Optimization and Computational Fluid Dynamics - Department of ...

4 Adjoint Methods for Shape Optimization 81
discretized and numerically solved to compute the adjoint variables’ field. In
discrete adjoint, the equations are obtained directly from the discretized flow
PDE’s. Concerning their requirements at the development stage, both approaches
have their advantages and disadvantages and the interested reader
may refer to [55] for more details. However, according to [43, 44] and the personal
experience of the authors, both approaches result in sensitivity derivatives
of the same accuracy.
In fluid mechanics, the adjoint method has been introduced by Pironneau,
[57] and applied to flows governed by elliptic PDE’s. Jameson was the first
to present the continuous adjoint formulation for the optimal design of aerodynamic
shapes [28, 29, 33] in transonic flows. These papers deal with the
inverse design of airfoils and wings in inviscid flows, while the extension to
viscous flows is found in [32] where the Navier-Stokes equations are used as
state equations.
Since 1988, Jameson has presented many publications on discrete and preferentially
on continuous adjoint approaches. Among them, we report on the
inverse design of wings in subsonic, transonic and supersonic flows using
multiblock structured grids [30, 59], minimization of drag and/or maximization
of lift in inviscid [35] and viscous flows [36], sonic boom reduction for
supersonic flows [1, 46], shock wave reduction in external aerodynamics [25],
unsteady aerodynamic design of isolated airfoils and wings [45, 47], aerodynamic
design of full aircraft configurations [34], multipoint drag minimization
[39] and multidisciplinary optimization [38, 40].
On the other hand, Giles made a considerable improvement in the development
of the discrete adjoint approach. The properties of solutions to the
adjoint equations are analyzed in [23, 24]. He also presented the analytical
solutions of the adjoint equations using Green’s functions [56], an exact approach
for the solution of the adjoint equations [22], a shock wave reduction
method in external aerodynamics [21] and the harmonic approach to turbomachinery
steady and unsteady designs [11, 14].
The discrete adjoint approach using unstructured grids was first presented
by Peraire in inviscid, [15] and viscous flows [16], for 2D and 3D [17], configurations.
Peraire also applied the adjoint approach to multipoint optimization
problems [18].
The continuous adjoint approach for unstructured grids has been developed
by Anderson [4, 5] for inviscid and viscous flows. The discrete adjoint
formulation for turbulent flows with the Spalart-Allmaras turbulence model
can be found in [2, 3]. Improvements of the discrete approach concerning the
handling of grid sensitivities can be found in [49, 50] where remeshing and
mesh movement strategies used at each optimization cycle are discussed.
The convergence of the iterative optimization algorithm, which in each cycle
solves the flow and adjoint equations followed by the updating of the design
variables, can be significantly improved by the so-called one-shot method
[61, 37]. The three systems of equations are solved simultaneously in one shot,
and the convergence is further accelerated using multigrid. Improvements of