Optimization and Computational Fluid Dynamics - Department of ...

82 Kyriakos C. Giannakoglou and Dimitrios I. Papadimitriou
the one-shot method can be found in [27, 26] where the three systems of equations
are solved with the same time-step. Another variation of the one-shot
method can be found in [12, 13].
A different approach, which under certain circumstances may lower the
total computational cost is the incomplete gradient method [42, 60]. Less
important terms in the gradient expression are omitted, reducing thus the
accuracy of sensitivity derivatives while increasing the overall efficiency.
The continuous adjoint approach may become unable to deal with functionals
not expressed in terms of pressure (inadmissible functionals). To handle
inadmissible functionals, the adjoint formulation needs to be carefully
modified [6, 9].
An adjoint approach which avoids the computation of sensitivities of grid
metrics over the flow field has been presented for inviscid [31] and viscous [53]
flows. According to this approach, the gradient depends on flow and adjoint
variables as well as grid sensitivities over the boundary only, irrespective of
whether the objective function is a boundary or field integral. The lack of internal
grid sensitivities in the gradient expression reduces the computational
cost and increases accuracy by avoiding repetitive grid remeshings at each
optimization cycle. Approaches that skip the computation of grid sensitivities
in the discrete adjoint approach are presented in [41, 51].
The computation of the exact Hessian matrix using adjoint approaches
and its use in shape optimization is rare in the literature. In aerodynamic
design, the Hessian matrix in the neighborhood of the optimal solution is
analyzed in [7], while a method to compute the Hessian matrix can be found
in [62], although the structural optimization is of concern.
This chapter presents a detailed framework for the development of discrete
and especially continuous adjoint methods developed by the authors. The development
covers inverse design and optimization problems (minimization of
viscous losses, expressed as either entropy generation or total pressure losses)
in internal aerodynamics. Applications in external aerodynamics (where the
method can be extended to other objectives such as drag minimization constrained
by the lift, etc.) have been worked out using the present approach but
are not included in this chapter in the interest of space. The Navier-Stokes
equations with the Spalart-Allmaras turbulence model are used as state equations.
The demonstration is restricted to internal aerodynamics (design of a
cascade airfoil and a duct). In addition to theory and examples on the computation
and use of gradients (in steepest descent, quasi-Newton methods like
BFGS, etc.), recent achievements on the formulation of adjoint-based exact
Hessian methods in aerodynamic optimization are presented.
The principles of the adjoint approaches, discrete and continuous, are first
presented. Since the discrete adjoint is based on the already discretized flow
equations, any further discussion would be useful only for a specific discretization
scheme. On the other hand, the continuous adjoint formulation is presented
in complete detail. Emphasis is laid on the formulation which leads to
an expression of the objective function gradient which is free of field integrals.