Giampietro Carpentieri

Adjoint-based aerodynamic shape optimization on unstructured meshes

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.

An Investigation over CFD-Based Models for the Identification of Nonlinear Unsteady Aerodynamics Responses

An investigation is performed to assess the importance of the nonlinear effects on the dynamic behavior of a profile in a transonic flow. The analysis is done by applying a system identification approach to a computational fluid dynamics (CFD) code, considered as a black-box system. The CFD code is based on the Euler equations. Both frequency and time domain approaches are used to evaluate the nonlinear effects on the response of different airfoils. The results show that for weak shocks the aerodynamic operator describing the dynamics of the profile around a steady (transonic) flow condition is linear. For strong shocks results obtained with linear models appear to be conservative.

Development of the Discrete Adjoint for a Three-Dimensional Unstructured Euler Solver

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-the-fly 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.

Improving the Efficiency of Aerodynamic Shape Optimization on Unstructured Meshes

In this paper the exact discrete adjoint of a finite volume formulation on unstructured meshes for the Euler equations in two dimensions is derived and implemented to support aerodynamic shape optimization. The accuracy of the discrete exact adjoint is demonstrated and compared with that of the approximate adjoint. A solution process for the adjoint equations, which is similar to that used for the flow equations, is modified to account for multiple functionals. An optimization framework, which couples an analytical shape parameterization to the flow/adjoint solver and to a Sequential Quadratic Programming optimization algorithm, is tested on constrained and unconstrained airfoil design cases. Preliminary results are also presented for a Sequential Linear Programming algorithm, which appears to be able to deal properly with constrained design in spite of its simplicity. Aerodynamic shape optimization can be efficiently performed by means of the adjoint method which enables functional gradients to be calculated at the price of roughly one additional flow computation. 1 This method is very attractive in its discrete approach where the adjoint problem is directly formulated on the discretized flow equations. The method is straightforward to understand; only some linear algebra is involved. However, the derivation of the discrete adjoint code can be challenging due to the complexity of differentiating the original discrete formulation. Hand–coding the adjoint may require a lot of human work. 2 Automatic Differentiation tools to derive the code 3–5 may be successfully applied. In this work the adjoint algorithm is hand–coded following a methodology outlined in previous work, 6,7 in the context of implicit solver development. The discrete adjoint implemented here is exact since the residual Jacobian is obtained from the exact differentiation of the residual vector. It gives gradients that are consistent with the ones obtained by finite difference applied to the original solver. In practice, to simplify the derivation of the adjoint code, different approximations can be made in the differentiation. In turn, those approximations can have a detrimental effect on the accuracy. 2 Some of those approximations are discussed here. The solution process for the adjoint problem, despite the linearity of the equations, can be the same time marching adopted in the flow solver. 2,4,8,9 This greatly simplifies the coding and gives a robust adjoint solver. Here an implicit time stepping is used which is modified to account for the solution of the adjoint of multiple functionals. In constrained shape optimization more than one functional is usually involved so that it is convenient to solve the adjoint problem simultaneously rather than sequentially. The flow and the adjoint solver are part of an optimization framework, which includes the shape parameterization and an optimization algorithm. The latter is in charge of driving the optimization process. The shape parameterization used here is based upon orthogonal Chebyshev polynomials. 10 This parameterization gives completeness in the design space and behaves smoothly during the optimization process even in

Aerodynamic shape optimization using the adjoint Euler equations

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 ...

Comparison of Exact and Approximate Discrete Adjoint for Aerodynamic Shape Optimization

The effect of approximations in the discrete adjoint on constrained shape optimization is investigated. The different approximations are compared with the exact discrete adjoint, which is capable of producing exact gradient/sensitivity information. The purpose of such a comparison is to understand whether or not approximate adjoint codes can be effective for shape optimization in spite of the error in the computed gradient.