Maximum drag profiles located in a flow the adjoint method based on the first variation with a material derivative and a shape derivative

Authors:
Kazunori Shinohara
Affiliations:
Japan Aerospace Exploration Agency JAXA, JAXA's Engineering Digital Innovation Center JEDI, 3-1-1 Yoshinodai, Chuuou, Sagamihara Kanagawa 252 5210, Japan. E-mail: shinohara@06.alumni.u-tokyo.ac.jp
Venue:
Journal of Computational Methods in Sciences and Engineering
Year:
2013

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Abstract

To obtain the maximum drag profiles for the surface of an object located in unsteady flow, a shape optimization algorithm based on an adjoint method is presented. The adjoint method is based on the Lagrange multiplier method. Using a first variation of the Lagrange function, stationary conditions can be derived. These conditions consist of state equations, adjoint equations, and sensitivity equations with boundary conditions. The sensitivity equations are derived based on the shape derivative and the material derivative. To achieve the optimal shape based on these stationary conditions, a smoothing technique, a constant volume technique, a node relocation technique, the SUPG/PSPG stabilized method, and the GPBi-CG solver are implemented in the shape optimization algorithm. Using this algorithm, under Stokes flow, we can obtain the Pironneau's result found in the literature where the optimal shape is of the rugby ball type. Under unsteady flow i.e. for a Reynolds number of 1000, this algorithm can also construct an optimal shape. Compared an initial cylindrical shape, the drag of the optimal shape under constant volume can be increased by about 218% in the case of a Reynolds number of 1000.