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Adjoint Methods for Shape Optimization

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Optimization and Computational Fluid Dynamics

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 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.

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Authors and Affiliations

  1. Parallel CFD & Optimization Unit, Lab. of Thermal Turbomachines, School of Mechanical Engineering, National Technical University of Athens, Greece

    Kyriakos C. Giannakoglou & Dimitrios I. Papadimitriou

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  1. Kyriakos C. Giannakoglou
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  2. Dimitrios I. Papadimitriou
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  1. Institut für Strömungstechnik und Thermodynamik (ISUT), Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106, Magdeburg, Germany

    Dominique Thévenin  & Gábor Janiga  & 

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Giannakoglou, K.C., Papadimitriou, D.I. (2008). Adjoint Methods for Shape Optimization. In: Thévenin, D., Janiga, G. (eds) Optimization and Computational Fluid Dynamics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72153-6_4

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  • DOI: https://doi.org/10.1007/978-3-540-72153-6_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-72152-9

  • Online ISBN: 978-3-540-72153-6

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