Optimization and Computational Fluid Dynamics - Department of ...

86 Kyriakos C. Giannakoglou and Dimitrios I. Papadimitriou
joint equations are schematically shown below.
DISCRETE CONTINUOUS
ADJOINT ADJOINT
Differential State Equations
Discretization
❄
Discrete State
Equations
Adjoint Operation
❄
Discrete Adjoint Equations
Adjoint Operation
❄
Differential Adjoint
Equations
Discretization
❄
Theoretically, the value of F can be estimated with the same accuracy
irrespective of the approach used [43, 44]. However, each approach has its
own advantages and disadvantages which influence users’ preferences. In the
discrete approach, the derivation of the discrete adjoint equations is more
cumbersome, unless automatic differentiation is used [58]. Once the discrete
adjoint equations have been found, they can be solved using the same solvers
as those used to solve the state equations. Using the continuous approach,
one might use different discretization schemes. This might become important,
since the adjoint equations can be derived and solved without having access
to the flow solver source code. However, in practice, it has been proven that
the accuracy of the continuous adjoint approach depends on the discretization
scheme which must be as close as possible to that used for the discretization
of the state equations [4, 5]. Note that differences in the gradient values,
computed by the two approaches, may also appear if the computational grid
is not adequate enough.
4.3 Inverse Design Using the Euler Equations
The continuous adjoint approach for the inverse design of aerodynamic shapes
in inviscid flow is now presented. The Euler equations are used as state equations
and these are written in vector form as
∂U
∂t
∂finv
k +
∂xk
= 0 (4.10)