Zhong-Hua Han

Hierarchical Kriging Model for Variable-Fidelity Surrogate Modeling

The efficiency of building a surrogate model for the output of a computer code can be dramatically improved via variable-fidelity surrogate modeling techniques. In this article, a hierarchical kriging model is proposed and used for variable-fidelity surrogate modeling problems. Here, hierarchical kriging refers to a surrogate model of a highfidelity function that uses a kriging model of a sampled lower-fidelity function as a model trend. As a consequence, the variation in the lower-fidelity data is mapped to the high-fidelity data, and a more accurate surrogate model for the high-fidelity function is obtained. A self-contained derivation of the hierarchical kriging model is presented. The proposed method is demonstrated with an analytical example and used for modeling the aerodynamic data of an RAE 2822 airfoil and an industrial transport aircraft configuration. The numerical examples show that it is efficient, accurate, and robust. It is also observed that hierarchical kriging provides a more reasonable mean-squared-error estimation than traditional cokriging. It can be applied to the efficient aerodynamic analysis and shape optimization of aircraft or any other research areas where computer codes of varying fidelity are in use.

A new cokriging method for variable-fidelity surrogate modeling of aerodynamic data

Cokriging is a statistical interpolation method for the enhanced prediction of a less intensively sampled primary variable of interest with assistance of intensively sampled auxiliary variables. In the geostatistics community it is referred to as two- or multi-variable kriging. In this paper, a new cokriging method is proposed and used for variable-fidelity surrogate modeling of aerodynamic data obtained with an expensive high-fidelity CFD code, assisted by data computed with cheaper lower-fidelity codes or by gradients computed with an adjoint version of the high-fidelity CFD code, or both. A self-contained derivation as well as the numerical implementation of this new cokriging method is presented and the comparison with the autoregressive model of Kennedy and O’Hagan is discussed. The developed cokriging method is validated against an analytical problem and applied to construct global approximation models of the aerodynamic coefficients as well as the drag polar of an RAE 2822 airfoil based on sampled CFD data. The numerical examples show that it is efficient, robust and practical for the surrogate modeling of aerodynamic data based on a set of CFD methods with varying degrees of fidelity and computational expense. It can potentially be applied in the efficient CFD-based aerodynamic analysis and design optimization of aircraft.

Alternative Cokriging Method for Variable-Fidelity Surrogate Modeling

Surrogate modeling plays an increasingly important role in different areas of aerospace engineering, such as erodynamic shape optimization, aerodynamic data production, structural design, and multidisciplinary design optimization of aircraft or spacecraft. Cokriging provides an attractive alternative approach to conventional kriging to improve the efficiency of building a surrogate model. It was initially proposed and applied in the geostatistics community for the enhanced prediction of less intensively sampled primary variables of interest with the assistance of intensively sampled auxiliary variables. As the underlying theory of cokriging is that of two-variable or multivariable kriging, it can be regarded as a general extension of (one-variable) kriging to a model that is assisted by auxiliary variables or secondary information. In an attempt to apply cokriging to the surrogate modeling problems associated with deterministic computer experiments, this article is motivated by the development of an alternative cokriging method to address the challenge related to the construction of the covariance matrix of cokriging [7]. Earlier work done by other authors related to this study can be found in the statistical community. For example, Kennedy and O’Hagan (KOH) proposed an autoregressive model to calculate the covariances and crosscovariances in the covariance matrix and developed a Bayesian approach to predict the output from an expensive high-fidelity simulation code with the assistance of lower-fidelity simulation codes. This Bayesian approach is identical to a form of cokriging suitable for computer experiments. Later, Qian andWu proposed a similar method, in which a random function (Gaussian process model) was used to replace the constant multiplicative factor of KOH’s method to account for the nonlinear scale change. KOH’s method was applied to multifidelity analysis and design optimization in the context of aerospace engineering by Forrester et al. and Kuya et al. More recently, Zimmerman and Han proposed a cokriging method with simplified cross-correlation estimation. In this article, we propose an alternative approach for the construction of the cokriging covariance matrix and develop a more practical cokriging method in the context of surrogate-based analysis and optimization. The developed cokriging method is validated against an analytical problem and applied to construct global approximation models of the aerodynamic coefficients as well as the drag polar of an RAE 2822 airfoil.

A Variable-Fidelity Modeling Method for Aero-Loads Prediction

A Variable-Fidelity Modeling (VFM) method has been developed as an efficient and accurate aerodynamic data modeling strategy. In this approach, a set of CFD methods with varying degrees of fidelity and computational expense is exercised to reduce the number of expensive high-fidelity computations. Kriging-based bridge functions are constructed to match the low- and high fidelity CFD data. The method is demonstrated by constructing a global approximation model of the aerodynamic coefficients of an RAE 2822 airfoil based on sampled data. The model is adaptively refined by inserting additional samples. It is shown that the method is promising for efficiently generating accurate aerodynamic models that can be used for the rapid prediction of aerodynamic data across the flight envelope.

Surrogate-Based Optimization

Surrogate-based optimization (Queipo et al. 2005, Simpson et al. 2008) represents a class of optimization methodologies that make use of surrogate modeling techniques to quickly find the local or global optima. It provides us a novel optimization framework in which the conventional optimization algorithms, e.g. gradient-based or evolutionary algorithms are used for sub-optimization(s). Surrogate modeling techniques are of particular interest for engineering design when high-fidelity, thus expensive analysis codes (e.g. Computation Fluid Dynamics (CFD) or Computational Structural Dynamics (CSD)) are used. They can be used to greatly improve the design efficiency and be very helpful in finding global optima, filtering numerical noise, realizing parallel design optimization and integrating simulation codes of different disciplines into a process chain. Here the term “surrogate model” has the same meaning as “response surface model”, “metamodel”, “approximation model”, “emulator” etc. This chapter aims to give an overview of existing surrogate modeling techniques and issues about how to use them for optimization.

Bilevel Adaptive Weighted Sum Method for Multidisciplinary Multi-Objective Optimization

The primary goal of this research is to develop a framework for dealing with multi-objective, multidisciplinary optimization problems with a large number of variables. The proposed method is expected to provide a relatively uniformly spaced, widely distributed Pareto front. To achieve this end, a novel integration of the adaptive weighted sum method within a concurrent subspace optimization framework is presented. In the bilevel framework of concurrent subspace optimization, the adaptive weighted sum is used to make tradeoffs among multiple, conflicting objectives. To obtain better distributed solutions, two modifications are made. First, an additional equality constraint in suboptimization for each expected solution is relaxed because it causes slow convergence within the bilevel optimization framework. The probability of entrapment in local minima can also be reduced. Second, the mesh of the Pareto front patches is modified due to the low efficiency of the original scheme. The proposed method is demonstrated with three multidisciplinary design optimization problems: 1) a numerical multidisciplinary design optimization test problem with a convex Pareto front, available within the NASA multidisciplinary design optimization Test Suite; 2) a test problem with a nonconvex Pareto front, which is not easily solved; and 3) a conceptual design of a subsonic passenger aircraft, which consists of two objectives, four design variables, five coupling behavior variables, seven constraints in aerodynamics, and weight discipline. The primary results show that the proposed method is promising with regard to obtaining a uniformly spaced, widely distributed, and smooth Pareto front and is applicable in the design of large-scale, complex engineering systems such as aircraft.

Coupled Aerodynamic/Structural Optimization of a Subsonic Transport Wing Using a Surrogate Model

[Abstract] Coupled aerodynamic and structural optimization is performed for the preliminary design of a high-subsonic transport-aircraft wing using surrogate models. The aerodynamic performance of wing/body combination in transonic flow is calculated with full-potential equation in conjunction with viscous correction method. Structural analysis is performed using finite-element method (FEM) to obtain stress and deform distribution. The span, taper ratio, sweep angle and linear twist angle are chosen as design variables that define the aerodynamic configuration of the wing. And another four representing thicknesses of spars and skin are selected as the design variables for structural discipline. After the aeroelastic analysis of the various candidate wings, the aerodynamic and structural performances are obtained such as the lift coefficient, the drag coefficient, and the deformation and equivalent stress of the wing. Based on these sample data, the approximation models for analyzing the aerodynamic and structural performances are established using surrogate models including quadratic response surface method (RSM), kriging model (KM) and radial-basis function (RBF) Network. The modeling accuracy is evaluated by numerical-error analysis. We aim to select the approximation models with best accuracy to replace the complicated and time-consuming analysis in optimization. It is found that KM and RSM has comparative high accuracy and both are more accurate than RBF. Multi-objective optimization for the supercritical wing is performed based on RSM, for maximizing lift-to-drag ratio and minimizing weight. And the optimization is constrained by lift, reference area, deform, equivalent stress. The performance of the optimal design is proven to be improved based on the initial design. And compared with the optimal design without considering aeroelastic effect, lift-to-drag ratio is increased by 5.77% and lift is increased by 19.55%. It is proven by practice that considering aeroelastic effect is necessary in priliminary design of aircraft when optimizing high-aspect-ratio wing.

Optimization of Active Flow Control over an Airfoil Using a Surrogate-Management Framework

An efficient method based on the surrogate-management framework has been excised to optimize the actuation parameters of active flow control over an airfoil via a synthetic jet. In this approach, sample points are chosen by the design of experiments method, and approximation models are built based on the sampled data obtained from unsteady Reynolds-averaged Navier―Stokes simulations. The accuracy of these approximation models is evaluated at some test points by comparing the approximated values with the accurate values obtained from unsteady Reynolds-averaged Navier―Stokes simulations. Three types of approximation models (quadratic response-surface model, kriging model, and radial-basis-function neutral network) are built from the same data set. The model with highest accuracy is chosen as the surrogate model to be used to replace the unsteady Reynolds-averaged Navier― Stokes analysis during optimization. The optimization objective is to maximize the lift coefficient of a NACA 0015 airfoil at given angles of attack (14 to 22°), with the jet momentum coefficient, nondimensional frequency, and jet angle being the design variables. The surrogate model is coupled with a simulated annealing genetic algorithm optimizer to efficiently obtain the global optimum. As a result of the optimization process, the lift coefficient at an angle of attack of 16° is increased by 16.9% and the corresponding drag is decreased by 13.4% with respect to the initial controlled flow. It is preliminarily shown that the presented method is efficient and applicable for optimization of active flow control via a synthetic jet.

Flight Performance Analysis of Hybrid Airship: Revised Analytical Formulation

A hybrid airship denotes one kind of aircraft that combines the use of aerodynamic and buoyant lift. It is supposed to achieve the best combination of the high-speed characteristics of the airplane and the heavy-lifting capacity of the airship. In this work, an improved flight performance analysis method for a hybrid airship is proposed, aiming to provide a set of new formulas that are more suitable for a hybrid airship. The new formulas for analyzing the steady and accelerated performances of a hybrid airship are derived in a systematic way. The main advantage of the new formulas is that the relationship between the flight performances of a hybrid airship and airplane is indicated in a clearer and simpler expression. Base on the derivation, the theoretical comparisons are performed to show the advantage and disadvantage of the hybrid airship. An example of estimating flight performance of a model hybrid airship is presented to preliminarily demonstrate and evaluate the developed method, which shows a reasonable result.

A Preconditioned Multigrid Method for Efficient Simulation of Three-dimensional Compressible and Incompressible Flows

Abstract To develop an efficient and robust aerodynamic analysis method for numerical optimization designs of wing and complex configuration, a combination of matrix preconditioning and multigrid method is presented and investigated. The time derivatives of three-dimensional Navier-Stokes equations are preconditioned by Choi-Merkle preconditioning matrix that is originally designed for two-dimensional low Mach number viscous flows. An extension to three-dimensional viscous flow is implemented, and a method improving the convergence for transonic flow is proposed. The space discretizaition is performed by employing a finite-volume cell-centered scheme and using a central difference. The time marching is based on an explicit Runge-Kutta scheme proposed by Jameson. An efficient FAS multigrid method is used to accelerate the convergence to steady-state solutions. Viscous flows over ONERA M6 wing and M100 wing are numerically simulated with Mach numbers ranging from 0.010 to 0.839. The inviscid flow over the DLR-F4 wing-body configuration is also calculated to preliminarily examine the performance of the presented method for complex configuration. The computed results are compared with the experimental data and good agreement is achieved. It is shown that the presented method is efficient and robust for both compressible and incompressible flows and is very attractive for aerodynamic optimization designs of wing and complex configuration.