Optimization and Computational Fluid Dynamics - Department of ...

4 Adjoint Methods for Shape Optimization 93
matrix with the right eigenvectors of the conservative Jacobian matrix Aknk,
L is the matrix with the right eigenvectors of the nonconservative Jacobian
matrix and ¯ Λ is the diagonal matrix with the eigenvalues of the Jacobian
matrix. The boundary condition at the inlet of the flow domain is imposed
by solving
since pt is constant at the inlet.
� (P ¯ Λ) T Ψ �T = 0 (4.36)
4.5.2 Minimization of Entropy Generation
From a practical point of view, it makes no difference to use either total
pressure losses or entropy generation as a measure of viscous losses in internal
flows. From the mathematical point of view, however, the use of entropy
generation has the additional feature that the objective function is a field
integral. Thus, F is defined as
�
�
∂s 1
F = ρuk dΩ =
∂xk T τkm
∂uk
dΩ . (4.37)
∂xm
Ω
The gradient of F can be computed using the direct approach, by expressing
the sensitivities of the finite volume dΩ as, [53],
δ(dΩ)
=
δbi
∂
� �
δxk
dΩ . (4.38)
∂xk δbi
� �
∂ δxk
The integral that includes is integrated by parts and the gradient
∂xk δbi
of F is, finally, given by
�
� �
δF 1 ∂uk ∂T ∂ 1
= − τkm dΩ − 2
δbi Ω T 2 ∂xm ∂bi Ω ∂xm T τkm
�
∂uk
dΩ
∂bi
�
1
− 2
T τkm
�
∂um δxl 1
nk dS +
∂xl δbi T τkm
∂uk δxl
nldS (4.39)
∂xm δbi
where ∂T
∂bi
fk = f inv
and ∂uk
∂bi
Sw
Ω
Sw
are computed using Eq. (4.14) (direct approach), with
. The adjoint approach, through Eq. (4.38) and integrations
as follows
k −fvis k
by parts, gives the final expression of δF
δbi