Keywords: adjoint methods, shape optimization, mesh morphing, filtered gradients, radial basis function, numerical aerodynamics, high-speed trains.

The operating conditions of modern high-speed trains require efficient aerodynamic shape optimization methods. In practice, industrial shape optimization often focuses on already pre-optimized geometries. At the same time, many of the interesting aerodynamic features can be investigated by means of computational fluid dynamics. Up to now, different numerical strategies were presented to determine the required shape modifications of an object in order to optimize its aerodynamic performance with regard to a certain objective function. Following Othmer's work [1] on the continuous adjoint formulation for the computation of sensitivities of steady, incompressible, ducted flows, the continuous approach for the optimization of steady, incompressible, external flows and the open source finite volume solver OpenFOAM [2] are used, in this paper, to solve the primal and the adjoint equations. The latter are derived from the steady, incompressible Reynolds-averaged Navier-Stokes equations (RANS).

Adjoint-based shape optimization is a gradient-based approach which provides shape modifications evaluated by means of sensitivity analysis. Gradient-based shape optimization methods are iterative procedures. To avoid re-meshing for each iteration, CAD-free mesh morphing using radial basis function interpolation suggested by de Boer et al. [3] and Bos [4] was employed to modify the initial computational grid according to the evaluated surface displacements for the next iteration. Prior to morphing, the sensitivities are smoothed using a Gaussian filter to improve the quality of the obtained new surface, see Stück et al. [5,6].

The objective of the study presented in this paper is to determine how far a shape optimization procedure using adjoint-based methods with the continuous approach in conjunction with filtered gradients and CAD-free mesh morphing based on radial basis function interpolation is capable of meeting the requirements of modern train development. Initial applications of the developed process chain consider shape optimization with regard to the generated pressure wave. The described approach was first tested for the flow around a sphere. It was shown, that the tools and the developed process chain consisting of numerical simulation, sensitivity analysis and mesh morphing work well. Actually, after one iteration no significant improvement of the objective function could be observed. Thus, more iterations will be required to improve the aerodynamics of the object with regard to the considered objective function. The presented optimization procedure is being applied on a train head car.

1

C. Othmer, "A Continuous Adjoint Formulation for the Computation of Topological and Surface Sensitivities of Ducted Flows", International Journal for Numerical Methods in Fluids, 58, 861-877, 2007.

2

Open FOAM: The Open Source CFD Toolbox, 2011. URL

3

A. de Boer, M.S. van der Schoot, H. Bijl, "Mesh Deformation based on Radial Basis Function Interpolation", Computers & Structures, 85, 784-795, 2007.

4

F.M. Bos, "Numerical Simulations of Flapping Foil and Wing Aerodynamics - Mesh Deformation using Radial Basis Functions", Dissertation, Technische Universiteit Delft, Ipskamp Drukkers B.V., 2009.

5

A. Stueck, Th. Rung, "Filtered Gradients for Adjoint-based Shape Optimization", AIAA-2011-3072, 20th AIAA Computational Fluid Dynamics Conference, Honolulu, Hawaii, June 27-30, 2011.

6

A. Stueck, Th. Rung, "Adjoint RANS with Filtered Shape Derivatives for Hydrodynamic Optimization", Computers & Fluids, 47(1), 22-32, 2011.

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