SCIENCE CHINA Technological Sciences • RESEARCH PAPER •
July 2013 Vol.56 No.7: 1790–1797 doi: 10.1007/s11431-013-5250-1
Multidisciplinary design optimization of adaptive wing leading edge SUN RuJie1, CHEN GuoPing1*, ZHOU Chen2, ZHOU LanWei1 & JIANG JinHui1 1
State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China; 2 Key Laboratory of Fundamental Science for National Defense—Advanced Design Technology of Flight Vehicle, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China Received April 12, 2013; accepted May 2, 2013; published online May 24, 2013
Adaptive wing can significantly enhance aircraft aerodynamic performance, which refers to aerodynamic and structural optimization designs. This paper introduces a two-step approach to solve the interrelated problems of the adaptive leading edge. In the first step, the procedure of airfoil optimization is carried out with an initial configuration of NACA 0006. On the basis of the combination of design of experiment (DOE), response surface method (RSM) and genetic algorithm (GA), an adaptive airfoil can be obtained whose lift-to-drag ratio is larger than the baseline airfoil’s at the given angle of attack and subsonic speed. The next step is to design a compliant structure to achieve the target airfoil shape, which is the optimization result of the previous step. In order to minimize the deviation of the deformed shape from the target shape, the load path representation topology method is presented. This method is developed by way of GA, with size and shape optimization incorporated in it simultaneously. Finally, a comparison study with the Solid Isotropic Material with Penalization (SIMP) method in Altair OptiStruct is conducted, and the results demonstrate the validity and effectiveness of the proposed approach. adaptive wing, multidisciplinary design optimization, aerodynamic optimization, structural optimization, genetic algorithm Citation:
Sun R J, Chen G P, Zhou C, et al. Multidisciplinary design optimization of adaptive wing leading edge. Sci China Tech Sci, 2013, 56: 17901797, doi: 10.1007/s11431-013-5250-1
1 Introduction In flight, the wing, especially the leading edge, plays a dominant role in the performance of the aerodynamics. Nowadays, aircraft wing is typically designed for the cruise condition, while it cannot satisfy the request of high lift-to-drag ratio in take-off and landing conditions. Thus, the adaptive wing concept has been put forward recently, which can change its shape real-time to adjust to different flying conditions. This issue mainly refers to two sections: aerodynamic optimization and structural optimization. Many relevant researches have been conducted respectively in recent *Corresponding author (email:
[email protected]) © Science China Press and Springer-Verlag Berlin Heidelberg 2013
decades. Since early 1980s, Air Force Research Labs (AFRL) of the United States conducted a project called Mission Adaptive Wing in a modified F-111 aircraft [1]. Using the mechanical systems to change the shape of the upper leading and trailing edges of the wing, this test proved its aerodynamic superiority over conventional ones. However, due to its drawbacks of increases in weight, complexity and mechanical performance, the program was prevented from further development. In 1994, Kota [2], firstly proposed a methodology of designing compliant mechanisms for adaptive wing. This method utilized compliant mechanism to transfer the energy from the actuator to deform the shape of leading and tailing edges into desired shapes. Starting in tech.scichina.com
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1998 [3], under the support of AFRL, the flexible system company developed a compliant variable camber wing leading edge, which could produce a 0−6° change in camber. The results showed a 51% increase in lift-to-drag ratio and a 25% increase in the lift coefficient for 6°. Saggere and Kota [4] proposed a novel approach to static shape control of adaptive structures. Kota and Hetrick [5] patented an adaptive wing with a compliant trailing edge. In 2006 [6], a flight test model with a variable geometry trailing edge was scheduled for performance test on a Scaled Composite White Knight Aircraft. The results indicated that the trailing edge could significantly extend 15% or more flying ranges through continuously optimizing the lift-to-drag ratio. The technology of aerodynamic shape optimization has been developed rapidly due to the progress of computational fluid dynamics. Methods in aerodynamic optimization can be categorized into two kinds: non-gradient and gradient based methods (GBM). GBM has been widely used in airfoil optimization design [7]. In non-gradient based methods, the typical ones are response surface method (RSM) and genetic algorithm (GA) [8]. Keane [9], Li et al. [10] and Sun [11] used RSM to optimize the airfoil of aircraft and wind turbine, respectively. Ma et al. [12] established a hierarchical multi-objective optimization platform for aerodynamic optimization design of a low Reynolds number airfoil. Liang et al. [13] adopted multi-objective GA for airfoil optimization represented by non-uniform rational B-spline. In another area of high-speed trains, aerodynamic optimization technology is also very important [14]. Yao et al. proposed a three-dimensional aerodynamic optimization design of highspeed train nose based on GA-GRNN [15] and the Kriging model [16]. The synthesis approaches in the field of topology optimization are generally classified into two categories: continuous and discrete optimization. In the area of continuous optimization, the widely-used methods are the homogenization [17] and the Solid Isotropic Material with Penalization (SIMP) [18]. Maute and Reich [19] established the topology optimization model of adaptive wing leading and trailing edges based on the SIMP approach. Liu et al. [20] proposed a guide-weight method to solve two kinds of topology optimization problems with multiple loads. The combination of the sensitivity filtering technique with the subset simulation was introduced by Wang [21]. For the discrete optimization, the most popular method is the ground structure method [22]. Lu and Kota [23, 24] and Liu and Ge [25] incorporated this discrete method into GA to demonstrate its application in adaptive wing structures. Wang et al. [26] proposed a global topology optimization algorithm based on subset simulation for the singular problem of trusses. However, the methods mentioned above may generate invalid structures, which contain gray areas or suffer from the disconnected structure issue. For these reasons, a different approach, based on the load path representation, was proposed [24, 27, 28]. By taking a novel approach to parameterize the
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design domain and define the design variables, this method was free of gray areas and disconnected structures effectively and efficiently. Santer and Pellegrino [29] also did further research about this approach and its applications in compliant structures. In this paper, a two-step optimization procedure is developed. This method tactfully combines aerodynamic and structural optimizations to make the final results more convincing. The first step is based on the iSIGHT platform. It optimizes the airfoil shape automatically through combining the software of Matlab, Gambit, Fluent with iSIGHT. Then, the method of load path representation, incorporated into GA, is described in details. It implements the topology, size and shape to be simultaneously optimized to synthesize the best internal structural configuration that matches the optimal shape, which is determined by the optimization result of the first step. A comparison study with the SIMP method is carried out to assess the effectiveness of the developed load path representation method, followed by a discussion and conclusions.
2 Aerodynamic optimization of an airfoil 2.1 Airfoil parameterization The airfoil profile is regenerated through altering the value of the control points during the optimal design of two-dimensional airfoil, while parametric description tends to be more advantageous. This paper uses the linear superposition method of analytic function to fit the airfoil profile, which is defined from the baseline airfoil profile, type function and its coefficients, as follows: n
y ( x ) yb ( x ) k f k ( x ),
(1)
k 1
where yb ( x ) is the baseline airfoil profile, n and k are the number of control points and the coefficients, which determine the airfoil profile. f k ( x ) is the type function, and
k f k ( x ) is the disturbance of the baseline airfoil. In this paper, the first quarter of the airfoil is just taken into consideration, so the type function is modified based on the Hicks-Henne type function, expressed as f1 ( x ) 0.25(1 x )0.25 (1 4 x )e20 x , f k ( x ) 0.25sin 3 (4 x )e ( k ) , k 1,
(2)
where e(k)=ln0.5/lnxk, 0xk1, 0 x 0.25, xk 4 x , xk (k=1, 2, 3, 4, 5) are set to be 0, 0.04, 0.1, 0.3, 0.6 by experience in this paper. 2.2
Integration of the optimization
In this paper, NACA 0006 airfoil is adopted as a baseline
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airfoil and the numerical solution of N-S equations is provided as flow solver. A C-type structured mesh around the airfoil is generated by using the commercial software Gambit. The commercial software Fluent is employed to compute the flow field, and the Spalart-Allmaras turbulence model is chosen. Simulations of the characteristics of the airfoil are carried out at Ma=0.6 and =4°. The design variables are the coefficients of the polynomial equation defining upper and lower airfoil profiles. Lift-to-drag ratio CL/CD is assumed to be the objective function, and lift coefficient CL is considered as the constraint condition. The designing process of the airfoil aerodynamic optimization is shown in Figure 1. 2.3
Optimization approaches
GA is a search heuristic that mimics the process of natural evolution. It is routinely used to generate valid solutions to optimization and search problems. GA is more likely to find the global optimal solution, but the optimization needs to cost a vast amount of time when GA is conducted separately. In order to improve the optimization efficiency, this paper puts forward a method based on the combination of Latin hypercube design of experiment (DOE), RSM and GA. First,
Figure 1
Designing process of airfoil aerodynamic optimization
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the data of variables in the design space are selected by using Latin hypercube experimental design techniques and the results of the selected data are obtained through a series of simulations. Then, the response surface model is generated through a regression analysis, usually using polynomial expressions. Finally, the optimization of the objective function using GA based on the response surface model is carried out to achieve the optimal results. 2.4
Optimization results
Table 1 shows the results of the optimization. In this table, the initial and optimized values of lift coefficient CL, drag coefficient CD, and lift-to-drag ratio CL/CD are presented. The pressure coefficient distributions and pressure contours of initial and optimized profiles are shown in Figures 2 and 3 and the optimized airfoil shape is shown in Figure 4. According to Figure 4, the camber of the leading edge has increased Table 1
Aerodynamic characteristics of pre- and aft-optimization Initial
Optimized
CL
0.52787
0.53434
CD
0.01198
0.00879
CL/CD
44.0626
60.7895
Figure 2 foils (b).
Pressure coefficient distributions of initial (a) and optimal air-
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Figure 3
Figure 4
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Pressure contours of initial (a) and optimal (b) airfoils.
Shape of the baseline and optimal airfoil.
compared with the baseline airfoil. Consequently, the shock wave has been weakened. For the case studied, the optimization leads to almost 40% increase in lift-to-drag ratio.
edge in subsonic and supersonic flying conditions have been obtained in Section 2. These two shape curves will be adopted as the initial and objective functions in the topology optimization. Before the optimization, the following problems should be defined. First, a parent lattice should be defined within the design domain, which is an experiential choice. The chosen lattice, adopted from ref. [29], is illustrated in Figure 5. It is a network modeled with beam elements. Then, four types of points, the input, output, control and fixed points, should be specified prior. The input point corresponds to the location where external actuation is applied. The output points are used to measure the shape change. The control points determine the shape of the generated structures. They are free to float within the design space to
3 Structural optimization 3.1
Description of optimization problem
In this section, the load path approach will be used to synthesize an adaptive airfoil structure. The material used is high strength aluminum, and its properties are listed in Table 2. As shown in Figure 4, the two ideal shapes of leading
Table 2
Material properties of the Aluminum 7075-T6 Mechanical properties
Value
Young’s modulus (MPa)
71700
Yield strength (MPa)
503
Possion’s ratio
0.33
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in Table 3. 3.3
Figure 5
The parent lattice of the compliant structure.
generate diverse structures. The fixed points are to ground the structures. Finally, the optimization objective is specified in terms of minimizing shape change to evaluate the difference between the deformed curve and target curve. The Least Square Error (LSE) deviation shown in eq. (3) is used to define the shape change. n
min f ( x) i 1
x
Tar, i
xDef ,i yTar,i yDef ,i , 2
where n is the number of output points;
x
Def , i
2
x
Tar,i
(3)
GA optimization process
Among optimization algorithms, GA is more efficient in searching the optimum in the whole solution space for its heuristic nature. GA generates the offspring and improves them through repetitive application of the selection, crossover and mutation operations, which have a crucial effect on the final results. Due to the difference from traditional methods, the reproduction schemes should be revised properly. A new procedure based on GA is proposed, which enables the topology, size and shape of the structure to be explored simultaneously. Discrete design variables, such as topology of load paths, and continuous design variables, including cross section dimensions of each beam member, locations of control points, input point and the external actuation, are all involved in the optimization process. The parameterization of these variables in GA may be represented as shown in Table 4. The constraints are (4)
0.0005 hb 0.0045,
(5)
0.0005 hi 0.0055, i 1 l ,
(6)
0.015 yf 0.025, xf 0.25,
(7)
LiLow Li LiUp , i 1 m,
(8)
, yDef ,i are the coordinates of the ith output point on
the target and deformed boundaries respectively. 3.2
0 F 250,
, yTar,i and
Load path representation and selection
Compared with the traditional ‘binary ground structure’ approach [23], the load path method has avoided the probability of invalid structures that are disconnected between the input point and the rest of the structure. The parameterization of the load path approach is to assign a binary parameter, not to the existence of individual members adopted in the traditional method, but to the existence of a path sequence to ensure the structural connectivity [27]. For a compliant structure, there are mainly three types of load paths: input to boundary control points; input to fixed points and fix to boundary control points. The selection of the load paths for the structure is not readily made. It is not practical to parameterize all possible load paths for the rapid increase in the number of load paths with the complexity of the structure. Proper methods are very essential to restrict the number systematically. The K shortest path method [30] has limited the number to some extent. To fulfill the condition that a load path should not contain a closed loop, the K shortest loopless path algorithm [31] is used to determine the load paths of the structure in this research. Given a unit weight of each single beam member, the total weight of a load path equals the number of members in it. Based on this method, the K shortest load paths from one point to any other different type points in Figure 5 can be obtained easily. For example, the 5 shortest load paths from point 10 to point 5 are listed
Table 3 The 5 shortest load paths from the input point to another different type point Path sequence
Weight
10 13 14 5
3
10 12 11 14 5
4
10 12 13 14 5
4
10 13 11 14 5
4
10 15 16 14 5
4
Table 4 Parameterization scheme of design variables Variable
Description
F
actuation magnitude
hb
boundary beam heights
hi
internal beam heights
(xf,yf)
actuation location
Li
boundary control points
(xi,yi)
internal control point’s location
pTopi
path existence or elimination
pSeqi
points along a load path
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xiLow xi xiUp , yiLow yi yiUp ,
i 1 n,
pTopi {0,1}, i 1 q, q
pTop i 1
i
1,
max s ,
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(9) (10) (11) (12)
where l, m, n, q are the numbers of internal beam elements, boundary control points, internal control points and the load paths. Their initial values are 25, 9, 6, 90 respectively. In this research, the widths of each beam remain the same, 5 mm, thus only the heights, hb and hi, are considered as design variables. For the boundary elements with the same height, they can be represented by only one variable, hb. L refers to the boundary control points, and they can move along the boundary. The internal control points can also float within the design space subjected to additional constraints, such as two or more elements should not cross over each other. In the process of iterations, a roulette wheel selection scheme is introduced to ensure fitter individuals to be selected for reproduction with a high probability. The selected parent populations then produce new offspring through two operations: crossover and mutation. The crossover strategy uses two-point crossover to create two children from two parents. For the dimensions of beam elements, locations of control points and the external actuation, the ordinary crossover operation is taken. However, for the load path, it should exchange both the path sequence and topology information between two selected parents randomly. The mutation strategy is used for topology of load paths, dimensions of beam elements, locations of control points and the external actuation. The crossover and mutation probability is 0.8 and 0.012 respectively, and the number of the individuals is 100. 3.4
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Figure 6 Optimal compliant structure (a) and comparison of the actual deformation and target shape (b).
Load path optimization results
Figure 6(a) illustrates the optimal structure of the leading edge that best achieved the request after several times of 1500 iterations. A static analysis under the force of 196.4 N at the given input location is performed. The result is shown in Figure 6(b), which obviously reveals that the actually deformed shape indeed matches the target shape. The LSE deviation of this structure is reduced from the initial 71.72 mm to the final 27.03 mm. In order to verify the feasibility of the optimization result and the effect of the actual deformation, the analysis of the optimal structure is conducted in the MSC/Nastran software. The simulation result is shown in Figure 7, and the deformation in MSC/Nastran is consistent with the result of Figure 6(b). In addition, the maximum stress is 395 MPa,
Figure 7 Displacements (a) and stresses (b) of the optimal structure analyzed in MSC/Nastran.
which is lower than the yield strength limit of 503 MPa. 3.5
Comparison with the SIMP method
In the previous section, the developed load path representation method is used to realize the topology optimization of an adaptive wing leading edge. In this section, the widelyused SIMP method implemented in Altair OptiStruct [32] is adopted to make comparison with the results obtained by
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Figure 8
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Topology result based on the SIMP method (a) and the actual deformation of the final solution (b).
the developed load path representation method. The SIMP method is a density-based topology optimization approach. In SIMP, the material density of individual element is used as the design variable, and varies continuously between 0 and 1 according to the following relationship [29]: n
E , E0 0
(13)
where the density is the design variable and E is the corresponding material stiffness. 0 and E0 represent the original material properties. Generally, the optimal solutions involve intermediate values and E, which are not meaningful in factual manufacturing. Therefore, the penalization technique needs to be introduced to penalize intermediate densities so that the final design can be represented by densities of 0 or 1 for each element. Herein n is the penalty. In the present study, a minimum member size control in Altair OptiStruct is used, and the penalty starts at 2 and is increased to 3 for the second and third iterative phases. To make a comparison study, the same optimization problem needs to be reformulated in Altair OptiStruct with LSE deviation as the objective and stresses as the constraints. Figure 8 indicates the SIMP topology optimization results. It is found that the final solution is quite free of intermediate densities, thus validating the penalization technique effectiveness. Comparing the results in Figure 8(a) with the load path representation solution in Figure 6(b), one can find that the optimal topologies obtained by these two methods are very close. For deformation comparison between Figures 8(b) and 7(a), the LSE deviation obtained by the SIMP method is 44.17 mm, which is significantly higher than that achieved with the load path representation method.
4 Conclusions A two-step approach for the optimizations of the adaptive
leading edge has been introduced in this paper. In the first step of aerodynamic optimization, a hybrid method, including DOE, RSM and GA, is adopted. This method effectively achieves an optimal airfoil, which can improve the lift-todrag ratio greatly under a given flying condition. The optimal airfoil obtained from the first step is regarded as the objective function of the second step optimization. During the optimization process of this step, the modified load path representation method and its incorporation into GA are adopted to design an optimal structural configuration, whose actual deformation matches the target airfoil shape. Taking a special parameterization scheme, this novel method can explore the topology, size and shape of the compliant structure simultaneously. Finally, a comparison study is conducted with the wellknown SIMP method implemented in Altair OptiStruct. The results confirm that the developed load path representation method is effective in solving topology optimization problems with the LSE deviation as the objective. This work was supported by the Aeronautical Science Foundation of China (Grant No. 2012ZA52001) and the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20123218120005). 1
2 3
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