Advances in Aerodynamic Shape Optimization

Abstract

The focus of CFD applications has shifted to aerodynamic design. This shift has been mainly motivated by the availability of high performance computing platforms and by the development of new and efficient analysis and design algorithms. In particular automatic design procedures, which use CFD combined with gradient-based optimization techniques, have had a significant impact on the design process by removing difficulties in the decision making process faced by the aerodynamicist.A fast way of calculating the accurate gradient information is essential since the gradient calculation can be the most time consuming portion of the design algorithm. The computational cost of gradient calculation can be dramatically reduced by the control theory approach since the computational expense incurred in the calculation of the complete gradient is effectively independent of the number of design variables. The foundation of control theory for systems governed by partial differential equations was laid by J.L. Lions [1]. The method was first used for aerodynamic design by Jameson in 1988 [2,3]. Since then, the method has even been successfully used for the aerodynamic design of complete aircraft configurations [4].In the present work a continuous adjoint formulation has been used to derive the adjoint system of equations, in which the adjoint equations are derived directly from the governing equations and then discretized. This approach has the advantage over the discrete adjoint formulation in that the resulting adjoint equations are independent of the form of discretized flow equations. The adjoint system of equations has a similar form to the governing equations of the flow, and hence the numerical methods developed for the flow equations [5,6,7] can be reused for the adjoint equations. Moreover, the gradient can be derived directly from the adjoint solution and the surface motion, independent of the mesh modification.In order to accelerate the convergence of the descent process the gradient is then smoothed implicitly via a second order differential equation. This is

Impact of Shape Parameterisation on Aerodynamic Optimisation of Benchmark Problem

“…The design variable gradients are calculated using the adjoint method which allows all the gradients to be calculated for a computational cost in the order of one flow solve. This is done by solving the adjoint equations [37] to calculate the sensitivity of drag coefficient with respect to the unit normal at each surface mesh point, i.e.…”

Smooth and robust solutions for Dirichlet boundary control of fluid–solid conjugate heat transfer problems

“…In the literature of aerodynamic shape optimization [40][41][42], the Jameson-type smoothing, which is based on introducing a specific weighted Sobolev norm, has been tested extensively. It directly applies a similar smoothing operator to the Steepest Descent direction without any modification to the objective function.…”

“…Numerical smoothing [40][41][42] has to be involved in the gradient computation to enhance the regularity of computed control profiles. (See Section 4.2.)…”

Geometric Comparison of Aerofoil Shape Parameterization Methods

“…The size and complexity of the resulting optimisation problem can however cause significant difficulties. For example, as all the point displacements are considered independently, the resulting sensitivities are often not smooth, which can present difficulties for flow solvers if not appropriately handled 17 . The large number of design variables involved can also lead to slow convergence rates and extremely expensive finite-difference gradient calculations.…”