International Journal of Aerospace Engineering

Volume 2016 (2016), Article ID 8092824, 15 pages

http://dx.doi.org/10.1155/2016/8092824

## Proper Orthogonal Decomposition as Surrogate Model for Aerodynamic Optimization

DIMEAS, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Received 18 December 2015; Revised 9 June 2016; Accepted 26 June 2016

Academic Editor: Ratneshwar Jha

Copyright © 2016 Valentina Dolci and Renzo Arina. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A surrogate model based on the proper orthogonal decomposition is developed in order to enable fast and reliable evaluations of aerodynamic fields. The proposed method is applied to subsonic turbulent flows and the proper orthogonal decomposition is based on an ensemble of high-fidelity computations. For the construction of the ensemble, fractional and full factorial planes together with central composite design-of-experiment strategies are applied. For the continuous representation of the projection coefficients in the parameter space, response surface methods are employed. Three case studies are presented. In the first case, the boundary shape of the problem is deformed and the flow past a backward facing step with variable step slope is studied. In the second case, a two-dimensional flow past a NACA 0012 airfoil is considered and the surrogate model is constructed in the (Mach, angle of attack) parameter space. In the last case, the aerodynamic optimization of an automotive shape is considered. The results demonstrate how a reduced-order model based on the proper orthogonal decomposition applied to a small number of high-fidelity solutions can be used to generate aerodynamic data with good accuracy at a low cost.

#### 1. Introduction

Despite flow nonlinearities and geometrical complexities, the level of maturity reached by the numerical methods of computational fluid dynamics enables performing detailed numerical simulations of problems of practical interest. However, there exist applications, such as aerodynamic shape optimization, which plays nowadays an important role in the design process for aerospace and automotive engineering, demanding multiple simulations to perform optimization loops. Optimization techniques can require the prediction of certain quantities, such as lift and drag, as functionals of the field variables associated with a parametric partial differential system describing the physical behaviour of the problem. The evaluation of the implicit relationship between the inputs (the problem parameters), which identify specific system configurations, and the outputs (quantities such as lift and drag) requires the solution of the partial differential system. Numerous optimization methods and models have been developed and can be distinguished between local, global, and hybrid methods, based on gradient, evolutionary, or genetic algorithms [1]. Whatever optimization algorithm is applied, many evaluations of the functional relationship are needed. To keep the requirements of computational resources within certain limits, reduced-order models (ROMs) (or surrogate models) can be employed. A ROM provides a rapid and reliable estimate of the input-output relationship, with a considerable reduction of the computational cost.

The choice of the particular ROM however is quite critical, as it must preserve the essential physics and predictive capability of the high-fidelity partial differential model. The ROM definition can be sample based, employing statistical analysis such as kriging, polynomial chaos expansions, or Principal Component Analysis, for global surrogate models [2], or based on dimensionality considerations: the solution of the partial differential problem evolves in a low-dimensional manifold induced by the parametric dependence; therefore, the high dimensionality of the discretization space can be reduced constructing an approximation of this manifold, leading to a reduced-basis method [3, 4].

The definition of the ROM used in this work relies on the proper orthogonal decomposition (POD), a statistical technique able to extract the essential physics of some input information. The statistical analysis of the data allows expressing the flow field in terms of a set of low rank basis vectors and the ROM is obtained using this reduced set. In literature, the POD appears in different equivalent forms such as Karhunen-Loève decomposition, Principal Component Analysis (PCA), and Singular Value Decomposition (SVD) [5]. According to POD, an optimal linear basis, named POD basis, may be interpreted as a solution of minimization of the projection error of the original system, equivalent to maximizing the energy in the projection [6]. Therefore, the optimality of the basis vector is in an energetic sense. If a problem is described by a representative number of high-fidelity calculations from which a set of basis vectors may be extracted, the singular values become rapidly small and a small number of basis vectors are sufficient to approximate the initial data. In this way, POD provides an efficient statistical tool to capture the dominant features of a model characterized by many degrees of freedom and to represent it to the desired accuracy by using a reduced set of modes. The ROM is derived by projecting the high-fidelity model onto a reduced space spanned by only some of POD modes.

The POD has been applied in many different fields: data compression, image processing, dynamical systems, and fluid mechanics. Its application in fluid dynamics was first introduced by Lumley [7] for the detection of coherent structures in a turbulent flow. In fluid dynamics, the POD is usually employed to find a basis for the projection of the Navier-Stokes equations and to obtain a ROM composed of a system of ordinary differential equations for the time dependent POD expansion coefficients [8]. Less commonly, the POD is applied in the frequency space [9] or in a parameter space. In this latter case, as an example, the POD can be used to describe flow fields around modified body shapes, using the information about the flow past few selected geometries of the body, which form the snapshots for the POD. Examples of this approach can be found in the works of LeGresley and Alonso [10], Bui-Thanh et al. [11], Mifsud et al. [12], and Tang et al. [13].

The application of POD-based ROMs in the parameter space for aerodynamic shape optimization is quite recent and still under active development. In the above-cited references, the focus is on high-speed inviscid flows, in the transonic and supersonic regimes. The main results of the present work are the extension of the POD application in the parameter space to low-speed viscous flows and the use of a low number of snapshots for the initial set. This reduction is possible for two reasons: firstly, the locations of the high-fidelity calculations in the parameter space are chosen using a central composite design exploiting the statistical inference theory, and secondly the errors of the surrogate model, with respect to the full model solution, have a rapid decay with the number of snapshots. The POD expansion coefficients are functions of the parameters and are continuously extended in the parameter space by the response surface method (RSM) [14] using different techniques, such as least-square regression and radial basis functions (RBF) method [15]. This approach is termed POD with interpolation (PODI) [16].

It is not trivial to underline that, with respect to the application of the response surface methodology (RSM), the PODI method is able to provide the description of the entire fields of the original variables from which the objective function is calculated. The information is much more complete and justifies the minimum rise of computational effort. With RSM, on the other hand, it is possible to obtain only the value of the objective function in the desired point. More detailed information about this comparison is presented in Section 3.

In the present work, the PODI is applied to three test cases. The first one is the low-speed viscous flow past a backward facing step, with the step slope as parameter. In the second case, the flow past a NACA 0012 airfoil is reconstructed by the ROM in the two-parameter space (Mach, angle of attack). Finally, in the third case, the POD is coupled to a genetic algorithm; four shape parameters are optimized in order to minimize the drag coefficient of an automotive shape. In all the applications, the high-fidelity calculations are obtained from steady solutions of the Navier-Stokes equations.

In the first part of the paper, POD, RSM, and the interpolation strategies are briefly described. The second part reports the results obtained for the three cases under study, and in the last section some concluding remarks are given.

#### 2. Proper Orthogonal Decomposition with Interpolation

##### 2.1. Proper Orthogonal Decomposition

Several POD methods can be found in literature: the Karhunen-Loève decomposition, the Principal Component Analysis, and the Singular Value Decomposition (SVD). It can be shown that they are all equivalent [6]. For steady-state problems, each high-fidelity calculation, the* snapshot*, is represented as a set of discrete data, a vector of dimension , and the number of grid points or cells. In this case, the POD described as SVD is more straightforward [17].

###### 2.1.1. SVD as Proper Orthogonal Decomposition

The central issue of the POD is to approximate the snapshots simultaneously by a single, normalized vector * as well as possible*, that is, to solve the optimization problemwhere .

In (1), is equal to the number of snapshots.

Considering the Lagrangian functional associated with (1), the following eigenvalue problem can be written:with the condition

The orthonormal vectors , found solving eigenvalue problem (3), are called* POD basis of rank *. The approximation of the snapshots , columns of , by the first vectors is* optimal among all rank ** approximations* to the columns of . This method is the so-called* method of snapshots* [18].

The choice of is of central importance for the POD. No a priori rule is available; rather, it is possible to apply a heuristic criterion based on the ratio between the modeled energy and the total energy contained in the system which can be expressed as

##### 2.2. Continuous Extension of the POD Projection Coefficients

The vector represents a vector of scalar functions of grid points (or cells), such as the primitive variables of the flow field. The POD is applied to each nondimensional variable to compute a distinct basis. As shown in the previous section, each snapshot can be expanded as, setting , where the projection coefficients are discrete functions in the parameter space, with values defined at the points corresponding to the individual snapshots . To use the derived ROM as a prediction tool, it is necessary to extend the discrete functions in continuous functions in the parameter space, and the field variable in a generic point of the parameter space may be approximated by the linear combination

The combination of the POD and the continuous extension of the projection coefficients is termed POD with interpolation (PODI) [16].

In the case of one-dimensional parameter spaces, the continuous extension is obtained by linear or spline interpolation. For multidimensional spaces, the method of the response surface is adopted. The response surface method (RSM) describes the continuous behaviour of a dependent variable by a set of simple basis functions [14]. Response surfaces are generally valid in a large region only in the case of few parameters; when a great number of parameters are involved, such as in the case of optimization problems, the RSM must be treated carefully. RSM has been originally developed for experimental data, employing regression techniques. In this way, random experimental fluctuations are smoothed out. When the data set is provided by numerical simulations, a response surface obtained with a regression method in general does not fit exactly the data, introducing an undesirable smoothing. Therefore, in this work, other than a least square regression technique, interpolating methods, based on radial basis functions, are studied.

It is important to underline that number and topology of the discretization should not vary in the high-fidelity calculations. In this way, the correct correlations between the different CFD fields, corresponding to different solutions, can be performed. Only the position of the nodes on the geometrical mesh can change, exploiting the procedure of morphing.

###### 2.2.1. Least Square Regression

In the framework of response surface methodology, least square regression can be a simple technique to estimate a best fit approximation of the POD coefficients .

The second-order model of a response surface in the space defined by the parameters can be written as

In (8), with are indicated the regressors of the model and is an estimate of the error of the approximation. The second-order model can be useful in the event of a strong curvature in the true input-output relation. On the other hand, there can be cases where a first-order expression can fit properly the system behaviour with the relation

In the present work, both regression types are used.

Applying relation (8) or relation (9) to the points where the coefficient is evaluated for specific settings of the parameters , we have the system in matrix notation:where , is the model matrix, , and contains the regressors of the model. System (10) can be solved through least square minimization, assuming that has zero mean:

Once the vector of the regressors is found, the response surface is defined (8) and the POD coefficients are known as continuous functions in the parameter space.

###### 2.2.2. Radial Basis Functions

Least squares method does not provide an exact fit of the computed values , which is provided if interpolation is used. One of the primary tools for interpolating multidimensional data is the radial basis functions (RBF) method. The guiding principle behind this general method is to use translations of a single basis function that depends only on the Euclidean distance from its center, therefore radially symmetric about its center, in order to create a multidimensional interpolant [15]. The model of response surface for the POD coefficients that can be built using RBF has the general formwith being the centers and the radial basis functions, each one weighted by the expansion coefficient . In our work, two typical radial basis functions are used: The Gaussian . The multiquadric .

The parameter is called shape parameter and it is related to the width of the basis function. In Figure 1 [19], the influence of the shape parameter can be understood for Gaussian e multiquadric basis.