Optimization and Computational Fluid Dynamics - Department of ...

80 Kyriakos C. Giannakoglou and Dimitrios I. Papadimitriou
flow solutions per optimization cycle. The computation of the exact Hessian
is demonstrated using both discrete and continuous approaches.
Test problems are solved using the proposed methods. They are used to
compare the so-computed first and second derivatives with those resulting
from the use of finite difference schemes. On the other hand, the efficiency of
the proposed methods is demonstrated by presenting and comparing convergence
plots for each test problem.
4.1 Introduction
Gradient-based methods, often used in aerodynamic optimization, must be
supported by a tool to compute the gradient of an objective function which
quantifies the efficiency of candidate solutions to the problem. Since this
text focuses on aerodynamic shape optimization only, candidate solutions
are considered to be 2D or 3D aerodynamic shapes (airfoils, blades, wings,
air inlets, etc.). The objective function is expressed in terms of flow variables
computed by numerically solving the flow equations in the corresponding
domains, with problem-specific boundary conditions. The adjoint approach,
based on control theory, is a means to compute the gradient required by a
gradient-based optimization method.
Generally, deterministic algorithms driven by the gradient of the objective
function converge to the global optimal solution much faster than evolutionary
algorithms, provided that the starting solution does not mislead them by
trapping the search around local optima. The reason for the superiority of the
adjoint method (despite the aforementioned risk) is that the gradient points
towards a better solution whereas the evolutionary algorithms are, generally,
unable to exploit local information during the solution refinement. On the
other hand, evolutionary algorithms have some other advantages [8, 20]. However,
the analysis of these algorithms and their performance is beyond the
scope of this chapter which is exclusively concerned with adjoint methods.
Aerodynamic problems usually involve a great number of design variables,
so the computation of gradients by means of finite difference schemes renders
deterministic algorithms very time-consuming. Compared to other methods,
such as the complex-variable approach [48], or the direct sensitivity analysis
(as it will become clear as this chapter develops, the direct approach is based
on the computation and use of the gradient of flow variables with respect to
the design variables as intermediate quantities), the adjoint approach is more
efficient to compute the gradient of the objective function. The total cost for
the gradient computation does not depend on the number of design variables
and is approximately equal to that of solving the flow equations.
Two adjoint approaches, namely the continuous and discrete, have been
developed. In continuous adjoint, the adjoint Partial Differential Equations
(PDE) are formed starting from the corresponding flow PDE’s and are then