International Journal of Aerospace Engineering

Volume 2018, Article ID 5421027, 12 pages

https://doi.org/10.1155/2018/5421027

## Static Aeroelastic Modeling and Rapid Analysis of Wings in Transonic Flow

School of Aeronautics, Northwestern Polytechnical University, No. 127 Youyi West Road, Xi’an 710072, China

Correspondence should be addressed to Yilang Liu; moc.361@2112121gnaliyuil

Received 12 September 2017; Revised 1 March 2018; Accepted 18 March 2018; Published 13 May 2018

Academic Editor: Christopher J. Damaren

Copyright © 2018 Jingyuan Yang et al. 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 fast static aeroelastic analysis method, coupling with the modal method and Kriging surrogate model, is proposed in this paper. The deflection of the wing is described by the modal method, and the Kriging surrogate model is utilized to model the generalized forces under different deformations, angles of attack, and Mach numbers in order to replace the CFD solver. We analyzed the static aeroelasticity of HIRENASD wing in transonic flow field by coupling with the generalized force model by the static equilibrium equation. The results were compared with those of the experimental data and the references, and the comparison shows that the method is useful for the small deflections. After enough training cases are finished, the high-accuracy aerodynamic force coefficients and wing deflection will be obtained rapidly, which will only take several seconds. This method is more time saving than the CFD/CSD method, when it comes to a large quantity of the static aeroelastic analyses. Hence, it has good perspective for engineering applications during the aircraft design period.

#### 1. Introduction

Most of the modern aircrafts have high-aspect-ratio swept wing [1] and use composite material [2], which makes the static aeroelasticity become more and more severe. For instance, the wing on the Boeing 787 Dreamliner will nominally deflect 10 feet at cruise [3]. For the high-aspect-ratio swept wing, it will give rise to the wing deflection and torsion because of the static aeroelasticity problems, which will cause the angle of attack at the local section becomes smaller, and changes the distribution of surface pressure [3]. For the transonic aircraft, the decrease of angle of attack at the local section will also influence the intensity of shock wave. Therefore, it is crucial to analyze the static aeroelasticity for modern aircrafts.

By CFD/CSD method, loosely coupling is usually utilized to increase the computational efficiency. Hence, we can analyze the structure and the flow field separately. When it comes to structural analysis, there are two methods: one is linear analysis method. The wing deflection can be depicted by the modal method, under the assumption of small deflection. The other is nonlinear method. Finite element analysis is usually applied. Mian et al. [4] used this method to analyze the nonlinear deflection of high-aspect-ratio wing. In addition, both the multibody method [5] and nonlinear aeroelastic scaling method [6] are also used. In this paper, we used the computational structural dynamics (CSD) to analyze the structural deflection. On the other hand, when it comes to aerodynamic analysis, there are also two methods, linear and nonlinear methods, which are similar to structural analysis methods. Traditional aerodynamic linear theory is usually applied in the linear analysis method. For certain conditions, linear theory works well. However, the conditions are limited to the subsonic and supersonic flows [7]. When it refers to nonlinear aerodynamic calculation in transonic flow, computational fluid dynamics (CFD) is usually applied at present. When it comes to the static aeroelastic analysis, the steady CFD solution of the rigid wing is calculated firstly. Next, we get the generalized force and calculate the deflection of wing to obtain the new boundary by coupling with the static equilibrium equation. After that, we change the wall boundary by grid deformation method [8] and use CFD to calculate the aerodynamic force on the deformed wing. The results can be obtained by repeating the above steps in order, until the deflection converges or diverges. However, when it comes to the complex three-dimensional configurations, this method is time-consuming and has high computational complexity.

More and more researchers now used the surrogate method in their studies [9–12], as the problems mentioned above exist in the fluid-structure coupling problems and optimum design. Surrogate methods can fit the nonlinear multiple-input/output function accurately [13, 14] and has high computational efficiency. Due to these advantages, more and more researchers utilized these methods to model the nonlinear unsteady aerodynamics [15–19]. Lindhorst et al. [20] combined the parameter reduction via proper orthogonal decomposition and system identification methods to model nonlinear unsteady two-dimensional aerodynamics. And the model can accurately predict the static and transient response of the airfoil. Furthermore, they demonstrated the application on a three-dimensional case [21], the high-Reynolds-number aerostructural dynamics (HIRENASD), and the model can capture the influences of nonlinear aerodynamic effects on the forces. Moreover, the model can be used in both static and transient aeroelastic investigations at a fixed Mach number [22]. Kou and Zhang [23] applied radial basis function neural network to model two-dimensional nonlinear aerodynamics. And the approach can capture both linear and nonlinear characteristic.

This paper proposed an approach to model the nonlinear steady aerodynamics in transonic flow. The wing deflection is described by the modal method under linear structural assumption, while the aerodynamic force of different deformed wing is obtained by the surrogate model. And the deflection is calculated by the generalized forces coupling with the static equilibrium equation. The approach is used to perform the static aeroelastic analysis of HIRENASD wing. The results were compared with those obtained by CFD/CSD method, which shows the validity and the accuracy of the approach. The model has acceptable accuracy for engineering. This approach is more efficient than the CFD/CSD method in the acceptable accuracy, when it comes to plenty of analysis.

#### 2. Introduction of Method

##### 2.1. CFD/CSD Coupling Method

The flow governing equations used to solve the aerodynamics can be written as where is the control volume, is the boundary of the control volume, is the outer normal vector of the control volume boundary, denotes the volume of the element, and denotes the surface area of each surface. (When it comes to two dimension, denotes the surface area, and denotes the length.) is the vector of conservative variables, is the vector of the inviscid fluxes, and denotes viscous fluxes. More details of , , and are shown in [24].

The CFD solver based on the steady Reynolds-Averaged Navier-Stoke (RANS) [25] equations has the ability to simulate the flow with viscous effects. An unstructured RANS solver based on the finite volume is used in this paper. The Spalart-Allmaras (S-A) turbulence model [26] works well in describing the viscosity in the transonic flow, in which shock wave exists. Thus, this model is used in all cases in this study.

Under the assumption of linear small deflection, the modal method is utilized to describe the wing deflection. Since there is no need to take the structural inertial force into consideration, the deflection is computed by the static equilibrium equation: where is the generalized displacement vector, is the generalized stiffness matrix, and is the generalized force vector.

To get rid of the dynamic pressure effect, the generalized forces are nondimensionalized by dynamic pressure, which are called as generalized force coefficients. where represents the generalized force coefficient vector and represents the dynamic pressure.

The new structural boundary can be depicted by the following equation:
where is the coordinate matrix of the surface grid nodes for the rigid wing, is the vector of modal coordinates, is the modal matrix of the surface mesh for the *i*th mode, and is the coordinate matrix of the grid nodes.

In aeroelastic analysis, the relaxation factor is often used to enhance the computational stability, since large deformation may result in negative volume of the grid. And the new structural boundary would be obtained in the following formulation:
where represents the coordinate matrix of the boundary grid nodes at the *j +* 1th iteration and represents the coordinate matrix of the boundary grid nodes at the *j*th iteration. represents the coordinate matrix of the grid nodes, which is obtained from the new wing deflection. It is calculated according to the generalized forces of the *j*th iteration and .

The grid is deformed according to the new boundary by spring analogy method. In addition, loosely coupling is usually utilized to increase the computational efficiency.

The process of using CFD/CSD method has already been depicted in Introduction.

##### 2.2. Kriging Model

The Kriging surrogate model [27] is a kind of model aimed at minimizing variance and constructing an unbiased estimation of the spatial distribution data via the statistical method of stochastic process. The functional expression [28] can be given as where is the regression model and is the regression parameters. is the nonparametric random function, and its statistical properties are written as

Mean value

Variance

Covariance
where and are the design sites, and is the function with parameter and represents the spatial relativity among the design sites. The spatial relativity between every two design sites is related to their spatial distance. Hence, it can be depicted by the following equation:
where is the number of design variables and is the distance between every two design sites. The concrete function is given as
where and are the coordinate values of the *i*th and *j*th design sites in the *k*th direction and is the constant parameter of the function in the *k*th direction.

Aimed to minimize , after the mathematical derivation, the predictor is computed as where and , where , which represents the vector of regression values for the design sites, and is the response array of the design sites. Since and are related to the design sites instead of the predictor sites and the predictor sites are only related to and , the predicted response will soon be obtained when is given.

We utilized the Gauss Function as **R** matrix:
where .

#### 3. The Static Aeroelastic Analysis Method

##### 3.1. Sampling Method

As the model is expected to calculate the generalized force coefficients at the different conditions of Mach numbers (), angles of attack (), and dynamic pressures (), model samples need to be chosen at three steps.

The first step is to choose some sets of and from a certain range by Latin hypercube sampling (LHS) [29]. The nonlinear effect in transonic flow occurs when either or changes. For example, the aerodynamic force would change nonlinearly even if either or increases linearly due to the effect of the shock wave. Hence, the nonlinear effect of the and should be taken into consideration. The next step is to choose several sets of dynamic pressures at each set of and . Then, the static aeroelasticity will be analyzed at each condition to obtain the corresponding equilibrium position. The third step is to choose several sets of generalized displacements by LHS from a certain range close to each equilibrium position, in order to ensure that the numerical value of each mode generalized displacement is limited to a certain range. The obtained generalized displacements need to be able to describe the real wing deformations and enable the deformations to change in a certain range, since the model needs to predict the aerodynamic forces for the wing of different deformations. This sampling method can satisfy the requirements, so we gained the generalized displacements in this way. Finally, CFD method is used to calculate the corresponding generalized force coefficients and aerodynamic force coefficients at each training case. Hence, the sum of training cases can be calculated by the following function: where represents the sum of training cases, represents the sampling sets of and , represents the sampling sets of , and represents the sampling sets of generalized displacements.

##### 3.2. Modeling Method

A model is required to calculate the generalized forces at different , , and deflections. Hence, besides a set of generalized displacements, and will also be input to the model. And the output is a set of the corresponding generalized force coefficients. The relation between the inputs and the output is given as where is the vector of generalized force coefficients and is the vector of generalized displacements. To get the generalized forces, we need to multiply the by the dynamic pressure, for the coefficients are nondimensionalized by the dynamic pressure. The forces would be used to calculate the deformation of wing, which will put forward the procedure. This model is utilized to replace the CFD flow solver and will be called many times during the static aeroelastic analysis. We named this model as elastic generalized force (EGF) model for convenience.

In addition, we produced another model to predict the lift and drag coefficients of the deformed wing. The concrete function can be written as where represents the lift coefficient and is the drag coefficient. Different from the output of EGF, the outputs of this model are lift and drag coefficients, while the inputs are the same as those of EGF. In order to distinguish the models, we called this model as elastic aerodynamic force (EAF) model.

*Remark 1. *A model is required to calculate the generalized force coefficients of the rigid wing, since the corresponding generalized forces are the initial forces to calculate the wing deflection. The inputs of this model are the Mach number and angle of attack, since the forces are only related to flow condition parameters. And the output is the same as that of EGF. The function can be given as
where is the vector of the generalized force coefficients for the rigid wing. In routine, we named this model as rigid generalized force (RGF) model.

Similar to EAF, we also produced another model to predict the aerodynamic force coefficient of the rigid wing. The inputs are same as those of RGF, while the outputs are lift and drag coefficients. Therefore, the function of this model can be depicted as where is the lift coefficient of the rigid wing and is the drag coefficient. This model will also be called only once. This model was named as rigid aerodynamic force (RAF) model in routine.

The flow charts of the analysis are shown in Figures 1 and 2.

*Remark 2. *Figures 1 and 2 are shown to illustrate the analysis procedure. The middle chart is the procedure of CFD/CSD method. At the beginning of the analysis, both the aerodynamic force of the rigid wing and the natural modes of the structure are needed and are not related to each other. Hence, the flow field and the structural analysis can be divided and computed separately. The 1st iteration of the flow field analysis consists of the red blocks ① and ②. Then, the flow field and structural analysis would be performed alternately until the wing deformation converges or diverges, as shown in the circle of the middle chart. Each iteration of flow field consists of the red blocks ③ and ②. However, the flow field analysis, shown by the red blocks, would cost a lot of time at each iteration. Therefore, the surrogate model is used to replace them.