Virtual Stackelberg game coupled with the adjoint method for aerodynamic shape optimization

We present a new continuous adjoint method for aerodynamic shape optimization using the Euler equations, which reduces the computational cost of the gradients by reducing the volume integral part of the adjoint gradient formula to a surface integral. The savings are particularly significant for three-dimensional aerodynamic shape optimization problems on general unstructured and overset meshes. In order to validate the concept, the new gradient equations have been tested for various aerodynamic shape optimization problems, including an inverse problem for threedimensional wing configurations, and drag minimization problems of a single-element airfoil and a three-dimensional wing-fuselage configuration. In order to assess their accuracy, the results are compared with finite-difference gradients, complex-step gradients, and gradients calculated by the previous adjoint method which includes a volume integral. ∗Thomas V. Jones Professor of Engineering, Department of Aeronautics and Astronautics, Stanford University, AIAA Fellow †Postdoctoral Fellow, Department of Aeronautics and Astronautics, Stanford University, AIAA Member

A new method for aerodynamic shape optimization using a genetic algorithm with real number encoding is presented. The algorithm is used to optimize three different problems, a simple hill climbing problem, a quasi-one-dimensional nozzle problem using an Euler equation solver and a three-dimensional transonic wing problem using a nonlinear potential solver. Results indicate that the genetic algorithm is easy to implement and extremely reliable, being relatively insensitive to design space noise.

A direct method for aerodynamic shape optimization based on the use of Bezier spline approximation is proposed. The method is tested as applied to the optimization of the supersonic part of an axisymmetric de Laval nozzle. The optimization results are compared with the exact solution obtained by the control contour method (variational nozzle) and with nozzles constructed using another direct method, namely, local linearization. It is shown that both direct optimization methods can be used on rather coarse grids without degrading the accuracy of the solution. The optimization procedure involves the isoperimetric condition that the surface area of the nozzle is given and fixed, which prevents the use of the control contour method. Optimization with allowance for viscosity is performed using the method. For fairly short maximum possible nozzle lengths in the range of Reynolds numbers under consideration, it is shown that allowance for viscosity does not improve the nozzle shape produced by optimization based on the Euler equations. The role of viscosity is reduced to the determination of an optimal length.

An efficient approach to optimization of real-life 3D aerodynamic shapes for minimum drag is presented. The method allows for substitution of fictitious surfaces for those parts of aircraft which are not subject to modification in the process of design. The approach essentially reduces the geometrical complexity of computational models thus making the optimization of complete aerodynamic configurations practically feasible and, eventually, increasing the accuracy and reliability of optimization. The optimization framework is that of OPTIMAS, the in-house IAI optimization tool based on Genetic Algorithms and full Navier-Stokes computations. The method is illustrated by example of wing shape optimization for the full configuration of generic business jet with close coupled wing-body-nacelle, and vertical/horisontal tails. The results indicate the applicability of the method to practical aerodynamic design.KeywordsAerodynamic PerformanceWing ShapeAerodynamic DesignAerodynamic ShapeMinimum DragThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper presents an aerodynamic shape optimization method based on a genetic algorithm with the compressible Euler or Navier-Stokes equations. A new algorithm to represent aerodynamic configurations is proposed to find optimum aerodynamic configurations efficiently. We solve three aerodynamic shape optimization problems. First, an inverse problem is solved to understand a basic character and to examine the possibility of the genetic algorithm. Second, a drag minimization problem is solved to find a streamlined body. Finally, L/D(ratio of lift to drag) maximization problem in a transonic flow is solved and the optimized configuration is compared with a supercritical airfoil. The results show that genetic algorithm is powerful and robust in solving aerodynamic shape optimization problems.

A robust and accurate method for aerodynamic shape optimization, previously developed by the authors, is extended to automatic multipoint design of aircraft configurations with under-the-wing-mounted nacelle. The shape is optimized for minimum drag at fixed lift subject to numerous geometrical and aerodynamic constraints. The method is driven by genetic algorithms and full Navier–Stokes computations combined with reduced-order-models approach. A recently developed optimization software based on the proposed method is applied to the design of a wing combined with fuselage and under-the-wing-mounted nacelle in a close coupling installation. A multipoint multiconstrained optimization carried out in a fully automatic mode achieved significant drag reduction within and beyond industry-realistic design conditions. The results indicate the ability of the developed automatic techniques to provide industrial-strength solutions that are not easily found through a traditional process of intelligent trial and error.

Research on POD-based CST airfoil parameterization method for aerodynamic shape optimization

Aerodynamic shape optimization of aircraft configurations often ignores stability considerations. To address this, a method for the computation of static, dynamic, and transient aircraft stability derivatives and their sensitivities for use in gradient-based optimization is introduced and evaluated. Computational fluid dynamics in the form of a three-dimensional structured-grid multiblock flow solver with both Euler and Reynolds-averaged Navier–Stokes equations is used. To compute the stability derivatives, a time-spectral formulation is used to compute an oscillating solution for the configuration of interest. From this oscillating solution, a series of linear regressions is performed to calculate the various stability derivatives. Because the solution is time dependent, it contains the information required to compute the transient, or “dot,” derivatives for the configuration. An adjoint method is used to compute the gradients of the stability derivatives of interest, enabling gradient-based stability-constrained aerodynamic shape optimization with respect to a large number of design variables. The computed stability derivatives are verified for an airfoil and validated for a generic unmanned combat aerial vehicle. The stability-constrained optimization of a wing demonstrates the viability and usefulness of the method for aircraft design optimization.

An investigation is presented that considers various aspects of an aerodynamic shape optimization framework. A two-dimensional aerofoil transonic zero-lift drag minimization test case is used to investigate the effect of dimensionality, shape deformation parameters, and optimizer on the results from the shape optimization process. A flexible control point-based parameterization is implemented which decouples the design variables from the surface, such that control point deformations determine the surface and volume mesh deformations in a unified manner. A gradient-based optimizer (feasible sequential quadratic programming) and global search algorithm (gravitational search algorithm) are tested on the constrained optimization case. The results show, as expected, that an increase in the number of dimensions produces a greater design space coverage and better optimization results, and the gradient-based method is prone to terminating in local optima or at constraint boundaries, so the global search algorithm is more reliable at locating optima. Efficient, reduced, and orthogonal shape deformation parameters are defined here by singular value decomposition extraction, and are shown to be particularly effective, demonstrating a 99.7% drag reduction for the case considered.

An aerodynamic shape optimization technique has been developed for helicopter rotor blades in hover based on a continuous adjoint method on unstructured meshes. For this purpose, the Euler flow solver and the adjoint sensitivity analysis were formulated on the rotating frame of reference. To handle the repeated evaluation of the flow solution and the sensitivity analysis efficiently, the flow and adjoint solvers were parallelized using a domain decomposition strategy. A solution-adaptive mesh refinement technique was adopted to better resolve the tip vortex. Applications were made to the aerodynamic shape optimization of the Caradonna-Tung rotor blade and the UH-60 rotor blade. The results showed that the present method is effective in determining optimum aerodynamic configurations of rotor blades in hover, which require minimum inviscid torque while maintaining the desired thrust level at transonic flight conditions.

This paper presents an approach for the rapid implementation of an adjoint solver for the ReynoldsAveraged Navier–Stokes equations with a Spalart–Allmaras turbulence model. Automatic differentiation is used to construct the partial derivatives required in the adjoint formulation. The resulting adjoint implementation is computationally efficient and highly accurate. The assembly of each partial derivative in the adjoint formulation is discussed. In addition, a coloring acceleration technique is presented to improve the adjoint efficiency. The RANS adjoint is verified with complex-step method using a flow over a bump case. The RANS-based aerodynamic shape optimization of an ONERA M6 wing is also presented to demonstrate the aerodynamic shape optimization capability. The drag coefficient is reduced by 19% when subject to a lift coefficient constraint. The results are compared with Euler-based aerodynamic shape optimization and previous work. Finally, the effects of the frozenturbulence assumption on the accuracy and computational cost are assessed.

Surrogate models are used to dramatically improve the design efficiency of numerical aerodynamic shape optimization, where high-fidelity, expensive computational fluid dynamics (CFD) is often employed. Traditionally, in adaptation, only one single sample point is chosen to update the surrogate model during each updating cycle, after the initial surrogate model is built. To enable the selection of multiple new samples at each updating cycle, a few parallel infilling strategies have been developed in recent years, in order to reduce the optimization wall clock time. In this article, an alternative parallel infilling strategy for surrogate-based constrained optimization is presented and demonstrated by the aerodynamic shape optimization of transonic wings. Different from existing methods in which multiple sample points are chosen by a single infill criterion, this article uses a combination of multiple infill criteria, with each criterion choosing a different sample point. Constrained drag minimizations of the ONERA-M6 and DLR-F4 wings are exercised to demonstrate the proposed method, including low-dimensional (6 design variables) and higher-dimensional problems (up to 48 design variables). The results show that, for surrogate-based optimization of transonic wings, the proposed method is more effective than the existing parallel infilling strategies, when the number of initial sample points are in the range from Nv to 8Nv (Nv here denotes the number of design variables). Each case is repeated 50 times to eliminate the effect of randomness in our results.

The performance of an aerodynamic shape optimization routine is greatly dependent on its geometry control system. This system must accurately parameterize the initial geometry and generate a flexible set of design variables for the optimization cycle. It must also generate new instances of the geometry based on the changes to the design variables dictated by the optimization routine. In response to changes in the geometry, it is also desirable to generate a new surface grid with the same topology as the original grid. This new surface grid can be used to perturb the associated volume grid. This paper presents two geometry control systems, a CAD-free system, and a CATIA V5 CAD-based system. The two systems provide practical tools for aerodynamic optimization. They also provide a basis for comparing CAD-free and CAD-based systems and understanding additional issues that need to be addressed in order to develop reliable optimization systems.

Aerodynamic shape optimization of large-span coal sheds for wind-induced effect mitigation using surrogate models

This work demonstrates the performance of Jetstream, a high-fidelity aerodynamic shape optimization methodology for three-dimensional turbulent flows. The geometry parameterization and mesh movement is accomplished using B-spline volumes and linear elasticity mesh movement. The Euler or Reynolds-averaged Navier-Stokes (RANS) equations are solved at each iteration using a parallel Newton-Krylov-Schur method. The equations are discretized in space using summation-by-parts operators with simultaneous approximation terms to enforce boundary and block interface conditions. The gradients are evaluated using the discrete-adjoint method to allow for gradient-based optimization using a sequential quadratic programming algorithm. The goal of this work is to investigate the performance of Jetstream for three test problems. The first problem is the drag minimization of a two-dimensional symmetric airfoil in transonic inviscid flow, under a geometric constraint that the airfoil have a thickness greater than or equal to that of a NACA 0012 airfoil. Although the shock waves are not quite eliminated, they are substantially weakened, such that the drag coefficient is reduced by 86% compared to the NACA 0012 airfoil. The second problem is drag minimization through optimizing the twist distribution of a three-dimensional wing characterized by NACA 0012 sections in subsonic inviscid flow, subject to a lift constraint. A nearly elliptical spanwise lift distribution is achieved by the optimized twist distribution, leading to a span efficiency factor of 0.98. The third problem is drag minimization through optimizing the sections and twist distribution of the blunt-trailing-edge Common Research Model wing in transonic turbulent flow, subject to lift and pitching moment constraints. For this case the optimization is performed based on the solution of the RANS equations, with the Spalart-Allmaras turbulence model fully coupled and linearized. The drag coefficient is reduced by eleven counts, or 6%, when analyzed on a fairly fine mesh.