Optimization and Computational Fluid Dynamics - Department of ...

Chapter 4
Adjoint Methods for Shape
Optimization
Kyriakos C. Giannakoglou and Dimitrios I. Papadimitriou
Abstract In aerodynamic shape optimization, gradient-based methods often
rely on the adjoint approach, which is capable of computing the objective
function sensitivities with respect to the design variables. In the literature
adjoint approaches are proved to outperform other relevant methods, such
as the direct sensitivity analysis, finite differences or the complex variable
approach. They appear in two different formulations, namely the continuous
and the discrete one, which are both discussed in this chapter.
In the first part, continuous and discrete approaches for the computation
of first derivatives are presented. The mathematical background for both approaches
is introduced. Based on it, adjoints for either inverse design problems
associated with inviscid or viscous flows or for the minimization of viscous
losses in internal aerodynamics are developed. The Navier-Stokes equations
are used as state equations. The elimination of field integrals expressed in
terms of variations in grid metrics leads to a formulation which is independent
of the grid type and can thus be employed with either structured or
unstructured grids. From the physical point of view, the minimization of viscous
losses in ducts or cascades is handled by minimizing either the difference
in total pressure between inlet and outlet (the objective function is, then, a
boundary integral) or the field integral of entropy generation. The discrete
adjoint approach is, practically, used to compare and cross-check the derivatives
computed by means of the continuous approach.
In the second part of this chapter, recent theoretical formulations on the
computation and use of the Hessian matrix in optimization problems are
presented. It is demonstrated that the combined use of the direct sensitivity
analysis for the first derivatives followed by the adjoint approach for second
derivatives may support the Newton method at the cost of N +2 equivalent
Kyriakos C. Giannakoglou · Dimitrios I. Papadimitriou
Parallel CFD & Optimization Unit, Lab. of Thermal Turbomachines, School of Mechanical
Engineering, National Technical University of Athens, Greece
(e-mail: kgianna@central.ntua.gr, e-mail: dpapadim@mail.ntua.gr)
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