Shock and Vibration

Volume 2017, Article ID 2564314, 16 pages

https://doi.org/10.1155/2017/2564314

## Aeroelastic Optimization Design for High-Aspect-Ratio Wings with Large Deformation

^{1}School of Aeronautic Science and Engineering, Beihang University, Beijing, China^{2}Chengdu Aircraft Design & Research Institute, Chengdu, China

Correspondence should be addressed to Yang Meng; nc.ude.aaub@ymmus

Received 17 May 2017; Revised 11 September 2017; Accepted 17 September 2017; Published 23 October 2017

Academic Editor: Enrico Zappino

Copyright © 2017 Changchuan Xie 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

This paper presents a framework of aeroelastic optimization design for high-aspect-ratio wing with large deformation. A highly flexible wing model for wind tunnel test is optimized subjected to multiple aeroelastic constraints. Static aeroelastic analysis is carried out for the beamlike wing model, using a geometrically nonlinear beam formulation coupled with the nonplanar vortex lattice method. The flutter solutions are obtained using the - method based on the static equilibrium configuration. The corresponding unsteady aerodynamic forces are calculated by nonplanar doublet-lattice method. This paper obtains linear and nonlinear aeroelastic optimum results, respectively, by the ISIGHT optimization platform. In this optimization problem, parameters of beam cross section are chosen as the design variables to satisfy the displacement, flutter, and strength requirements, while minimizing wing weight. The results indicate that it is necessary to consider geometrical nonlinearity in aeroelastic optimization design. In addition, optimization strategies are explored to simplify the complex optimization process and reduce the computing time. Different criterion values are selected and studied for judging the effects of the simplified method on the computing time and the accuracy of results. In this way, the computing time is reduced by more than 30% on the premise of ensuring the accuracy.

#### 1. Introduction

Highly flexible wings, crucial for high-altitude long-endurance (HALE) unmanned aerial vehicles (UAVs), are characterized by light weight with high aspect ratios. Although the slender wings can maximize the lift-to-drag ratio, they may undergo large deformation of about 25% of wing semispan during normal flight load, exhibiting geometrically nonlinear behaviors. Patil and Hodges [1] studied the aeroelasticity of a HALE aircraft and found that large deformation induced by flexibility will change the aerodynamic load distribution. This will result in significant changes in aeroelastic and flight dynamic responses of the aircraft. Thus, the aeroelastic analysis based on small deformation assumption may be improper. Shearer and Cesnik [2] also studied the nonlinear dynamic response of a flexible aircraft. The nonlinear structural response is governed by the 6-DOF rigid-body motions coupled with aeroelastic equations. They compared three solutions: rigid, linearized, and nonlinear models and highlighted the use of the latter formulation to model the flexible wings. All these studies show that the geometrical nonlinearity due to the flexibility should be properly accounted for a nonlinear aeroelastic formulation. Such flexible wings can be treated as a statically nonlinear but dynamic linear system. Then the question of the dynamic stability of the statically nonlinear aeroelastic system may be addressed by a linear dynamic perturbation analysis about this nonlinear static equilibrium [3–5]. Patil and Hodges [6] studied the nonlinear effects and concluded that the dominant geometrically nonlinear effect comes from the nonzero steady-state curvature. Thereby, a linear analysis for curved wing structures can accurately predict the effect of nonlinear behavior on the linearized stability.

Optimization design of aircraft structures subjected to aeroelastic constraints is not new now in preliminary stage of modern aircraft structural design. The aeroelastic optimization process, in essence, is a unified, multidisciplinary design process which can overcome barriers between different discipline groups while reducing the design cycle time. Previous researches have mainly optimized aircraft structures from the perspectives of both static and dynamic constraints such as stress, displacement, modal frequency, and flutter constraints; see, for example, the previous publications by Sikes et al. [7], Patil [8], Battoo and de Visser [9], Maute and Allen [10], Tischler, and Venkayya [11]. However, these researches have not taken into account the geometric nonlinearity induced by large deformation. It may lead to an inaccurate modeling of aircraft in both structural and aerodynamic perspectives. Butler et al. [12, 13] presented aeroelastic optimization of high-aspect-ratio wings subjected to a minimum flutter speed constraint based on a dynamic stiffness matrix method. Yet, the basic assumption in solving the dynamic problems is linear. Previous researches have already shown that geometrical nonlinearities must be properly considered in aeroelastic analysis for flexible wings. In general, few studies have attempted to deal with the geometric nonlinearities in aeroelastic optimization design for flexible wings with large deformation. In this paper, numerical studies will demonstrate the necessity of considering nonlinear effects in aeroelastic optimization. Moreover, nonlinear aeroelastic optimization strategies are explored in order to simplify the complex optimization process while reducing the computing time.

The main purpose of the present work is to establish a complete framework of aeroelastic optimization design for high-aspect-ratio wing with large deformation. The geometrical nonlinearities in present theoretical modeling are considered from three aspects: structural dynamic, aerodynamics, and fluid-structure interface. The incremental finite element method is used to solve the geometrically nonlinear structural problems, and aerodynamic loads on nonplanar wings are calculated by the nonplanar vortex lattice method. The unsteady aerodynamic forces in flutter analysis are calculated by the nonplanar doublet-lattice method. A surface spline interpolation method [14] is used to exchange force and displacement information between the structure model and aerodynamic model. In this way, the aeroelastic formulation is naturally obtained by combining these three aspects together. A wing model for wind tunnel test is optimized subject to displacement, torsion, flutter, and strength constraints. The nonlinear aeroelastic analysis, as a key part in optimization design, is integrated into the whole optimization process to obtain the static equilibrium configuration and flutter speed. To improve efficiency of nonlinear aeroelastic optimization process, some simplifications in static and dynamic aeroelastic analysis are studied and the results indicate that such changes can greatly reduce the computing time.

#### 2. Theoretical Formulations

A complete theoretical framework of nonlinear aeroelastic analysis has been discussed in [3, 6]. An introduction is presented here, followed by the optimization formulation for achieving the optimum wing shape.

##### 2.1. Geometrical Nonlinear Elasticity

Geometric nonlinearities are based on the kinematic description of the body and the strain on the wing should be defined in terms of local displacement of the wing for dynamic motions. These result in the nonlinear geometric equations including the quadric term of the displacement differential, and the nonlinear force equilibrium equation established on the deformed state of the structure. Geometric nonlinear effects are prominent in two different aspects: geometric stiffening due to initial displacements and stresses and follower forces due to a change in loads as a function of displacements. Both of these two factors are considered in this paper and solved with the nonlinear incremental finite element method [15]. The updated Lagrange formulation (ULF) is used in this work and the main equations are presented below.

The relationship between strain and displacement iswhere means the partial derivative of displacement with respect to the coordinate at time . The stress tensor at time satisfieswhere is direction cosine of small aeroelement at time and is the corresponding surface force. The linear elastic constitution can be described as follows:where is the elastic tensor, which has a different form for isotropic or anisotropic material.

The strain can be decomposed into a linear part and a nonlinear part :

The stress is decomposed by increments, where represents the stress at time , and is the incremental stress to be calculated at each time step.

The integral equation is established by linearization in each incremental step:where is a vector of incremental external force, including the aerodynamic force, gravity, and engine thrust at time , and is the element volume. Using the following shape functions, the relationship between strain and displacements is performed as follows:

Substituting them into (6) yields the element governing equation for static problems:where is the equivalent inner force of the structure. The stiffness matrix can be decomposed into a linear part and nonlinear part . The nonlinear part is related to the deformed configuration, load condition, and strain, which should be updated in each computation step.

The corresponding dynamic equation can be expressed as follows: where is the instant mass matrix at time . and are the structural displacement vector and acceleration vector, respectively.

According to previous researches [1, 3], an assumption of small-amplitude vibration around nonlinear static equilibrium state is suitable for aeroelastic stability problem such as flutter analysis. Ignoring the damping effect, there is where is the large static deformation from (8) and is a small vibration deformation. According to (9) and the static equilibrium condition, the vibration equation of the dynamic system reduces towhere is the inertial matrix of the structure at the static equilibrium. is the corresponding stiffness matrix. Both of them are functions of static deformation and vary with different static equilibrium states. From (11), the mode shapes and frequencies are deduced. Combined with nonlinear aerodynamic computation, the flutter boundary can be determined by - method. However, the flutter boundary is only predicted under a certain static equilibrium state. Different static equilibrium states have different flutter boundaries and the exact boundary should be searched iteratively to make the flutter speed be consistent with the static equilibrium state.

##### 2.2. Nonplanar Aerodynamics

Two methods are used to obtain the aerodynamic loads: nonplanar vortex lattice method (VLM) [16] for solving steady aerodynamic loads in static aeroelastic analysis and doublet-lattice method (DLM) [17, 18] for solving unsteady aerodynamic loads in dynamic stability analysis. The lifting surface mesh in both of them can fit the actual wing surface and be consistent with the structural nonlinearities caused by large deformation.

###### 2.2.1. Nonplanar Vortex Lattice Method

In static aeroelastic analysis, the steady aerodynamic load is calculated by nonplanar vortex lattice method, which is based on full potential equations without any linearization. The exact boundary condition is satisfied on the actual wing surface. A Cartesian coordinate system is established for aeroelastic analysis. The -axis points from the nose to the tail along the free stream, the -axis points to the right side, and -axis is determined by the right-hand rule. In order to solve the steady aerodynamic load, the lifting surface is ideally discretized into trapezoidal panels, with two side edges parallel to the undistributed flow as shown in Figure 1. This spatial surface mesh can reflect the actual wing surface that has camber and various platform shapes well. Some typical panel elements are shown in Figure 2. Each vortex ring consists of four segments of a vortex line, and the leading segment is placed on the chord line. The aerodynamics of the panel act on the midpoint of this segment (represented by “” in Figure 2). The collocation point (represented by “” in Figure 2) is located at the center of the three-quarters chord line, and the actual boundary condition will be implemented at this point.