Transonic Airfoil Design Research Articles

Multi-fidelity design optimization of transonic airfoils using shape-preserving response prediction

A computationally efficient methodology for transonic airfoil design optimization is presented. Our approach exploits a corrected physics-based low-fidelity surrogate that replaces, in the optimization process, an accurate but computationally expensive highfidelity airfoil model. Correction of the low-fidelity model is achieved by aligning its corresponding airfoil surface pressure distributions with that of the high-fidelity model using a shape-preserving response prediction technique. The presented method is applied to airfoil lift maximization in two-dimensional inviscid transonic flow, subject to constraints on shock induced pressure drag and airfoil cross-sectional area. More than a 90% reduction in high-fidelity function calls is achieved when compared to direct highfidelity model optimization.

Inverse aerodynamic design applications using the MGM hybrid formulation

The well-known modified Garabedian–Mcfadden (MGM) method is an attractive alternative for aerodynamic inverse design, for its simplicity and effectiveness (P. Garabedian and G. Mcfadden, Design of supercritical swept wings, AIAA J. 20(3) (1982), 289–291; J.B. Malone, J. Vadyak, and L.N. Sankar, Inverse aerodynamic design method for aircraft components, J. Aircraft 24(2) (1987), 8–9; Santos, A hybrid optimization method for aerodynamic design of lifting surfaces, PhD Thesis, Georgia Institute of Technology, 1993). Owing to these characteristics, the method has been the subject of several authors over the years (G.S. Dulikravich and D.P. Baker, Aerodynamic shape inverse design using a Fourier series method, in AIAA paper 99-0185, AIAA Aerospace Sciences Meeting, Reno, NV, January 1999; D.H. Silva and L.N. Sankar, An inverse method for the design of transonic wings, in 1992 Aerospace Design Conference, No. 92-1025 in proceedings, AIAA, Irvine, CA, February 1992, 1–11; W. Bartelheimer, An Improved Integral Equation Method for the Design of Transonic Airfoils and Wings, AIAA Inc., 1995). More recently, a hybrid formulation and a multi-point algorithm were developed on the basis of the original MGM. This article discusses applications of those latest developments for airfoil and wing design. The test cases focus on wing-body aerodynamic interference and shock wave removal applications. The DLR-F6 geometry is picked as the baseline for the analysis.

Inverse design of transonic airfoils using genetic algorithm and a new parametric shape method

The inverse design of transonic airfoils is carried out using genetic algorithm and different shape parameterization methods. A new technique for airfoil shape parameterization is presented considering the features of the transonic flow. The method is then applied to several airfoil inverse design problems with defined geometry or surface pressure distributions. The influence of parametric shape methods on the accuracy and computational cost of the design process are investigated for inviscid and viscous flow problems. The fitness function evaluation is carried out using an unstructured grid finite volume flow solver. It is shown that the new shape parameterization method is able to reduce the total computational time by about 50% in the inverse design of transonic airfoils, particularly when viscous flow calculations are concerned.

Iterative Response Surface Based Optimization Scheme for Transonic Airfoil Design

An improved response surface based optimization technique is presented for two-dimensional airfoil design at transonic speed. The method is based on an iterative scheme where least-square fitted quadratic polynomials of objective function and constraints are repeatedly corrected locally, about the current minimum, by adding the actual function value to the data set used to construct the polynomials. When no further cost function reduction is achieved, the design domain upon which the optimization is initially performed is changed, preserving its initial size, by updating the center point with the position of the last minimum found. The optimization is then conducted by using the same approximations built over the initial design space, which are again iteratively corrected until convergence to a given tolerance. To construct the response surfaces, the design space is explored by using a uniform Latin hypercube, aiming at reducing the bias error, in contrast with previous techniques based on D-optimality criterion. The geometry is modeled by using the PARSEC parameterization

A Kriging-based probabilistic optimization method with an adaptive search region

An adaptive search region method for design optimization is proposed. The validity of the current search region is checked by investigating the probabilistic distribution of the design variable, and if the search region is inadequate, it is changed adaptively. To ease the computational burden, evaluation of the objective function, in both probabilistic analysis of the distribution of design variables and optimization, is performed using a Kriging model. The present method was validated by application to transonic airfoil design. Even when starting with an inadequate initial search region, it was possible to obtain an airfoil with good aerodynamic performance using the present method. Functional analysis of variance (ANOVA) was performed to identify the effects of each design variable on the objective function and constraints. The results indicated that the relevant definition of the search region is essential to obtain correct information from ANOVA.

Multi-Objective Optimization Using Kriging Model and Data Mining

In this study, a surrogate model is applied to multi-objective aerodynamic optimization design. For the balanced exploration and exploitation, each objective function is converted into the Expected Improvement (EI) and this value is used as fitness value in the multi-objective optimization instead of the objective function itself. Among the non-dominated solutions about EIs, additional sample points for the update of the Kriging model are selected. The present method was applied to a transonic airfoil design. Design results showed the validity of the present method. In order to obtain the information about design space, two data mining techniques are applied to design results: Analysis of Variance (ANOVA) and the Self-Organizing Map (SOM).

Data Mining for Aerodynamic Design Space

Analysis of variance (ANOVA) and self-organizing map (SOM) were applied to data mining for aerodynamic design space. These methods make it possible to identify the effect of each design variable on objective functions. ANOVA shows the information quantitatively, while SOM shows it qualitatively. Furthermore, ANOVA can show the effects of interaction between design variables on objective functions and SOM can visualize the trade-offs among objective functions. This information will be helpful for designers to determine the final design from non-dominated solutions of multi-objective problems. These methods were applied to two design results: a fly-back booster in reusable launch vehicle design, which has 4 objective functions and 71 design variables, and a transonic airfoil design performed with the adaptive search region method.

Aerodynamic Optimization Design with Kriging Model

This study applied a Kriging model to optimization of a constrained aerodynamic design problem. The objective function and all constraint functions are considered statistically independent to avoid treating the complicated multivariate normal distribution in the constrained optimization problem. In the Kriging model of objective functions, expected improvements (EI) is calculated and in the Kriging model of constraint function, the probabilities of satisfying the constraints are calculated. Based on these values, an efficient exploration of the global optimum is performed. Two data mining techniques are used to investigate the information of design space such as the relationship between objective function and design variables: Functional Analysis of Variance (ANOVA), and Self-Organizing Map (SOM). ANOVA shows information quantitatively, while SOM shows it qualitatively. Based on the information, elimination of design variables with little effect on objective function is performed. The present method is applied to two-dimensional (2D) transonic airfoil design. The results showed the validity of the present method.

Transonic airfoil design and optimisation by using vibrational genetic algorithm

In this study, the power of vibrational genetic algorithm (VGA) for transonic airfoil design and optimisation problems, which are generally characterized by multi‐model topology in the design parameter space, has been introduced. This type of problem is characterized by search and computational time to achieve satisfying solutions. In order to obtain more robust and faster algorithm, vibration concept, our earlier study, is further developed. This developed VGA is coupled with a full potential flow‐field solver for inverse design and airfoil optimisation problem in transonic case. The performance of this implemented strategy is compared with that offered by a classical or more commonly used genetic algorithm.

Inverse Transonic Airfoil Design Using Parallel Simulated Annealing and Computational Fluid Dynamics

Inverse Transonic Airfoil Design Using Parallel Simulated Annealing and Computational Fluid Dynamics

Inverse transonic airfoil design using parallel simulated annealing and computational fluid dynamics

Inverse transonic airfoil design using parallel simulated annealing and computational fluid dynamics

Inverse and Direct Airfoil Design Using a Multiobjective Genetic Algorithm

Some of the advantages and drawbacks of genetic algorithms applications to aerodynamic design are demonstrated. A numerical procedure for the aerodynamic design of transonic airfoils by means of genetic algorithms, with single-point, multipoint, and multiobjective optimization capabilities, is presented. In the ® rst part, an investigation on the relative ef® ciency of different genetic operators combinations is carried out on an aerodynamic inverse design problem. It is shown how an appropriate tuning of the algorithm can provide improved performances, better adaption to design space size and topology, and variables cross correlation. In the second part, the multiobjective approach to design is introduced. The problem of the optimization of the drag rise characteristics of a transonic airfoil is addressed and dealt with using a single point, a multipoint, and a multiobjective approach. A comparison between the results obtained using the three different strategies is ® nally established, showing the advantages of multiobjective optimization.

Dual-Point Design of Transonic Airfoils Using the Hybrid Inverse Optimization Method

A new dual-point design procedure has been developed to improve the aerodynamic performance of transonic airfoils at two design points. A target pressure distribution at each design point was optimized by a genetic algorithm. Single-point design was performed to make an airfoil that produced given target pressures and satisfied specified geometric constraints at each design point. Following this, several intermediate airfoils were made by the hybrid inverse optimization method. A redesigned airfoil of dual-point design was found using the weighted average of intermediate airfoil shapes, based on the linear variation assumption. Two examples of dual-point design to improve the performance at an off-design condition are presented.

An inverse method for the design of transonic airfoils and wings

A design method for transonic airfoils and wings based on the solution of the Euler/Navier-Stokes equations is described. The applied design strategy is an inverse design method based on the work of Takanashi. The difference between the computed pressure distribution of a given geometry and the prescribed target pressure distribution is iteratively reduced by the solution of an inverse formulated transonic small perturbation equation (TSP-equation). In this design method an analysis code is required, such as the Euler/Navier-Stokes code CEVCATS/FLOWer developed at DLR. It is shown that Takanashi's method must be modified in order to ensure the convergence of the design in transonic flow. In addition a smoothing algorithm based on Bézier curves is used to obtain a smooth surface. In order to estimate the accuracy and convergence of the design method, a redesign of a known transonic airfoil and wing is conducted. In all designed cases very accurate results are obtained within a small number of design cycles.

Genetic algorithms applied to the aerodynamic design of transonic airfoils

Introduction T HIS work focuses on the application of genetic algorithms to the design of shockless transonic airfoils. The genetic algorithm selects a number of airfoils from a population by means of the fitness function related to the airfoil performance, and then it applies a mixture of crossover and random mutation operators over the selected individuals to generate a new population. The process is iterated until convergence criteria are met. Genetic algorithms offer an effective answer to complex optimization problems as they are able to effectively explore a very large space of potential solutions. Successful applications to complex design problems in the aerospace field can be found in Refs. 1-3.

Two-point transonic airfoil design using optimization for improved off-design performance

A two-point, aerodynamic design method is presented that improves the aerodynamic performance of transonic airfoils over a range of the flight envelope. It couples an Euler flow solver and a numerical optimization tool. The major limitation of single-point design is the poor off-design performance. Two-point design is used to extend the optimized performance range over more of the desired flight envelope. The method is applied to several transonic flow design points, and the results are compared to single-point design results. The secondary design points are chosen by varying the Mach number and the angle of attack. The two-point designs perform better than the single-point design over the design-point range.

Transonic airfoil design by constrained optimization

A N aerodynamic design method is developed which cou- ples flow analysis and numerical optimization to find an airfoil shape with improved aerodynamic performance. The flow analysis code is based on the coupled Euler and bound- ary-layer equations in order to include the rotational, viscous physics of transonic flows. The numerical optimization pro- cess searches for the best feasible design for the specified design objective and design constraints. The method is dem- onstrated with several examples at transonic flow conditions. Contents The optimization process is performed with a commercially available constrained optimization tool.3 The sensitivity of the flow to the perturbation is calculated by finite differences. The effectiveness and efficiency of the design process are influenced by many factors: the number and the shape of the base functions, the number and the tolerance of the con- straints, the flow model and the grid used for flow analyses, and the flight condition at the design point. Design Demonstration The objective of the present design is to produce minimum drag at a specified transonic flight condition. Inequality con- straints are imposed on lift, pitching moment, and cross-sec- tional area of the optimized airfoil. The lift and the area of the optimized airfoil should not be smaller than those of the original airfoil, and the pitching moment should not increase in absolute value. Also imposed are side constraints which limit the magnitude of the design variables. Side constraints are important because a large geometry change can cause boundary-layer separation leading to a termination of the flow solver.

An inverse method with regularity condition for transonic airfoil design

It is known from Lighthill's exact solution of the incompressible inverse problem that in the inverse design problem the surface pressure distribution and the free stream speed cannot both be prescribed independently. This implies the existence of a constraint (regularity condition) on the prescribed pressure distribution. The same constraint exists at compressible speeds. In this paper, an inverse design method for transonic airfoil is presented. In the method, the target pressure distribution contains a free parameter that is adjusted during the computation to satisfy the regularity condition derived in this paper. A few design results are presented here in order to demonstrate the capability of the method.

Transonic airfoil design procedure utilizing a Navier-Stokes analysis code

An iterative procedure for transonic airfoil design utilizing a Navier-Stokes analysis code to attain arbitrarily specified pressure distributions is presented. The concept of the present design procedure depends on the fact that an inverse design method of the residual-correction type can be combined both with a Navier-Stokes analysis code for a wide range of flow regimes and with conventional full potential analysis codes. The logical validity of the concept is presented and numerical examples of supercritical and low-speed designs are presented. Since the Navier-Stokes code is used in the analysis mode of the procedure, shock wave and viscous effects (including trailing-edge separation), which are difficult to evaluate with the conventional potential design methods, are properly evaluated and effectively incorporated in the design procedure.

Transonic Airfoil Analysis and Design Using Cartesian Coordinates

An inverse numerical technique for designing transonic airfoils having a prescribed pressure distribution is presented. The method uses the full potential equation, inverse boundary conditions, and Cartesian coordinates. It includes simultaneous airfoil update and utilizes a direct-inverse approach that permits a logical method for controlling trailing edge closure. The method can also be used for the analysis of flowfields about specified airfoils. Comparison with previous results shows that accurate results can be obtained with a Cartesian grid. Examples show the application of the method to design aft-cambered and other airfoils specifically for transonic flight.