Aerodynamic Shape Optimization Problems Research Articles

Continuous and discrete adjoint approaches for aerodynamic shape optimization with low Mach number preconditioning

AbstractDiscrete and continuous adjoint approaches for use in aerodynamic shape optimization problems at all flow speeds are developed and assessed. They are based on the Navier–Stokes equations with low Mach number preconditioning. By alleviating the large disparity between acoustic waves and fluid speeds, the preconditioned flow and adjoint equations are numerically solved with affordable CPU cost, even at the so‐called incompressible flow conditions. Either by employing the adjoint to the preconditioned flow equations or by preconditioning the adjoint to the ‘standard’ flow equations (under certain conditions the two formulations become equivalent, as proved in this paper), efficient optimization methods with reasonable cost per optimization cycle, even at very low Mach numbers, are derived. During the mathematical development, a couple of assumptions are made which are proved to be harmless to the accuracy in the computed gradients and the effectiveness of the optimization method. The proposed approaches are validated in inviscid and viscous flows in external aerodynamics and turbomachinery flows at various Mach numbers. Copyright © 2007 John Wiley & Sons, Ltd.

Discrete Adjoint-Based Approach for Optimization Problems on Three-Dimensional Unstructured Meshes

A comprehensive strategy for developing and implementing discrete adjoint methods for aerodynamic shape optimization problems is presented. By linearizing each procedure in the entire optimization problem, transposing each linearization, and reversing the sequential order of operations, the adjoint of the complete optimization problem, including flow equations and mesh motion equations is constructed in a modular and verifiable fashion. This construction is also shown to produce minimal memory overheads, and retain the same convergence characteristics of the original analysis problem in the sensitivity analysis. These techniques are implemented in a three-dimensional unstructured multigrid NavierStokes solver, and demonstrated on a transonic drag reduction problem for a wing body configuration.

On the use of metamodel-assisted, multi-objective evolutionary algorithms

This article is concerned with the optimal use of metamodels in the context of multi-objective evolutionary algorithms which are based on computationally expensive function evaluations. The goal is to capture Pareto fronts of optimal solutions with the minimum computational cost. In each generation during the evolution, the metamodels act as filters that distinguish the most promising individuals, which will solely undergo exact and costly evaluations. By means of the so-called inexact pre-evaluation phase, based on continuously updated local metamodels, most of the non-promising individuals are put aside without aggravating the overall cost. The gain achieved through this technique is amazing in single-objective problems. However, with more than one objective, noticeable performance degradation occurs. This article scrutinizes the role of metamodels in multi-objective evolutionary algorithms and proposes ways to overcome expected weaknesses and improve their performance. Minimization of mathematical functions as well as aerodynamic shape optimization problems are used for demonstration purposes.

Efficient Hierarchical Parallel Genetic Algorithms using Grid computing

In this paper, we present an efficient Hierarchical Parallel Genetic Algorithm framework using Grid computing (GE-HPGA). The framework is developed using standard Grid technologies, and has two distinctive features: (1) an extended GridRPC API to conceal the high complexity of the Grid environment, and (2) a metascheduler for seamless resource discovery and selection. To assess the practicality of the framework, a theoretical analysis of the possible speed-up offered is presented. An empirical study on GE-HPGA using a benchmark problem and a realistic aerodynamic airfoil shape optimization problem for diverse Grid environments having different communication protocols, cluster sizes, processing nodes, at geographically disparate locations also indicates that the proposed GE-HPGA using Grid computing offers a credible framework for providing a significant speed-up to evolutionary design optimization in science and engineering.

Evolutionary optimization using a new radial basis function network and the adjoint formulation

This article aims at extending previously published ideas on the formulation and use of low-cost surrogate evaluation tools in the context of optimization methods based on evolutionary algorithms (EAs). Our goal is to minimize the cost of solving optimization problems with computationally expensive evaluations. A search algorithm is proposed which brings together computational fluid dynamics tools, namely flow and adjoint equation solvers, new radial basis function networks (RBFNs) and standard EAs. The new RBFNs involve additional control parameters which allow their training on patterns for which both responses and their gradients are available. In aerodynamic shape optimization problems, the gradient can be computed through the adjoint method. Despite the known role of adjoint methods, i.e. that of computing local search directions, in the proposed method they are used to enrich the available information for the training of the surrogate evaluation models, through providing the objective function gradient for each and every pattern. Based on a number of preselected samples, with known responses and gradients, the proposed RBFN is trained and used as the exclusive evaluation tool during the evolutionary search. A small number of cycles is required so as to capture the global optimal solution. A cycle includes the exact evaluation of the outcome of the evolutionary search, the RBFN update after retraining it on the enriched database, and a new search based on the updated RBFN. The method application is demonstrated through single- and multi-objective mathematical problems as well as the inverse design of a peripheral compressor cascade.

Aerodynamic shape design using evolutionary algorithms and new gradient-assisted metamodels

Aerodynamic shape design and optimization problems based on evolutionary algorithms and surrogate evaluation tools, i.e., the so-called metamodels, have recently found widespread use. Using metamodels, trained either separately from or during the optimization loop, a considerable reduction in the overall computing cost can be achieved. To support metamodel-based evolutionary algorithms, a class of new metamodels which utilize both known responses and response gradients for their training is proposed. The new gradient-assisted metamodels are extensions of standard multi-layer perceptrons and radial basis function networks. To demonstrate the prediction capabilities of the proposed metamodels and investigate different implementation modes within search algorithms along with the relevant CPU cost, a number of 2D and 3D aerodynamic shape (namely airfoils and turbomachinery blades) design problems are analyzed. Single- and two-objective problems, aiming at designing shapes that reproduce known pressure distributions at specific operating points, are considered. The exact evaluation tool is a numerical solver of the compressible fluid flow equations. The necessary gradient of the objective function is obtained by formulating and numerically solving adjoint equations.

Simultaneous Pseudo-Timestepping for Aerodynamic Shape Optimization Problems with State Constraints

This paper presents a numerical method for constrained aerodynamic shape optimization problems. It is based on simultaneous pseudo-timestepping in which stationary states are obtained by solving the pseudo-stationary system of equations representing the state, costate, and design equations. The main advantages of this method are that it blends in nicely with a previously existing pseudo-timestepping method for the states only, and that a preconditioner can be used for convergence acceleration which stems from the reduced SQP methods. Design examples for drag reduction with constant lift and drag reduction with constant pitching moment for an RAE2822 airfoil are included. The overall cost of computation is 4--8 times that of the forward simulation runs depending on the grid size and the number of design parameters.

Simultaneous pseudo‐time stepping for aerodynamic shape optimization

AbstractIn this paper we discuss about numerical methods for aerodynamic shape optimization problems. These problems require efficient CFD techniques to solve the state (as well as costate) equations and fast algorithms for solving the optimization problems. Both of these are independent active areas of research since long time. Wide range of applications in science and engineering involve solution of optimization problems where the governing PDEs appear as constraints. Therefore, merging the two for the purpose of practical applicability is relatively new. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Reduction of the Adjoint Gradient Formula for Aerodynamic Shape Optimization Problems

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. To validate the concept, the new gradient equations have been tested for various aerodynamic shape-optimization problems, including an inverse problem for three-dimensional wing configurations, and drag-minimization problems of a single-element airfoil and a three-dimensional wing-fuselage configuration

Reduction of the Adjoint Gradient Formula in the Continuous Limit

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

Hierarchical Evolutionary Algorithms and Its Application in Transonic Airfoil Optimization in Aerodynamics

AbstractHierarchical evolutionary algorithms based on genetic algorithms (GAs) and Nash strategy of game theory are proposed to accelerate the optimization process and implemented in transonic aerodynamic shape optimization problems. Inspired from the natural evolution history that different periods with certain environments have different criteria for the evaluations of individuals' fitness, a hierarchical fidelity model is introduced to reach high optimization efficiency. The shape of an NACA0012 based airfoil is optimized in maximizing the lift coefficient under a given transonic flow condition. Optimized results are presented and compared with the single model results and traditional GA.

Newton-Krylov Algorithm for Aerodynamic Design Using the Navier-Stokes Equations

A Newton‐Krylov algorithm is presented for two-dimensional Navier‐Stokes aerodynamic shape optimization problems. The algorithm is applied to both the discrete-adjoint and the discrete e ow-sensitivity methods for calculating the gradient of the objective function. The adjoint and e ow-sensitivity equations are solved using a novel preconditioned generalized minimum residual (GMRES)strategy. Together with a complete linearization of the discretized Navier‐Stokes and turbulence model equations, this results in an accurate and efecient evaluation of the gradient. Furthermore, fast e ow solutions are obtained using the same preconditioned GMRES strategy in conjunction with an inexact Newton approach. The performance of the new algorithm is demonstrated for several design examples,includinginversedesign,lift-constraineddragminimization, liftenhancement, and maximization of lift-to-dragratio. In all examples, the normof the gradientisreduced by several ordersof magnitude, indicating that alocalminimumhasbeen obtained. Bytheuseoftheadjoint method,thegradient isobtained infromone-e fth to one-half of the time required to converge a eow solution.

Real-Coded Adaptive Range Genetic Algorithm and Its Application to Aerodynamic Design.

Real-coded Adaptive Range Genetic Algorithms (ARGAs) have been developed. The real-coded ARGAs possess both advantages of the binary-coded ARGAs and the use of the floating point representation to overcome the problems of having a large search space that requires continuous sampling. First, the efficiency and the robustness of the proposed approach are demonstrated by test functions. Then the proposed approach is applied to an aerodynamic airfoil shape optimization problem. The results confirm that the real-coded ARGAs consistently find better solutions than the conventional real-coded Genetic Algorithms do. The designed airfoil shape is considered to be the global optimal and thus ensures the feasibility of the real-coded ARGAs in aerodynamic designs.

Analysis of the Hessian for aerodynamic optimization: inviscid flow

In this paper, we analyze inviscid aerodynamic shape optimization problems governed by the full potential and the Euler equations in two and three dimensions. The analysis indicates that minimization of pressure-dependent cost functions results in Hessians whose eigenvalue distributions are identical for the full potential and the Euler equations. However, the optimization problems in two and three dimensions are inherently different. While the two-dimensional optimization problems are well posed, the three dimensional ones are ill-posed. Oscillations in the shape up to the smallest scale allowed by the design space can develop in the direction perpendicular to the flow, implying that a regularization is required. The analysis also gives an estimate of the Hessian’s condition number which implies that the problems at hand are ill-conditioned. Infinite dimensional approximations for the Hessians are constructed and preconditioners for gradient-based methods are derived from these approximate Hessians. Numerical results are given for the small disturbance potential equation in two and in three space dimensions.

A Preconditioning Method for Shape Optimization Governed by the Euler Equations

We consider a classical aerodynamic shape optimization problem subject to the compressible Euler flow equations. The gradient of the cost functional with respect to the shape variables is derived with the adjoint method at the continuous level. The Hessian (second order derivative of Ihe cost functional with respect to the shape variables) is approximated also at the continuous level, as first introduced by Arian and Ta'asan [1], The approximation of the Hessian is used to approximate the Newton step which is essential to accelerate the numerical solution of the optimization problem. The design space is discretized in the maximum dimension, i.e., the location of each point on the intersection of the computational mesh with the airfoil is taken to be an independent design variable. We give numerical examples for 86 design variables in two different flow speeds and achieve an order of magnitude reduction in the cost functional at a computational effort of a full solution of the analysis partial differential equation (PDE).