In this paper, the exact discrete adjoint of an unstructured finite-volume formulation of the Euler equations in two dimensions is derived and implemented. The adjoint equations are solved with the same implicit scheme as used for the flow equations. The scheme is modified to efficiently account for multiple functionals simultaneously. An optimization framework, which couples an analytical shape parameterization to the flow/adjoint solver and to algorithms for constrained optimization, is tested on airfoil design cases involving transonic as well as supersonic flows. The effect of some approximations in the discrete adjoint, which aim at reducing the complexity of the implementation, is shown in terms of optimization results rather than only in terms of gradient accuracy. The shape-optimization method appears to be very efficient and robust.

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The discrete adjoint of a reconstruction-based unstructured finite volume formulation for the Euler equations is derived and implemented. The matrix-vector products required to solve the adjoint equations are computed on-thefly by means of an efficient two-pass assembly. The adjoint equations are solved with the same solution scheme adopted for the flow equations. The scheme is modified to efficiently account for the simultaneous solution of several adjoint equations. The implementation is demonstrated on wing and wing-body configurations.

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Purpose -An aerodynamic shape optimization algorithm is presented, which includes all aspects of the design process: parameterization, flow computation and optimization. The purpose of this paper is to show that the Class-Shape-Refinement-Transformation method in combination with an Euler/adjoint solver provides an efficient and intuitive way of optimizing aircraft shapes. Design/methodology/approach -The Class-Shape-Transformation method was used to parameterize the aircraft shape and the flow was computed using an in-house Euler code. An adjoint solver implemented into the Euler code was used to compute the required gradients and a trust-region reflective algorithm was employed to perform the actual optimization. Findings -The results of two aerodynamic shape optimization test cases are presented. Both cases used a blended-wing-body reference geometry as their initial input. It was shown that using a two-step approach, a considerable improvement of the lift-to-drag ratio in the order of 20-30 per cent could be achieved. The work presented in this paper proves that the CSRT method is a very intuitive and effective way of parameterizating aircraft shapes. It was also shown that using an adjoint algorithm provides the computational efficiency necessary to perform true three-dimensional shape optimization. Originality/value -The novelty of the algorithm lies in the use of the Class-ShapeRefinement-Transformation method for parameterization and its coupling to the Euler and adjoint codes.