguishing feature stems from the (field or
boundary) integral used to define the objective function.
4.5.1 Minimization of Total Pressure Losses
In contrast to inverse design problems, F is defined by integrals along the
inlet and outlet boundaries of the domain, (Si,So) instead of the parameterized
wall boundary. Consequently, F and bi are “non-collocated”. For this
reason, the way inlet/outlet boundary conditions for the adjoint variables are
imposed is described below [54].
The total pressure losses functional is defined as
� � �
F = ptdS = ptdS − ptdS (4.32)
Si,o
and its gradient is simply given by
δF
δbn
=
�
δpt
dS −
δbi
Si
Si
�
So
So
δpt
dS (4.33)
δbi
since the inlet and exit boundaries remain fixed. Using the adjoint approach,
the gradient of the objective function can be expressed as follows
�
�
δF � � T δ(nkdS) δ(nmdS)
=− Ψ f k + ΨΛqm
δbi Sw δbi Sw δbi
� � �
∂uk ∂Ψm+1
− μ +
Sw ∂xl ∂xk
∂Ψk+1
� �
∂Ψn+1 δxl
+λδkm nmdS
∂xm ∂xn δbi
�
− Ψ T
� inv ∂f k −
∂xm
∂fvis
�
k δxm
nkdS . (4.34)
∂xm δbi
Sw
The field adjoint equations are given by Eq. (4.29) and are similar to the
ones used in viscous inverse designs. The solid wall conditions for the Ψ
components that correspond to the velocity components are homogeneous
Dirichlet. Dirichlet or Neumann conditions are imposed for ΨΛ, depending
on the wall condition on temperature.
The adjoint boundary conditions over the outlet of the flow domain are
imposed by solving
� (P ¯ Λ) T Ψ �T +
� �T ∂pt
L = 0 (4.35)
∂V
for the characteristic flow variables directed outwards. ∂pt
∂V is the derivative of
the total pressure with respect to the nonconservative variables V , P is the