International Journal of Aerospace Engineering

Volume 2016 (2016), Article ID 4706925, 16 pages

http://dx.doi.org/10.1155/2016/4706925

## Aerodynamic Optimization Based on Continuous Adjoint Method for a Flexible Wing

College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Received 6 April 2016; Accepted 10 October 2016

Academic Editor: Pier Marzocca

Copyright © 2016 Zhaoke Xu and Jian Xia. 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

Aerodynamic optimization based on continuous adjoint method for a flexible wing is developed using FORTRAN 90 in the present work. Aerostructural analysis is performed on the basis of high-fidelity models with Euler equations on the aerodynamic side and a linear quadrilateral shell element model on the structure side. This shell element can deal with both thin and thick shell problems with intersections, so this shell element is suitable for the wing structural model which consists of two spars, 20 ribs, and skin. The continuous adjoint formulations based on Euler equations and unstructured mesh are derived and used in the work. Sequential quadratic programming method is adopted to search for the optimal solution using the gradients from continuous adjoint method. The flow charts of rigid and flexible optimization are presented and compared. The objective is to minimize drag coefficient meanwhile maintaining lift coefficient for a rigid and flexible wing. A comparison between the results from aerostructural analysis of rigid optimization and flexible optimization is shown here to demonstrate that it is necessary to include the effect of aeroelasticity in the optimization design of a wing.

#### 1. Introduction

Thanks to the rapid development of computer performance and Computational Fluid Dynamics (CFD), CFD methods have been increasingly used in aerodynamic analysis of aircrafts. In the meantime, aerodynamic shape optimization based on CFD technology and optimization strategies has been more and more interesting to aircraft designers. Continuous adjoint formulation method was developed by Jameson to design aircrafts in transonic speeds, which can significantly reduce computational cost of aerodynamic shape optimization [1–5]. Subsequently, continuous adjoint method was utilized by Anderson and Venkatakrishnan on unstructured grid [6]. And recently Economon et al. in Stanford University have developed an open-source CFD code named SU2 [7–12], which is an integrated computational environment consisting of multi-physics simulation and design. The main module of SU2 is aerodynamic shape optimization on the basis of continuous adjoint method and Reynolds-averaged Navier-Stokes (RANS) equations. Now SU2 has also developed aerodynamic shape optimization based on discrete adjoint method [13, 14]. Discrete adjoint method was derived from the discrete flow equations by Baysal et al. [15, 16]. Subsequently discrete adjoint method has been developed rapidly [17–22]. Discrete adjoint equations which come from discrete flow equations can provide a little more accurate gradients than continuous adjoint equations, but discrete adjoint equations are more difficult to be derived and implemented numerically. Continuous adjoint equations are directly derived from the governing equations, which can be solved with similar methods of the governing equations and implemented more easily. However, since the coupling between aerodynamics and structures is so tight that these two disciplines cannot be isolated from each other especially for flexible wings, aerodynamic shape optimization alone may lead to the performance that is not appropriate in realistic flight conditions. But successive discipline optimization may lead to suboptimal results of coupled systems, so MultiDisciplinary Optimization (MDO) is essential to aircraft design. Relatively low fidelity models have been used in relevant disciplines in conceptual design, but these models cannot accurately capture nonlinear phenomena such as wave drag. So MDO of CFD and CSM (Computational Structural Mechanics) has been more and more attractive to aircraft researchers. MultiDisciplinary design optimization based on Kriging and response surface methodology has been a UAV wing and supersonic fighter wing [23, 24]. Martins et al. [25–27] have developed systematic aerostructural optimization design codes, which use coupled high-fidelity sensitivity analysis based on discrete adjoint method, and have been applied to design full aircraft configurations with coupled adjoint method. Mavriplis et al. [28–30] have developed time-dependent aeroelastic shape optimization based on discrete adjoint method.

However, aerostructural optimization that has been performed so far mostly is grounded on discrete adjoint method. In present work, aerodynamic optimization on the basis of continuous adjoint method including the effect of aeroelasticity is developed using FORTRAN 90, which is for the purpose of the implementation of coupled aerostructural adjoint optimization in the future. The structural model of the wing is created consisting of two spars, 20 ribs, and skin. Aerostructural analysis is performed with high-fidelity models, with Euler equations on the aerodynamic side, and with a linear shell element model on the structure side. This shell element is suitable for thin and thick shell problems and is easy to implement, and in particular this shell element can deal with the shell problems with intersections which are common in the structural model of aircrafts. Thin-Plate Spline (TPS) method is applied to ensure the interpolation between aerodynamics and structures consistent and conservative [31]. sequential quadratic programming (SQP) method, a gradient-based optimizer, is used to search for the optimal point in a coupled system consisting of aerodynamics and structures.

The paper begins with a description of aerodynamic analysis in Section 2, where the flow governing equations and the solution method are described. Section 3 presents structural analysis method with the theory of the shell element and structural design of the wing. The interpolation method between aerodynamic and structural disciplines is introduced in Section 4. The derivation of the continuous adjoint method is shown in Section 5. Design strategies containing parameterization methods, mesh deformation, and optimization algorithm are presented in Section 6. In the end, Section 7 presents verification of sensitivities of ONERA M6 and the optimization results for rigid and flexible DPW-W1.

#### 2. Aerodynamic Analysis

The governing equations of 3D steady inviscid compressible flow arewhere is conservative variable vector and is convective flux vector. They are defined aswhere is density, , , and are Cartesian velocity components, is total energy per unit mass, , , and are Cartesian components of unit normal vector, is static pressure, is total enthalpy, and is contravariant velocity, which is represented as

The finite-volume method of HLLC (Harten, Lax, van Leer, Contact) scheme is applied to solve Euler equations, and in order to reduce the computational time, an efficient implicit parallel hybrid LU-SGS (Lower-Upper Symmetric Gauss-Seidel) method [32] is used here.

#### 3. Structural Analysis

##### 3.1. Shell Element

In the structural analysis of the present work, a linear quadrilateral shell element [33] is applied to model the wing structure. This four-node quadrilateral shell element with 5/6 nodal degrees of freedom is constructed based on Hellinger-Reissner variational principle with independent displacements and stress resultants. The element stiffness matrix is integrated analytically with the advantage of fast stiffness computation. The shell element can get rid of locking which many shell theories may encounter and it can deal with both thin and thick shell problems. And in essence this shell element can be implemented easily for the linear isotropic aeroelasticity.

With the stationary condition, the variational formulation based on Hellinger-Reissner principle iswhere and are displacement vector and virtual displacement vector, respectively, and are independent stress resultants and virtual stress resultants, respectively, is a vector containing membrane strains, curvatures, and shear strains, is corresponding virtual vector, is the constitutive matrix, is translational displacement vector, is volume load in , and is the boundary load on a part of boundary . The membrane strains , curvatures , and shear strains are evaluated, respectively, as follows:where where is the nodal coordinate vector, is a unit vector normal to the shell midsurface, is the deviation of , and are isoparametric coordinates with the value , is the Jacobian matrix with local coordinates to isoparametric coordinates and local coordinates are evaluated with the center of the elements, and , , , and are the middle nodes of relevant edges of the elements [33]. From (5) it is shown that the shear strains are evaluated with independent interpolation functions, while the others are interpolated from displacements. These interpolation functions are adopted to fulfill the bending patch test. The interpolation of the stress resultants is defined as follows:where and are the second and third order unit matrices, respectively, the vector contains 14 parameters in total and 8 for the constant part and the others for the nonconstant part of the stress field, respectively, and are the coordinates of the center of gravity of the element, and is the parameter of the Jacobian matrix with and [33]. After a series of transformation, (4) can be written aswhere nele denotes the number of elements, is the element displacement vector, and is the element load vector which comes from the external virtual work. The matrices and are, respectively,

In view of (9), we know that the matrix only contains polynomials of and , so it can be evaluated analytically. While the matrix can only be carried out analytically in the projected flat surface [33]. As the matrix is evaluated with the flat surface, then the corresponding transformation should be adopted to get the sound stiffness matrix.

In order to analyze the shell with intersections, a second transformation should be utilized [33]. The drilling degree of freedom is not available for the nodes on intersections. So the nodes on intersections have six degrees of freedom and the other nodes have five. The intersections can usually be seen in the internal structure of the wing, such as the intersection between the spars, the ribs, and the skin. Eventually the stiffness matrix of the shell element takes the form as follows [33]:where denotes block parameters of the element stiffness matrix .

In the analysis of static aeroelasticity, structural equilibrium equation should be solved to obtain the structural deformation:where is the global stiffness matrix which is assembled with the element stiffness matrix, is the displacement vector of nodes, and is nodal force vector which is interpolated from the aerodynamic forces. Equation (11) is a classical linear equation set, and here it is solved using GMRES [34] method to get the values of the nodal displacements. The global stiffness matrix is sparse so Compressed Sparse Row (CSR) which is popular for storing general sparse matrix is used here to store it.

##### 3.2. Structural Design

The wing designed here is the wing-alone geometry called DPW-W1 which was developed for the Third AIAA Drag Prediction Workshop [35]. Figure 1 shows the overview of DPW-W1 (dimensions in m) and the geometric quantities are located in Table 1. The structural model is not necessary to represent exactly the internal components but is sufficient to capture the features of the wing. Furthermore since the realistic and precise structural model may have too many degrees of freedom, it is appropriate to simplify the model with the identical attributes. As the structural model of the DPW-W1 wing is not delivered, we create a structural model conforming to the wing which is a typical structure for a modern wing. The model used here consists of front and rear spars, ribs, and skin. There are 20 ribs distributed evenly along the span of the wing, not including the rib at the root for wing-body attachment. The front spar is located at 10% and the rear spar at 60% of the chord from the leading edge. In order to investigate appropriate effect of aerostructural deformation, the thickness of the two spars is adjusted to make sure that the displacement of the wing tip is about 3% of the semispan. Figure 2 shows the structural model of the wing. The skin is mirrored to the left to have a clear vision of the interior structure. The structural model of the wing is constructed with 1452 shell elements in total. The material used here for the structural model is assumed to be 7000 series of aluminium alloy with the properties shown in Table 2.