Optimization and Computational Fluid Dynamics - Department of ...

5 Efficient Deterministic Approaches for Aerodynamic Shape Optimization
�
139
δI(X,w,Z)= δi(X,w,Z)dV
V
V
� �
∂i(X,w,Z)
=
δX +
∂X
∂i(X,w,Z)
δw +
∂w
∂i(X,w,Z)
�
δZ dV .(5.49)
∂Z
The derivation is identical to what has already been seen, and gives the
adjoint equations
� � �
∂i ∂R ∂S
+ ψT + φT dV =0, (5.50)
∂w ∂w ∂w
V
�
For the gradient we get
V
V
� �
∂i ∂R ∂S
+ ψT + φT dV =0. (5.51)
∂Z ∂Z ∂Z
δI(X,w,Z)=
� � �
∂i(X,w,Z) T ∂R(X,w,Z) T ∂S(X,w,Z)
δX + ψ δX + φ δX dV (5.52)
∂X
∂X
∂X
It can be shown that Eq. (5.50) is equivalent to the equation
� ��∂ψ�T ∂F
∂ξ ∂w +
� � �
T
∂ψ ∂G
dV = 0 (5.53)
∂η ∂w
V
and the boundary condition (in the case of the drag)
ψ2nx + ψ3ny + nx cos(α)+ny sin(α) − n T φ =0. (5.54)
Note that the structural adjoint variables appear only in the boundary condition
(5.54), while the adjoint flow equation (5.53) is unchanged. This implies
that in order to implement the coupling, only the boundary condition treatment
in the FLOWer code has to be modified. Equation (5.42) represents
the structural adjoint equation and its boundary conditions. The structural
equation reads in the case of linear elasticity
S(X,w,Z)=K · Z − a = 0 (5.55)
where K is the symmetric stiffness matrix and a is the aerodynamic force.
The derivative ∂S
∂Z in (5.42) can thus be replaced by K and the product φT K
by Kφ. In this way, the same solver can be used for the structural direct and
adjoint equation, with different boundary conditions, given by the first and
second term in Eq. (5.42). The first term is reduced to a surface integral by